In this paper, we investigate the seepage flow problem of non-Newtonian fluids through a porous medium. The pressure fields of flow through a porous medium of a non-Newtonian fluid with …
In this paper, we investigate the seepage flow problem of non-Newtonian fluids through a porous medium. The pressure fields of flow through a porous medium of a non-Newtonian fluid with fractional derivative model are described by fractional partial differential equations. A kind of powerful analytical method, called Homotopy Perturbation Method (HPM) is also introduced to obtain the exact solutions of the problem. The objective is to propose alternative method of solution, which does not require small parameters, avoid linearization and physically unrealistic assumptions. The results show that the proposed method is very efficient and convenient and can readily be applied to a large class of problems.
In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSF-UM) …
In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSF-UM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.
In this paper, we introduce an analytical solution of the fractional derivative gas transport equation using the power-series technique. We present a new universal transform, namely, generalized Boltzmann change of …
In this paper, we introduce an analytical solution of the fractional derivative gas transport equation using the power-series technique. We present a new universal transform, namely, generalized Boltzmann change of variable which depends on the fractional order, time and space. This universal transform is employed to transfer the partial differential equation into an ordinary differential equation. Moreover, the convergence of the solution has been investigated and found that solutions are unconditionally converged. Results are introduced and discussed for the universal variable and other physical parameters such as porosity and permeability of the reservoir; time and space.
This thesis is devoted to Darcy Brinkman Forchheimer (DBF) equations with a non standard boundary conditions. We prove first the existence of different type of solutions (weak, strong) of the …
This thesis is devoted to Darcy Brinkman Forchheimer (DBF) equations with a non standard boundary conditions. We prove first the existence of different type of solutions (weak, strong) of the stationary DBF problem in a simply connected domain with boundary conditions on the normal component of the velocity field and the tangential component of the vorticity. Next, we consider Brinkman Forchheimer (BF) system with boundary conditions on the pressure in a non simply connected domain. We prove the well-posedness and the existence of a strong solution of this problem. We establish the regularity of the solution in the L^p spaces, for p >= 2.The approximation of the non stationary DBF problem is based on the pseudo-compressibility approach. The second order's error estimate is established in the case where the boundary conditions are of type Dirichlet or Navier. Finally, the finite elements Galerkin Discontinuous method is proposed and the convergence is settled concerning the linearized DBF problem and the non linear DBF system with a non standard boundary conditions.
This paper is devoted to deriving several fractional-order models for multiphase flows in porous media, focusing on some special cases of the two-phase flow. We derive the mass and momentum …
This paper is devoted to deriving several fractional-order models for multiphase flows in porous media, focusing on some special cases of the two-phase flow. We derive the mass and momentum conservation laws of multiphase flow in porous media. The mass conservation-law has been developed based on the flux variation using Taylor series approximation. The fractional Taylor series's advantage is that it can represent the non-linear flux with more accuracy than the first-order linear Taylor series. The divergence term in the mass conservation equation becomes of a fractional type. The model has been developed for the general compressible flow, and the incompressible case is highlighted as a particular case. As a verification, the model can easily collapse to the traditional mass conservation equation once we select the integer-order. To complete the flow model, we present Darcy's law (momentum conservation law in porous media) with time/space fractional memory. The modified Darcy's law with time memory has also been considered. This version of Darcy's law assumes that the permeability diminishes with time, which has a delay effect on the flow; therefore, the flow seems to have a time memory. The fractional Darcy's law with space memory based on Caputo's fractional derivative is also considered to represent the nonlinear momentum flux. Then, we focus on some cases of fractional time memory of two-phase flows with countercurrent-imbibition mechanisms. Five cases are considered, namely, traditional mass equation and fractional Darcy's law with time memory; fractional mass equation with conventional Darcy's law; fractional mass equation and fractional Darcy's law with space memory; fractional mass equation and fractional Darcy's law with time memory; and traditional mass equation and fractional Darcy's law with spatial memory.
This paper presents solutions of the fractional partial differential equation (fPDE) for analysing water movement in soils. The fPDE explains processes equivalent to the concept of symmetrical fractional derivatives (SFDs) …
This paper presents solutions of the fractional partial differential equation (fPDE) for analysing water movement in soils. The fPDE explains processes equivalent to the concept of symmetrical fractional derivatives (SFDs) which have two components: the forward fractional derivative (FFD) and backward fractional derivative (BFD) of water movement in soils with the BFD representing the micro-scale backwater effect in porous media. The distributed-order time-space fPDE represents water movement in both swelling and non-swelling soils with mobile and immobile zones with the backwater effect operating at two time scales in large and small pores. The concept of flux-concentration relation is now updated to account for the relative fractional flux of water movement in soils.
Prediction of the non-linear flow in porous media is still a major scientific and engineering challenge, despite major technological advances in both theoretical and computational thermodynamics in the past two …
Prediction of the non-linear flow in porous media is still a major scientific and engineering challenge, despite major technological advances in both theoretical and computational thermodynamics in the past two decades. Specifically, essential controls on non-linear flow in porous media are not yet definitive. The principal aim of this paper is to develop a meaningful and reasonable quantitative model that manifests the most important fundamental controls on low velocity non-linear flow. By coupling a new derivative with fractional order, referred to conformable derivative, Swartzendruber equation and modified Hertzian contact theory as well as fractal geometry theory, a flow velocity model for porous media is proposed to improve the modeling of Non-linear flow in porous media. Predictions using the proposed model agree well with available experimental data. Salient results presented here include (1) the flow velocity decreases as effective stress increases; (2) rock types of “softer” mechanical properties may exhibit lower flow velocity; (3) flow velocity increases with the rougher pore surfaces and rock elastic modulus. In general, the proposed model illustrates mechanisms that affect non-linear flow behavior in porous media.
We propose two friendly analytical techniques called Adomian decomposition and Picard methods to analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid. This fluid is assumed to be squeezed between …
We propose two friendly analytical techniques called Adomian decomposition and Picard methods to analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid. This fluid is assumed to be squeezed between two circular plates passing through porous medium channel with slip and no‐slip boundary conditions. A single fractional order nonlinear ordinary differential equation is obtained by means of similarity transformation with the help of the fractional calculus definitions. The resulting fractional boundary value problems are solved by the proposed methods. Convergence of the two methods’ solutions is confirmed by obtaining various approximate solutions and various absolute residuals for different values of the fractional order. Comparison of the results of the two methods for different values of the fractional order confirms that the proposed methods are in a well agreement and therefore they can be used in a simple manner for solving this kind of problems. Finally, graphical study for the longitudinal and normal velocity profiles is obtained for various values of some dimensionless parameters and fractional orders.
Geo-materials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores, which …
Geo-materials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores, which is commonly referred to as double porosity. To complicate things, the pore-network at each scale exhibits different permeability, and these networks are connected through fissure and conduits. Although some models are available in the literature, they lack a strong theoretical basis. This paper aims to fill this lacuna by providing the much needed theoretical foundations of the flow in porous media which exhibit double porosity/permeability. We first obtain a mathematical model for double porosity/permeability using the maximization of rate of dissipation hypothesis, and thereby providing a firm thermodynamic underpinning. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the double porosity/permeability model satisfy. These properties are important in their own right as well as serve as good (mechanics-based) a posteriori measures to assess the accuracy of numerical solutions. We also present several canonical problems and obtain the corresponding analytical solutions, which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding solutions under Darcy equations.
Geomaterials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with …
Geomaterials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with different hydromechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flows in such media, they lack a strong theoretical basis. This paper aims to fill this gap in knowledge by providing the theoretical foundation for the flow of incompressible single-phase fluids in rigid porous media that exhibit double porosity/permeability. We first obtain a mathematical model by combining the theory of interacting continua and the maximization of rate of dissipation (MRD) hypothesis, and thereby provide a firm thermodynamic underpinning. The governing equations of the model are a system of elliptic partial differential equations (PDEs) under a steady-state response and a system of parabolic PDEs under a transient response. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the model satisfy. We also present several canonical problems with analytical solutions which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding ones under Darcy equations.
This paper presents the solutions of fractional Drinfeld-Sokolov-Wilson (DSW) equations
 that occur in shallow water flow models using the residual power series method.
 The fractional derivatives and integrals are considered …
This paper presents the solutions of fractional Drinfeld-Sokolov-Wilson (DSW) equations
 that occur in shallow water flow models using the residual power series method.
 The fractional derivatives and integrals are considered in the conformable sense. In
 addition, surface plots of the solutions are given. The solutions and results show that
 the present method is very efficient and effective due to the lack of a need for complex
 calculations and that the method also has a wide range of practicability in the resolution
 of partial differential fractional equations.
In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville …
In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville and Grunwald-Letnikov fractional derivatives, two modified alternating direction methods: a modified alternating direction implicit Euler method and a modified Peaceman-Rachford method, are proposed for solving the 2D-NCSF-FD in uniform media. The stability and consistency, thus convergence of the two methods in a bounded domain are discussed. Finally, numerical results are given.
Based on an examination of K data from four different sites, a new stochastic fractal model, fractional Laplace motion, is proposed. This model is based on the assumption of spatially …
Based on an examination of K data from four different sites, a new stochastic fractal model, fractional Laplace motion, is proposed. This model is based on the assumption of spatially stationary ln( K ) increments governed by the Laplace PDF, with the increments named fractional Laplace noise. Similar behavior has been reported for other increment processes (often called fluctuations) in the fields of finance and turbulence. The Laplace PDF serves as the basis for a stochastic fractal as a result of the geometric central limit theorem. All Laplace processes reduce to their Gaussian analogs for sufficiently large lags, which may explain the apparent contradiction between large‐scale models based on fractional Brownian motion and non‐Gaussian behavior on smaller scales.
This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional …
This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.
In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated …
In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated in the form of convergent series with fast computable components. The numerical results show that the approach is correct, accurate and easy to implement when applied to fractional differential equations.
This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. When their fractional powers in time and in multi-fractional space are …
This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. When their fractional powers in time and in multi-fractional space are specified to unit integer values, the developed fractional equations of continuity and momentum for incompressible and compressible fluid flow reduce to the classical Navier-Stokes equations. As such, these fractional governing equations for fluid flow may be interpreted as generalizations of the classical Navier-Stokes equations. The derived governing equations of fluid flow in fractional differentiation framework herein are nonlocal in time and space. Therefore, they can quantify the effects of initial and boundary conditions better than the classical Navier-Stokes equations. For the frictionless flow conditions, the corresponding fractional governing equations were also developed as a special case of the fractional governing equations of incompressible flow. When their derivative fractional powers are specified to unit integers, these equations are shown to reduce to the classical Euler equations. The numerical simulations are also performed to investigate the merits of the proposed fractional governing equations. It is shown that the developed equations are capable of simulating anomalous sub- and super-diffusion due to their nonlocal behavior in time and space.
The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this …
The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.
Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical …
Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical diffusion theory is no longer suitable. The differential equations with fractional order have recently been proved to be powerful tools for describing anomalous diffusion. Nevertheless, the analysis methods and numerical methods for fractional differential equations are still in the stage of exploration. In the paper, we consider the Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition, respectively. We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme. It is found that the numerical results are in satisfactory agreement with the analytical solution. Through the robustness analysis, it is also found that the diffusion processes on fractals are more sensitive to the spectral dimension than to the anomalous diffusion exponent.
In this paper we consider variational iteration method to investigate solution of Kuramoto-Sivashinsky equations. Comparison of the results of this method obtained just in 2-iterations with RBF based mesh -free …
In this paper we consider variational iteration method to investigate solution of Kuramoto-Sivashinsky equations. Comparison of the results of this method obtained just in 2-iterations with RBF based mesh -free method and local continuous Galerkin methods, shows the efficiency of this method. Numerical experiments are included to show the efficiency of this method.
An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation …
An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).
In this paper, we numerically investigate the chaotic behaviors of a new fractional-order system. We find that chaotic behaviors exist in the fractional-order system with order less than 3. The …
In this paper, we numerically investigate the chaotic behaviors of a new fractional-order system. We find that chaotic behaviors exist in the fractional-order system with order less than 3. The lowest order we find to have chaos is 2.4 in such system. In addition, we numerically simulate the continuances of the chaotic behaviors in the fractional-order system with orders from 2.7 to 3. Our investigations are validated through numerical simulations.
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann–Liouville derivative. The efficiency and power of this approach are …
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.
This work analyses the nonlinear coupled axial–torsional vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two-second order partial differential equations that govern the nonlinear coupled axial–torsional vibration for …
This work analyses the nonlinear coupled axial–torsional vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two-second order partial differential equations that govern the nonlinear coupled axial–torsional vibration for such nanotube are derived. First, these equations are reduced to ordinary differential equations using the Galerkin method and then solved using homotopy perturbation method (HPM) to obtain the nonlinear natural frequencies in coupled axial–torsional vibration mode. It is found that the obtained frequencies are complicated due to coupling between two vibration modes. The dependence of boundary conditions, vibration modes and nanotubes geometry on the nonlinear coupled axial–torsional vibration characteristics of SWCNTs are studied in detail. It was shown that boundary conditions and maximum initial vibration velocity have significant effects on the nonlinear coupled axial–torsional vibration response of SWCNTs. It was also seen that unlike the linear model if the maximum vibration velocity increases, the natural frequencies of vibration increases too. To show the effectiveness and ability of this method, the results obtained with HPM are compared with the fourth-order Runge-Kutta numerical results and good agreement is observed. To the knowledge of authors, the results given herein are new and can be used as a foundation work for future work.
In this article, we applied the variational iteration method along with a Padé approximation (VIM-Padé) to obtain the analytical approximate solution for the motion of a spherical particle in a …
In this article, we applied the variational iteration method along with a Padé approximation (VIM-Padé) to obtain the analytical approximate solution for the motion of a spherical particle in a plane Couette flow. We studied the effects of different flow parameters on the velocity field. It is examined that the present analytical technique is extremely efficient and easy to apply for such a problem.
In this article, we study an approximate analytical solution of linear and nonlinear time-fractional order Klein–Gordon equations by using a recently developed semi analytical method referred as fractional reduced differential …
In this article, we study an approximate analytical solution of linear and nonlinear time-fractional order Klein–Gordon equations by using a recently developed semi analytical method referred as fractional reduced differential transform method with appropriate initial condition. In the study of fractional Klein–Gordon equation, fractional derivative is described in the Caputo sense. The validity and efficiency of the aforesaid method are illustrated by considering three computational examples. The solution profile behavior and effects of different fraction Brownian motion on solution profile of the three numerical examples are shown graphically.
In this article, conformable fractional form of Schrodinger equation has been presented. Then in this formalism two different and well-known potential have been come in. Wave function of these potential …
In this article, conformable fractional form of Schrodinger equation has been presented. Then in this formalism two different and well-known potential have been come in. Wave function of these potential are obtained in terms of Heun function and energy eigen values of each case is determined as well.
Based on the basic idea of the homotopy perturbation method which was proposed by Jihuan He, a target controllable image segmentation model and the corresponding multiscale wavelet numerical method are …
Based on the basic idea of the homotopy perturbation method which was proposed by Jihuan He, a target controllable image segmentation model and the corresponding multiscale wavelet numerical method are constructed. Using the novel model, we can get the only right object from the multiobject images, which is helpful to avoid the oversegmentation and insufficient segmentation. The solution of the variational model is the nonlinear PDEs deduced by the variational approach. So, the bottleneck of the variational model on image segmentation is the lower efficiency of the algorithm. Combining the multiscale wavelet interpolation operator and HPM, a semianalytical numerical method can be obtained, which can improve the computational efficiency and accuracy greatly. The numerical results on some images segmentation show that the novel model and the numerical method are effective and practical.
We develop a new application of the Mittag‐Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed …
We develop a new application of the Mittag‐Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.
This paper suggests a novel coupling method of homotopy perturbation and Laplace transformation for fractional models. This method is based on He’s homotopy perturbation, Laplace transformation and the modified Riemann-Liouville …
This paper suggests a novel coupling method of homotopy perturbation and Laplace transformation for fractional models. This method is based on He’s homotopy perturbation, Laplace transformation and the modified Riemann-Liouville derivative. However, all the previous works avoid the term of fractional order initial conditions and handle them as a restricted variation. In order to overcome this shortcoming, a fractional Laplace homotopy perturbation transform method (FLHPTM) is proposed with modified Riemann-Liouville derivative. The results from introducing a modified Riemann-Liouville derivative, fractional order initial conditions and Laplace transform in the cases studied show the high accuracy, simplicity and efficiency of the approach.
In this paper, we are development the new modified variation iteration method for solving nonlinear equation this method used Laplace transformation, the technique solve nonlinear problem without He's polynomial approximation …
In this paper, we are development the new modified variation iteration method for solving nonlinear equation this method used Laplace transformation, the technique solve nonlinear problem without He's polynomial approximation analytical solution of nonlinear equation by the Laplace transform method, is applied to numerical solution of eighth-order boundary value problems in one step Analytical results are given for several examples to illustrate the implementation and efficiency of the method. Comparison that solution by Numerical solution solving by Mathematica Software, result also show that numerical scheme is very effective
Nowadays, many researchers have considerable attention to fractional calculus as a useful tool for modeling of different phenomena in the world. In this work, we investigate the sum‐type singular nonlinear …
Nowadays, many researchers have considerable attention to fractional calculus as a useful tool for modeling of different phenomena in the world. In this work, we investigate the sum‐type singular nonlinear fractional q integro‐differential equations with m ‐point boundary value problem. The existence of positive solutions is obtained by the properties of the Green function, standard Caputo q derivative, Riemann–Liouville fractional q integral, and a fixed point theorem on a real Banach space , which has a partial order by using a cone . The proofs are based on solving the operators equation. By providing seven algorithms, four tables, and three figures, we give two numerical examples to illustrate our main result.
This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to …
This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.
Purpose This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are …
Purpose This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Design/methodology/approach Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique. Findings HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability. Originality/value The essential idea of this method is to introduce a homotopy parameter, say p , which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.
The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a …
The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.
The differintegration or fractional derivative of complex order ν , is a generalization of the ordinary concept of derivative of order n , from positive integer ν = n to …
The differintegration or fractional derivative of complex order ν , is a generalization of the ordinary concept of derivative of order n , from positive integer ν = n to complex values of ν , including also, for ν = − n a negative integer, the ordinary n ‐th primitive. Substituting, in an ordinary differential equation, derivatives of integer order by derivatives of non‐integer order, leads to a fractional differential equation, which is generally a integro‐differential equation. We present simple methods of solution of some classes of fractional differential equations, namely those with constant coefficients (standard I) and those with power type coefficients with exponents equal to the orders of differintegration (standard II). The fractional differential equations of standard I (II), both homogeneous, and inhomogeneous with exponential (power‐type) forcing, can be solved in the Liouville (Riemann) systems of differintegration. The standard I (II) is linear with constant (non‐constant) coefficients, and some results are also given for a class of non‐linear fractional differential equations (standard III).