In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading âŠ
In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading of Zika virus. The artificial neural network procedures (ANNPs) have been applied to solve the SEIR mathematical model for spreading of the Zika virus together with the hybridization efficiency of global swarming and local search schemes. The global particle swarm optimization (PSO) and local search active-set algorithm (ASA) have been proposed to solve the model. An error based objective function is presented for the SEIR differential model and then optimized by the hybrid computing efficiency of PSO-ASA. Five neurons, fifteen variables of each class and ten numbers of trials have been used to solve the SEIR mathematical model for spreading of the Zika virus. The correctness of the proposed computing ANNPs-PSO-ASA is observed by using the comparison of the obtained and reference solutions along with the performances of the absolute error, ranges around 10â06 to 10â08. The reliability of the designed computing ANNPs-PSO-ASA technique is observed by using the statistical operator performances on single/multiple trials for the SEIR system for spreading of the Zika virus dynamics.
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique âŠ
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reactionâdiffusion model. The new approach is called the Laplace-residual power series method (L-RPSM), which imitates the residual power series method in determining the coefficients of the series solution. The proposed method is also adapted to find an approximate series solution that converges to the exact solution of the nonlinear time-fractional reactionâdiffusion equations. In addition, the method has been applied to many examples, and the findings are found to be impressive. Further, the results indicate that the L-RPSM is effective, fast, and easy to reach the exact solution of the equations. Furthermore, several actual and approximate solutions are graphically represented to demonstrate the efficiency and accuracy of the proposed method.
The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In âŠ
The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convectionâdiffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.
This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from âŠ
This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.
This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space âŠ
This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space L2(â1,1)ĂC[0,T],(T<1). The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equationâs solution are discussed using Banachâs fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y.
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize âŠ
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, KleinâGordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approachâs reliability, capability, and efficiency.
In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractionalâorder Kawahara and modified Kawahara equations. The method convergence is controlled âŠ
In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractionalâorder Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractionalâorder and the wellâknown approach of least squares. Fractionalâorder derivatives are taken in the Caputo sense with numerical values in the closed interval [0, 1]. The suggested method is directly applied to fractionalâorder Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature.
The natural transform decomposition method (NTDM) is a relatively new transformation method for finding an approximate differential equation solution. In the current study, the NTDM has been used for obtaining âŠ
The natural transform decomposition method (NTDM) is a relatively new transformation method for finding an approximate differential equation solution. In the current study, the NTDM has been used for obtaining an approximate solution of the fractionalâorder generalized perturbed ZakharovâKuznetsov (GPZK) equation. The method has been tested for three nonlinear cases of the fractionalâorder GPZK equation. The absolute errors are analyzed by the proposed method and the qâhomotopy analysis transform method (qâHATM). 3D and 2D graphs have shown the proposed methodâs accuracy and effectiveness. NTDM gives a muchâclosed solution after a few terms.
In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on âŠ
In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.
A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models âŠ
A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models in fuzzy conditions. In this paper, an explicit finite difference method has been developed and applied to solve a fuzzy time-fractional cancer tumor model. The impact of using the fuzzy time-fractional derivative has been examined under the double parametric form of fuzzy numbers rather than using classical time derivatives in fuzzy cancer tumor models. In addition, the stability of the proposed model has been investigated by applying the Fourier method, where the net killing rate of the cancer cells is only time-dependent, and the time-fractional derivative is Caputoâs derivative. Moreover, certain numerical experiments are discussed to examine the feasibility of the new approach and to check the related aspects. Over and above, certain needs in studying the fuzzy fractional cancer tumor model are detected to provide a better comprehensive understanding of the behavior of the tumor by utilizing several fuzzy cases on the initial conditions of the proposed model.
Due to numerous applications, stretched flows got much attention now days. Current inspection discusses the involvement of thermal and mass transport in Williamson material past over a bi-directional surface. The âŠ
Due to numerous applications, stretched flows got much attention now days. Current inspection discusses the involvement of thermal and mass transport in Williamson material past over a bi-directional surface. The surface in stretched along [Formula: see text] and [Formula: see text] axies and flow occupies the region [Formula: see text] Heat transport is modeled via modified heat flux model (MHFM), whereas, generalized mass flux has been used in transportation of mass. The theory of boundary layer (BL) has been utilized on modeling the conservation laws with certain important considerations. Afterwards, the obtained ODEs have been approximated after transformation via optimal homotopy analysis procedure (OHAP). The convergence of used scheme is shown through error analysis table. Efficiency and authenticity of the code is shown by comparative study. Applications of magnetic field (variable), Williamson number, and index number make reduction in flow. The maximum amount of production in thermal energy is obtained using large values of Brownian motion and thermophoresis. Such considered model is used in the applications like improvement in thermal energy, recovery in petroleum, adjusting cooling (devices), and energy devices.
The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have âŠ
The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.
A numerical method is proposed to approximate the numeric solutions of nonlinear Fisherâs reaction diffusion equation with finite difference method. The method is based on replacing each terms in the âŠ
A numerical method is proposed to approximate the numeric solutions of nonlinear Fisherâs reaction diffusion equation with finite difference method. The method is based on replacing each terms in the Fisherâs equation using finite difference method. The proposed method has the advantage of reducing the problem to a nonlinear system, which will be derived and solved using Newton method. FTCS and CN method will be introduced, compared and tested.
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means âŠ
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized to introduce the fuzzy fractional derivatives. The proposed method provides a systematic scheme based on generalized Taylor expansion and minimization of the residual error function, so as to obtain the coefficients values of a fractional series based on the given initial data of triangular fuzzy numbers in the parametric form. The obtained approximated solutions are provided within an appropriate radius to the requisite domain in the form of rapidly convergent fractional series according to their parametric form. The methodâs performance and applicability are verified by applying it on some numerical examples. The impact of r-levels and fractional order Îł is presented quantitatively and graphically, showing the coincidence between the exact and the fuzzy approximated solutions. Moreover, for reliability and accuracy, our obtained results are numerically compared with the exact solutions and with results obtained using other methods described in the literature. This indicates that the proposed approach overcomes the difficulties that appear in other approaches to create fractional series solutions for varied uncertain natural problems arising within the fields of applied physics and engineering.
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which âŠ
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.
In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness âŠ
In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the VolterraâHammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.
The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet âŠ
The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet neural network (MWNN). The MRS is divided into two classes, infected I(Ξ) and Robots R(Ξ) . The design of the fitness function is presented by using the differential MRS and then optimized by the hybrid of the global swarming computational particle swarm optimization (PSO) and local active set procedure (ASP). For the exactness of the AI based MWNN-PSOIPS, the comparison of the results is presented by using the proposed and reference solutions. The reliability of the MWNN-PSOASP is authenticated by extending the data into 20 trials to check the performance of the scheme by using the statistical operators with 10 hidden numbers of neurons to solve the MRS.
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work âŠ
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.
A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel.We transform the Volterra Fredholm integral equations to a system of âŠ
A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel.We transform the Volterra Fredholm integral equations to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials.This lead to a system of algebraic equations with Legendre or Chebychev coefficients.Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained.Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.
The fractional LakshmananâPorsezianâDaniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of âŠ
The fractional LakshmananâPorsezianâDaniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of the space-time fractional LPD equation with the Kerr law of nonlinearity. By considering the conformable derivatives, the governing model is translated into integer-order differential equations with the aid of an appropriate complex traveling wave transformation. Dynamic behavior and phase portrait of traveling wave solutions are investigated. Further, various types of bright and kinked soliton solutions under definite parametric settings are discussed. Moreover, graphical representations of the obtained solution of the diverse fractional order are depicted to naturally illustrate the constructed solution.
This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude âŠ
This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified BenjaminâBonaâMahony equation, fractional ZakharovâKuznetsovâBenjaminâBonaâMahony equation and fractional (2+1)-dimensional KadomtsevâPetviashviliâBenjaminâBonaâMahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.
Abstract In this article, we consider existence and unique of solutions of linear mixed integral equations of third, second and first kinds. Then, we use the collection method to discuss âŠ
Abstract In this article, we consider existence and unique of solutions of linear mixed integral equations of third, second and first kinds. Then, we use the collection method to discuss numerical solutions by employing Chebyshev and Legendre polynomials. To support our results, we use Mable 18 to compute errors in all existing cases.
<abstract> <p>Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an âŠ
<abstract> <p>Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (<bold>MIE</bold>) of the second kind in the space $ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right], T &lt; 1. $ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (<bold>V-FIE</bold>), with the help of the Chebyshev polynomials (<bold>CPs</bold>), and then use the results in the general <bold>MIE</bold> to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the <bold>FI</bold> term is considered in position, in $ {L}_{2}\left[-\mathrm{1, 1}\right], $ and its kernel takes a logarithmic form multiplied by a general continuous function. While the <bold>VI</bold> term in time, in $ C\left[0, T\right], T &lt; 1, $ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.</p> </abstract>
We present an improvement of the numerical method based on Toeplitz matrices to solve the Volterra Fredholm Integral equation of the second kind with singular kernel. The kernel function đŠ âŠ
We present an improvement of the numerical method based on Toeplitz matrices to solve the Volterra Fredholm Integral equation of the second kind with singular kernel. The kernel function đŠ (s,t) is moderately smooth on [a, b] Ă [0, T] except possibly across the diagonal s = t. We transform the Volterra integral equations to a system of Fredholm integral equations of the second kind which will be solved by Toeplitz matrices method. This lead to a system of algebraic equations. Thus, by solving the matrix equation, the approximation solution is obtained.
Abstract In this study, a numerical scheme is proposed for the fifth order (FO) singular differential model (SDM), FO-SDM. The solutions of the singular form of the differential models are âŠ
Abstract In this study, a numerical scheme is proposed for the fifth order (FO) singular differential model (SDM), FO-SDM. The solutions of the singular form of the differential models are always considered difficult to solve and huge important in astrophysics. A neural network study together with the hybrid combination of global particle swarm optimization and local sequential quadratic programming schemes is provided to find the numerical simulations of the FO-SDM. An objective function is constructed using the differential FO-SDM along with the boundary conditions. The correctness of the scheme is observed by providing the comparison of the obtained and exact solutions. The justification of the proposed scheme is authenticated in terms of absolute error (AE), which is calculated in good measures for solving the FO-SDM. The efficiency and reliability of the stochastic approach are observed using the statistical performances to solve the FO-SDM.
Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical âŠ
Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.
A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy âŠ
A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.
This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients âŠ
This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients of sequences, and establishes certain convolution theorems. The convolution theorems are utilized to accomplish q-equivalence classes of generalized distributions called q-Boehmians. Consequently, the q-gamma operators are therefore extended to the generalized spaces and performed to coincide with the classical integral operator. Further, the generalized q-gamma integral is shown to be linear, sequentially continuous and continuous with respect to some involved convergence equipped with the generalized spaces.
The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two âŠ
The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.
This paper studies three time-fractional models that arise in plasma physics: the modified KortewegâdeVriesâZakharovâKuznetsov equation, the stochastic potential KortewegâdeVries equation, and the forced KortewegâdeVries equation. These equations are significant in âŠ
This paper studies three time-fractional models that arise in plasma physics: the modified KortewegâdeVriesâZakharovâKuznetsov equation, the stochastic potential KortewegâdeVries equation, and the forced KortewegâdeVries equation. These equations are significant in plasma physics for modeling nonlinear ion acoustic waves and thus helping us to understand wave dynamics in plasmas. We introduce a new approach that relies on a new fractional expansion in the natural transform space and residual power series method to construct analytical solutions to the governing models. We investigate the theoretical analysis of the proposed method for these equations to expose this approachâs applicability, efficiency, and effectiveness in constructing analytical solutions to the governing equations. Moreover, we present a comparative discussion between the solutions derived during the work and those given in the literature to confirm that the proposed approach generates analytical solutions that rapidly converge to exact solutions, which proves the effectiveness of the proposed method.
In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisherâs equation using a right starting space function and a unique analytic-numeric technique referred to as the âŠ
In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisherâs equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series approach. The generalized Taylorâs formula and the Laplace transform operator are coupled in the aforementioned method, where the coefficients, obtained through fractional expansion in the Laplace space, are determined by applying the limit concept. In order to validate and illustrate the theoretical methodology of the LRPS technique, as well as to show its effectiveness, adaptability, and superiority in solving various types of nonlinear time and space fractional differential equations, numerical experiments are generated. The obtained analytical solutions are compatible with the precise solutions and concur with those proposed by the other approaches. The outcomes show that the Laplace residual power series strategy is incredibly successful, straightforward to implement, and well suited for handling the complexity of nonlinear problems.
In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisherâs equation using a right starting space function and a unique analytic-numeric technique referred to as the âŠ
In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisherâs equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series approach. The generalized Taylorâs formula and the Laplace transform operator are coupled in the aforementioned method, where the coefficients, obtained through fractional expansion in the Laplace space, are determined by applying the limit concept. In order to validate and illustrate the theoretical methodology of the LRPS technique, as well as to show its effectiveness, adaptability, and superiority in solving various types of nonlinear time and space fractional differential equations, numerical experiments are generated. The obtained analytical solutions are compatible with the precise solutions and concur with those proposed by the other approaches. The outcomes show that the Laplace residual power series strategy is incredibly successful, straightforward to implement, and well suited for handling the complexity of nonlinear problems.
This paper studies three time-fractional models that arise in plasma physics: the modified KortewegâdeVriesâZakharovâKuznetsov equation, the stochastic potential KortewegâdeVries equation, and the forced KortewegâdeVries equation. These equations are significant in âŠ
This paper studies three time-fractional models that arise in plasma physics: the modified KortewegâdeVriesâZakharovâKuznetsov equation, the stochastic potential KortewegâdeVries equation, and the forced KortewegâdeVries equation. These equations are significant in plasma physics for modeling nonlinear ion acoustic waves and thus helping us to understand wave dynamics in plasmas. We introduce a new approach that relies on a new fractional expansion in the natural transform space and residual power series method to construct analytical solutions to the governing models. We investigate the theoretical analysis of the proposed method for these equations to expose this approachâs applicability, efficiency, and effectiveness in constructing analytical solutions to the governing equations. Moreover, we present a comparative discussion between the solutions derived during the work and those given in the literature to confirm that the proposed approach generates analytical solutions that rapidly converge to exact solutions, which proves the effectiveness of the proposed method.
The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two âŠ
The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.
This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients âŠ
This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients of sequences, and establishes certain convolution theorems. The convolution theorems are utilized to accomplish q-equivalence classes of generalized distributions called q-Boehmians. Consequently, the q-gamma operators are therefore extended to the generalized spaces and performed to coincide with the classical integral operator. Further, the generalized q-gamma integral is shown to be linear, sequentially continuous and continuous with respect to some involved convergence equipped with the generalized spaces.
In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading âŠ
In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading of Zika virus. The artificial neural network procedures (ANNPs) have been applied to solve the SEIR mathematical model for spreading of the Zika virus together with the hybridization efficiency of global swarming and local search schemes. The global particle swarm optimization (PSO) and local search active-set algorithm (ASA) have been proposed to solve the model. An error based objective function is presented for the SEIR differential model and then optimized by the hybrid computing efficiency of PSO-ASA. Five neurons, fifteen variables of each class and ten numbers of trials have been used to solve the SEIR mathematical model for spreading of the Zika virus. The correctness of the proposed computing ANNPs-PSO-ASA is observed by using the comparison of the obtained and reference solutions along with the performances of the absolute error, ranges around 10â06 to 10â08. The reliability of the designed computing ANNPs-PSO-ASA technique is observed by using the statistical operator performances on single/multiple trials for the SEIR system for spreading of the Zika virus dynamics.
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize âŠ
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, KleinâGordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approachâs reliability, capability, and efficiency.
This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space âŠ
This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space L2(â1,1)ĂC[0,T],(T<1). The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equationâs solution are discussed using Banachâs fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y.
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique âŠ
Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reactionâdiffusion model. The new approach is called the Laplace-residual power series method (L-RPSM), which imitates the residual power series method in determining the coefficients of the series solution. The proposed method is also adapted to find an approximate series solution that converges to the exact solution of the nonlinear time-fractional reactionâdiffusion equations. In addition, the method has been applied to many examples, and the findings are found to be impressive. Further, the results indicate that the L-RPSM is effective, fast, and easy to reach the exact solution of the equations. Furthermore, several actual and approximate solutions are graphically represented to demonstrate the efficiency and accuracy of the proposed method.
A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models âŠ
A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models in fuzzy conditions. In this paper, an explicit finite difference method has been developed and applied to solve a fuzzy time-fractional cancer tumor model. The impact of using the fuzzy time-fractional derivative has been examined under the double parametric form of fuzzy numbers rather than using classical time derivatives in fuzzy cancer tumor models. In addition, the stability of the proposed model has been investigated by applying the Fourier method, where the net killing rate of the cancer cells is only time-dependent, and the time-fractional derivative is Caputoâs derivative. Moreover, certain numerical experiments are discussed to examine the feasibility of the new approach and to check the related aspects. Over and above, certain needs in studying the fuzzy fractional cancer tumor model are detected to provide a better comprehensive understanding of the behavior of the tumor by utilizing several fuzzy cases on the initial conditions of the proposed model.
In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness âŠ
In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the VolterraâHammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.
In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on âŠ
In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.
A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy âŠ
A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.
<abstract> <p>Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an âŠ
<abstract> <p>Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (<bold>MIE</bold>) of the second kind in the space $ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right], T &lt; 1. $ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (<bold>V-FIE</bold>), with the help of the Chebyshev polynomials (<bold>CPs</bold>), and then use the results in the general <bold>MIE</bold> to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the <bold>FI</bold> term is considered in position, in $ {L}_{2}\left[-\mathrm{1, 1}\right], $ and its kernel takes a logarithmic form multiplied by a general continuous function. While the <bold>VI</bold> term in time, in $ C\left[0, T\right], T &lt; 1, $ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.</p> </abstract>
This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude âŠ
This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified BenjaminâBonaâMahony equation, fractional ZakharovâKuznetsovâBenjaminâBonaâMahony equation and fractional (2+1)-dimensional KadomtsevâPetviashviliâBenjaminâBonaâMahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.
The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In âŠ
The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convectionâdiffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.
This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from âŠ
This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.
Abstract In this article, we consider existence and unique of solutions of linear mixed integral equations of third, second and first kinds. Then, we use the collection method to discuss âŠ
Abstract In this article, we consider existence and unique of solutions of linear mixed integral equations of third, second and first kinds. Then, we use the collection method to discuss numerical solutions by employing Chebyshev and Legendre polynomials. To support our results, we use Mable 18 to compute errors in all existing cases.
The fractional LakshmananâPorsezianâDaniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of âŠ
The fractional LakshmananâPorsezianâDaniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of the space-time fractional LPD equation with the Kerr law of nonlinearity. By considering the conformable derivatives, the governing model is translated into integer-order differential equations with the aid of an appropriate complex traveling wave transformation. Dynamic behavior and phase portrait of traveling wave solutions are investigated. Further, various types of bright and kinked soliton solutions under definite parametric settings are discussed. Moreover, graphical representations of the obtained solution of the diverse fractional order are depicted to naturally illustrate the constructed solution.
The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have âŠ
The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work âŠ
The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.
Abstract In this study, a numerical scheme is proposed for the fifth order (FO) singular differential model (SDM), FO-SDM. The solutions of the singular form of the differential models are âŠ
Abstract In this study, a numerical scheme is proposed for the fifth order (FO) singular differential model (SDM), FO-SDM. The solutions of the singular form of the differential models are always considered difficult to solve and huge important in astrophysics. A neural network study together with the hybrid combination of global particle swarm optimization and local sequential quadratic programming schemes is provided to find the numerical simulations of the FO-SDM. An objective function is constructed using the differential FO-SDM along with the boundary conditions. The correctness of the scheme is observed by providing the comparison of the obtained and exact solutions. The justification of the proposed scheme is authenticated in terms of absolute error (AE), which is calculated in good measures for solving the FO-SDM. The efficiency and reliability of the stochastic approach are observed using the statistical performances to solve the FO-SDM.
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means âŠ
In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized to introduce the fuzzy fractional derivatives. The proposed method provides a systematic scheme based on generalized Taylor expansion and minimization of the residual error function, so as to obtain the coefficients values of a fractional series based on the given initial data of triangular fuzzy numbers in the parametric form. The obtained approximated solutions are provided within an appropriate radius to the requisite domain in the form of rapidly convergent fractional series according to their parametric form. The methodâs performance and applicability are verified by applying it on some numerical examples. The impact of r-levels and fractional order Îł is presented quantitatively and graphically, showing the coincidence between the exact and the fuzzy approximated solutions. Moreover, for reliability and accuracy, our obtained results are numerically compared with the exact solutions and with results obtained using other methods described in the literature. This indicates that the proposed approach overcomes the difficulties that appear in other approaches to create fractional series solutions for varied uncertain natural problems arising within the fields of applied physics and engineering.
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which âŠ
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.
Due to numerous applications, stretched flows got much attention now days. Current inspection discusses the involvement of thermal and mass transport in Williamson material past over a bi-directional surface. The âŠ
Due to numerous applications, stretched flows got much attention now days. Current inspection discusses the involvement of thermal and mass transport in Williamson material past over a bi-directional surface. The surface in stretched along [Formula: see text] and [Formula: see text] axies and flow occupies the region [Formula: see text] Heat transport is modeled via modified heat flux model (MHFM), whereas, generalized mass flux has been used in transportation of mass. The theory of boundary layer (BL) has been utilized on modeling the conservation laws with certain important considerations. Afterwards, the obtained ODEs have been approximated after transformation via optimal homotopy analysis procedure (OHAP). The convergence of used scheme is shown through error analysis table. Efficiency and authenticity of the code is shown by comparative study. Applications of magnetic field (variable), Williamson number, and index number make reduction in flow. The maximum amount of production in thermal energy is obtained using large values of Brownian motion and thermophoresis. Such considered model is used in the applications like improvement in thermal energy, recovery in petroleum, adjusting cooling (devices), and energy devices.
In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractionalâorder Kawahara and modified Kawahara equations. The method convergence is controlled âŠ
In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractionalâorder Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractionalâorder and the wellâknown approach of least squares. Fractionalâorder derivatives are taken in the Caputo sense with numerical values in the closed interval [0, 1]. The suggested method is directly applied to fractionalâorder Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature.
The natural transform decomposition method (NTDM) is a relatively new transformation method for finding an approximate differential equation solution. In the current study, the NTDM has been used for obtaining âŠ
The natural transform decomposition method (NTDM) is a relatively new transformation method for finding an approximate differential equation solution. In the current study, the NTDM has been used for obtaining an approximate solution of the fractionalâorder generalized perturbed ZakharovâKuznetsov (GPZK) equation. The method has been tested for three nonlinear cases of the fractionalâorder GPZK equation. The absolute errors are analyzed by the proposed method and the qâhomotopy analysis transform method (qâHATM). 3D and 2D graphs have shown the proposed methodâs accuracy and effectiveness. NTDM gives a muchâclosed solution after a few terms.
Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical âŠ
Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.
The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet âŠ
The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet neural network (MWNN). The MRS is divided into two classes, infected I(Ξ) and Robots R(Ξ) . The design of the fitness function is presented by using the differential MRS and then optimized by the hybrid of the global swarming computational particle swarm optimization (PSO) and local active set procedure (ASP). For the exactness of the AI based MWNN-PSOIPS, the comparison of the results is presented by using the proposed and reference solutions. The reliability of the MWNN-PSOASP is authenticated by extending the data into 20 trials to check the performance of the scheme by using the statistical operators with 10 hidden numbers of neurons to solve the MRS.
A numerical method is proposed to approximate the numeric solutions of nonlinear Fisherâs reaction diffusion equation with finite difference method. The method is based on replacing each terms in the âŠ
A numerical method is proposed to approximate the numeric solutions of nonlinear Fisherâs reaction diffusion equation with finite difference method. The method is based on replacing each terms in the Fisherâs equation using finite difference method. The proposed method has the advantage of reducing the problem to a nonlinear system, which will be derived and solved using Newton method. FTCS and CN method will be introduced, compared and tested.
A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel.We transform the Volterra Fredholm integral equations to a system of âŠ
A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel.We transform the Volterra Fredholm integral equations to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials.This lead to a system of algebraic equations with Legendre or Chebychev coefficients.Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained.Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.
We present an improvement of the numerical method based on Toeplitz matrices to solve the Volterra Fredholm Integral equation of the second kind with singular kernel. The kernel function đŠ âŠ
We present an improvement of the numerical method based on Toeplitz matrices to solve the Volterra Fredholm Integral equation of the second kind with singular kernel. The kernel function đŠ (s,t) is moderately smooth on [a, b] Ă [0, T] except possibly across the diagonal s = t. We transform the Volterra integral equations to a system of Fredholm integral equations of the second kind which will be solved by Toeplitz matrices method. This lead to a system of algebraic equations. Thus, by solving the matrix equation, the approximation solution is obtained.
Nonclassical quantum mechanics along with dispersive interactions of free particles, long-range boson stars, hydrodynamics, harmonic oscillator, shallow-water waves, and quantum condensates can be modeled via the nonlinear fractional Schrödinger equation. âŠ
Nonclassical quantum mechanics along with dispersive interactions of free particles, long-range boson stars, hydrodynamics, harmonic oscillator, shallow-water waves, and quantum condensates can be modeled via the nonlinear fractional Schrödinger equation. In this paper, various types of optical soliton wave solutions are investigated for perturbed, conformable space-time fractional Schrödinger model competed with a weakly nonlocal term. The fractional derivatives are described by means of conformable space-time fractional sense. Two different types of nonlinearity are discussed based on Kerr and dual power laws for the proposed fractional complex system. The method employed for solving the nonlinear fractional resonant Schrödinger model is the hyperbolic function method utilizing some fractional complex transformations. Several types of exact analytical solutions are obtained, including bright, dark, singular dual-power-type soliton and singular Kerr-type soliton solutions. Moreover, some graphical simulations of those solutions are provided for understanding the physical phenomena.
The article introduces the fractional modified Sprott A chaotic system with its thorough dynamical analysis such as solution of system, Lyapunov dynamics, route to chaos etc. Penta-compound combination anti-synchronization is âŠ
The article introduces the fractional modified Sprott A chaotic system with its thorough dynamical analysis such as solution of system, Lyapunov dynamics, route to chaos etc. Penta-compound combination anti-synchronization is introduced and applied on twelve chaotic systems of fractional order using adaptive SMC. The system is also controlled about a randomly chosen point and The synchronization technique is illustrated as application to secure communication. The simulations are performed using MATLAB software.
In quantum field theory, the fractional Kundu-Eckhaus and massive Thirring models are nonlinear partial differential equations under fractional sense inside nonlinear Schrödinger class. In this study, approximate analytical solutions of âŠ
In quantum field theory, the fractional Kundu-Eckhaus and massive Thirring models are nonlinear partial differential equations under fractional sense inside nonlinear Schrödinger class. In this study, approximate analytical solutions of such complex nonlinear fractional models are acquired by means of conformable residual power series method. This method presents a systematic procedure for constructing a set of periodic wave series solutions based on the generalization of conformable power series and gives the unknown coefficients in a simple pattern. By plotting the solutions behavior of the models; the convergence regions in which the solutions coincide to each other are checked for various fractional values. The approximate solutions generated by the proposed approach are compared with the exact solutions -if exist- and the approximate solutions obtained using qHATM and LADM. Numerical results show that the proposed method is easy to implement and very computationally attractive in solving several complex nonlinear fractional systems that occur in applied physics under a compatible fractional sense.
All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel âŠ
All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The NewtonâRaphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.
In this paper, we considered a mixed integral equation (MIE) of the second kind in the space L 2 [â b , b ] Ă C [0, T ], T âŠ
In this paper, we considered a mixed integral equation (MIE) of the second kind in the space L 2 [â b , b ] Ă C [0, T ], T < 1. The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method ( TMM ), we obtain a linear algebraic system ( LAS ) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.
In this article, a class of generalized telegraph and Cattaneo timeâfractional models along with Robin's initialâboundary conditions is considered using the adaptive reproducing kernel framework. Accordingly, a relatively novel numerical âŠ
In this article, a class of generalized telegraph and Cattaneo timeâfractional models along with Robin's initialâboundary conditions is considered using the adaptive reproducing kernel framework. Accordingly, a relatively novel numerical treatment is introduced to investigate and interpret approximate solutions to telegraph and Cattaneo models of timeâfractional derivatives in Caputo sense. This treatment optimized solutions relying on the Sobolev spaces and Schmidt orthogonalization process that can be directly implemented to generate Fourier expansion at a rapid convergence rate, in which the arbitrary kernel functions satisfy Robin's homogeneous conditions. Furthermore, the solution is displayed in a fractional series formula in complete Hilbert spaces without any restrictive hypothesis on the desired issues. The effectiveness, validity, and potentiality of the proposed procedure are demonstrated by testing some applications. The graphical consequences indicate that the method is superior, accurate, and convenient in solving such fractional models.
Mathematical modeling of fractional resonant Schrödinger equations is an extremely significant topic in the classical of quantum mechanics, chromodynamics, astronomy, and anomalous diffusion systems. Based on conformable residual power series, âŠ
Mathematical modeling of fractional resonant Schrödinger equations is an extremely significant topic in the classical of quantum mechanics, chromodynamics, astronomy, and anomalous diffusion systems. Based on conformable residual power series, a novel effective analytical approach is considered to solve classes of nonlinear time-fractional resonant Schrödinger equation and nonlinear coupled fractional Schrödinger equations under conformable fractional derivatives. The solution methodology lies in generating an infinite conformable series solution with reliable wave pattern by minimizing the residual error functions. The main motivation for using this approach is high accuracy convergence and low computational cost compared to other existing methods. In this orientation, the competency and capacity of the proposed method are examined by implementing several numerical applications. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features in feasibility, stability, and suitability for dealing with many fractional models emerging in physics and optics using the new conformable derivative.
In this paper, we study the existence of a unique solution of nonlinear mixed integral equation (NMIE) of the third kind in position and time in the space L2[0,1]ĂC[0,T], T<1. âŠ
In this paper, we study the existence of a unique solution of nonlinear mixed integral equation (NMIE) of the third kind in position and time in the space L2[0,1]ĂC[0,T], T<1. Moreover, the stability of the solution is discussed. Using a quadratic method, the NMIE is transformed into a system of NIEs in position. Using a collocation method with aid of two different polynomials, Hermite and Laguerre polynomials, we have two different nonlinear algebraic systems (NAS). The estimate of the error, in each numerical method is discussed. Many applications, for the NMIE of the first, second and third kind with continuous kernels in position and time, are considered. In addition, by considering different times of the proposed method and using Mable 18, many numerical results are computed. Moreover, the error estimate, in each case, is calculated.
Abstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with âŠ
Abstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a CaputoâFabrizio fractional derivative. By means of such an approach, we utilize the GramâSchmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>[</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>]</mml:mo> </mml:math> . We investigate and discuss stability and convergence of the proposed method. The n -term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the CaputoâFabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.
The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a âŠ
The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, $0<\alpha\leq1$ , concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which âŠ
This article investigates the local fractional generalized KadomtsevâPetviashvili equation and the local fractional KadomtsevâPetviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.
Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations (FPDEs) corresponding to different applications in science âŠ
Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations (FPDEs) corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the (n â 1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.
A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the âŠ
A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.
New technique model is used to solve the mixed integral equation (\textbf{MIE}) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function âŠ
New technique model is used to solve the mixed integral equation (\textbf{MIE}) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function and continuous kernel in time, in the space $L_{2} (\Omega )\times C[0,T],\, 0\leq T<1$, $\Omega$ is the domain of integration and $T$ is the time. The integral equation arises in the treatment of various semi-symmetric contact problems, in one, two, and three dimensions, with mixed boundary conditions in the mechanics of continuous media. The solution of the \textbf{MIE }when the kernel of position takes the potential function form, elliptic function form, Carleman function and logarithmic kernel are discussed and obtain in this work. Moreover, many special cases are derived.
The present study is related to present a novel design of intelligent solvers with a neuro-swarm heuristic integrated with interior-point algorithm (IPA) for the numerical investigations of the nonlinear SITR âŠ
The present study is related to present a novel design of intelligent solvers with a neuro-swarm heuristic integrated with interior-point algorithm (IPA) for the numerical investigations of the nonlinear SITR fractal system based on the dynamics of a novel coronavirus (COVID-19). The mathematical form of the SITR system using fractal considerations defined in four groups, 'susceptible (S)', 'infected (I)', 'treatment (T)' and 'recovered (R)'. The inclusive detail of each group along with the clarification to formulate the manipulative form of the SITR nonlinear model of novel COVID-19 dynamics is presented. The solution of the SITR model is presented using the artificial neural networks (ANNs) models trained with particle swarm optimization (PSO), i.e., global search scheme and prompt fine-tuning by IPA, i.e., ANN-PSOIPA. In the ANN-PSOIPA, the merit function is expressed for the impression of mean squared error applying the continuous ANNs form for the dynamics of SITR system and training of these networks are competently accompanied with the integrated competence of PSOIPA. The exactness, stability, reliability and prospective of the considered ANN-PSOIPA for four different forms is established via the comparative valuation from of Runge-Kutta numerical solutions for the single and multiple executions. The obtained outcomes through statistical assessments verify the convergence, stability and viability of proposed ANN-PSOIPA.
We study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $\psi (x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these âŠ
We study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $\psi (x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to $t_i = x_i^0, i=1,2$. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.
Abstract In this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional âŠ
Abstract In this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83â96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated âł-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some âł-series fractional differential equations.
Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the âŠ
Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the GaussâLegendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.
In this paper, the existence and uniqueness of solution of singular Hammerstein-Volterra integral equation (H-VIE) are considered. Toeplitz matrix (TMM) and product Nystrom method (PNM) to solve the H-VIE with âŠ
In this paper, the existence and uniqueness of solution of singular Hammerstein-Volterra integral equation (H-VIE) are considered. Toeplitz matrix (TMM) and product Nystrom method (PNM) to solve the H-VIE with singular logarithmic kernel are used. The absolute error is calculated.
This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional FokkerâPlanck model. The fractional derivative is âŠ
This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional FokkerâPlanck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional FokkerâPlanck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.
The NewellâWhiteheadâSegel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, âŠ
The NewellâWhiteheadâSegel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion, and convection system. In this analysis, a recent numeric-analytic technique, called the fractional residual power series (FRPS) approach, was successfully employed in obtaining effective approximate solutions to the NewellâWhiteheadâSegel equation of the fractional sense. The proposed algorithm relies on a generalized classical power series under the Caputo sense and the concept of an error function that systematically produces an analytical solution in a convergent fractional power series form with accurately computable structures, without the need for any unphysical restrictive assumptions. Meanwhile, two illustrative applications are included to show the efficiency, reliability, and performance of the proposed technique. Plotted and numerical results indicated the compatibility between the exact and approximate solution obtained by the proposed technique. Furthermore, the solution behavior indicates that increasing the fractional parameter changes the nature of the solution with a smooth sense symmetrical to the integer-order state.
In this paper, an efficient numerical method is used for solving 2D-mixed VolterraâFredholm integral equations. This method is based on 2D-orthonormal Bernstein polynomials (2D-OBPs) together with collocation method. This approach âŠ
In this paper, an efficient numerical method is used for solving 2D-mixed VolterraâFredholm integral equations. This method is based on 2D-orthonormal Bernstein polynomials (2D-OBPs) together with collocation method. This approach is applied to convert the problem under study into a system of algebraic equations which can be solved by using a convenient numerical method. Several useful theorems are proved which are concerned with the convergence and error estimate associated to the suggested scheme. Finally, by comparing the values of absolute error achieved from this method with values of absolute error obtained from other previous methods, we show that this method is very accurate and efficient.
The modeling of fuzzy fractional integro-differential equations is a very significant matter in engineering and applied sciences. This paper presents a novel treatment algorithm based on utilizing the fractional residual âŠ
The modeling of fuzzy fractional integro-differential equations is a very significant matter in engineering and applied sciences. This paper presents a novel treatment algorithm based on utilizing the fractional residual power series (FRPS) method to study and interpret the approximated solutions for a class of fuzzy fractional Volterra integro-differential equations of order 0 < ÎČ â€ 1 which are subject to appropriate symmetric triangular fuzzy conditions under strongly generalized differentiability. The proposed algorithm relies upon the residual error concept and on the formula of generalized Taylor. The FRPS algorithm provides approximated solutions in parametric form with rapidly convergent fractional power series without linearization, limitation on the problemâs nature, and sort of classification or perturbation. The fuzzy fractional derivatives are described via the Caputo fuzzy H -differentiable. The ability, effectiveness, and simplicity of the proposed technique are demonstrated by testing two applications. Graphical and numerical results reveal the symmetry between the lower and upper r -cut representations of the fuzzy solution and satisfy the convex symmetric triangular fuzzy number. Notably, the symmetric fuzzy solutions on a focus of their core and support refer to a sense of proportion, harmony, and balance. The obtained results reveal that the FRPS scheme is simple, straightforward, accurate and convenient to solve different forms of fuzzy fractional differential equations.
Abstract Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by a Petrov-Galerkin finite element method. The results show that any local âŠ
Abstract Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by a Petrov-Galerkin finite element method. The results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are determined by the system itself and are independent of the initial values. Comparing with other studies, the numerical scheme used in this paper is satisfactory with regard to its accuracy and stability. It has the advantage of being much more concise.
1. Integrals of the form where Ï( x ) is a function with a limited number of turning-points in the range of integration and k is a constant which may âŠ
1. Integrals of the form where Ï( x ) is a function with a limited number of turning-points in the range of integration and k is a constant which may take up large values, frequently occur in investigations in mathematical physics, and their computation by quadratures is often desirable.
In this paper, optimal homotopy asymptotic method (OHAM) has been extended to the solution of fractional order RLW Equation. Results obtained by proposed method are compared with homotopy perturbation method âŠ
In this paper, optimal homotopy asymptotic method (OHAM) has been extended to the solution of fractional order RLW Equation. Results obtained by proposed method are compared with homotopy perturbation method (HPM) and veriational iterational method (VIM). The results reveal that the proposed method is very effective and convenient for solving nonlinear partial differential equations of fractional order.
In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep âŠ
In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The âŠ
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0â€Î±<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.