In this article, we introduce a new homotopy perturbation method (NHPM) for solving non-linear problems, such that it can be converted a non-linear differential equations to some simple linear differential.We …
In this article, we introduce a new homotopy perturbation method (NHPM) for solving non-linear problems, such that it can be converted a non-linear differential equations to some simple linear differential.We will solve linear differential equation by using analytic method that it is better than the variational iteration method and to find parameter , we use projection method, which is easier and decrease computations in comparison with similar works.Also in some of the references perturbation method are depend on small parameter but in our proposed method it is not depend on small parameter, finally we will solve some example for illustrating validity and applicability of the proposed method.
In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) . The homotopy perturbation method …
In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) . The homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.
In this talk, we briefly describe the basic ideas and applications of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Compared to other analytic approximation methods, …
In this talk, we briefly describe the basic ideas and applications of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Compared to other analytic approximation methods, the HAM has some advantages. First, unlike perturbation techniques, the HAM has nothing to do with any small/large physical parameters so that it works for more problems, especially for those without small/large physical parameters. Besides, unlike all other methods, the HAM provides us a simple way to guarantee the convergence of solution series. In addition, the HAM provides us great freedom to choose equation-type and solution expression of high-order equations so that it is easy to obtain approximations at rather high order. Due to these advantages, the HAM have been successfully applied to solve lots of nonlinear problems in science, engineering, finance and so on. By means of the HAM, some classical problems have been solved with much better results. Especially, some new concepts have been proposed and some new solutions have been found by means of the HAM, mainly because a truly new method always brings us something new/different!
In this paper, we present a modification to so-called homotopy perturbation method for solving linear and non-linear integral equations. This method gives an approximate analytic solution to the equations (usually …
In this paper, we present a modification to so-called homotopy perturbation method for solving linear and non-linear integral equations. This method gives an approximate analytic solution to the equations (usually the exact solution of the equations). Some numerical examples presented to show the accuracy and efficiency of the method.
According to Nonlinear Fredholm differential and integral equation，it is proposed that the improved homotopy perturbation method is used to solve in this paper, and apply the numerical examples to compare …
According to Nonlinear Fredholm differential and integral equation，it is proposed that the improved homotopy perturbation method is used to solve in this paper, and apply the numerical examples to compare the advantages among homotopy perturbation method, Adomian decomposition method and improved homotopy perturbation method . The results show that improved homotopy perturbation method is more effective.
We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. …
We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.
In this paper, a homotopy perturbation method (HPM) is extended and applied for solving system of nonlinear equations of n-dimensional with n-variables. Also, numerical examples are used to show the …
In this paper, a homotopy perturbation method (HPM) is extended and applied for solving system of nonlinear equations of n-dimensional with n-variables. Also, numerical examples are used to show the performance of the presented method, on a series of examples published in the literature, and to compare with other literature methods.
As an effective method for solving linear and nonlinear equations, the homotopy perturbation method is usually applied to solving relevant problems. We analyze 74 studies on the application of the …
As an effective method for solving linear and nonlinear equations, the homotopy perturbation method is usually applied to solving relevant problems. We analyze 74 studies on the application of the homotopy perturbation method and present a comprehensive review of them with the conclusion obtained: (1) Homotopy perturbation method is generally applied to solving the problems of ordinary differential equations; (2) Homotopy perturbation method is usually combined with the technology of transform when it is used to solve more complicated equations; (3) By comparing homotopy perturbation method with other similar methods, many researchers sought that homotopy perturbation method is rapidly convergent, highly accurate, computational simple; (4) Some studies point out that when homotopy perturbation method is applied, some parameters including the number of terms, time span, time step must be prescribed carefully. Finally, two suggestions on the further study of the application of the HPM are provided.
In this work, we apply the new homotopy perturbation method (NHPM) to get accurate results for solving systems of nonlinear equations of Emden–Fowler type, we indicate that our method (NHPM) …
In this work, we apply the new homotopy perturbation method (NHPM) to get accurate results for solving systems of nonlinear equations of Emden–Fowler type, we indicate that our method (NHPM) is equivalent to the variational iteration method (VIM) with a specific convex. Four examples are given to illustrate our proposed methods. The method is easy to carry out and gives very accurate solutions for solving linear and nonlinear differential equations.
In this paper, we use the modified homotopy perturbation method to solving the system of linear equations. We show that this technique enables us to find the exact solution of …
In this paper, we use the modified homotopy perturbation method to solving the system of linear equations. We show that this technique enables us to find the exact solution of the system of linear equations. This technique is independent of the auxiliary parameter and auxiliary operator. Our results can be viewed as a novel improvement and an extension of the previously known results.
In this paper, the application of homotopy perturbation method (HPM) is extended to the perturbation problems. The HPM yields an analytical solution in terms of a rapidly convergent infinite power …
In this paper, the application of homotopy perturbation method (HPM) is extended to the perturbation problems. The HPM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. The efficiency of the HPM is examined by several illustrative examples. Comparisons with the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving perturbation problems.
In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the …
In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons between the results of the presented method and exact solutions are made. The results reveal that the HPM is very effective and convenient for solving such kind of equations.
In this paper, a new analytical approach based on homotopy perturbation Sumudu transform method (HPSTM) to a two-dimensional viscous flow with a shrinking sheet is presented. The series solution is …
In this paper, a new analytical approach based on homotopy perturbation Sumudu transform method (HPSTM) to a two-dimensional viscous flow with a shrinking sheet is presented. The series solution is obtained by HPSTM coupled with Pade approximants to handle the condition at infinity. The HPSTM is a combined form of the Sumudu transform method, homotopy perturbation method and He’s polynomials. This scheme finds the solution without any discretization or restrictive assumptions and avoids round-off errors. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. doi: 10.14456/WJST.2014.39
In this paper, an iterative numerical solution of the higher dimensional (3+1) physically important nonlinear evolutionary equation is studied by using the Homotopy Perturbation Method (HPM). For this purpose, the …
In this paper, an iterative numerical solution of the higher dimensional (3+1) physically important nonlinear evolutionary equation is studied by using the Homotopy Perturbation Method (HPM). For this purpose, the Jumbo-Miwa (JM) equation is analyzed with HPM and the available exact solution obtained by Homogenous Balance Method will be compared to show the accuracy of the proposed numerical algorithm. The result approves the effectiveness and accuracy of the HPM.
In this paper, we study the nonlinear behaviour of multi-component plasma. For this an efficient technique, called Homotopy perturbation Sumudu transform method (HPSTM) is introduced. The power of method is …
In this paper, we study the nonlinear behaviour of multi-component plasma. For this an efficient technique, called Homotopy perturbation Sumudu transform method (HPSTM) is introduced. The power of method is represented by solving the time fractional Kersten-Krasiloshchik coupled KdV-mKdV nonlinear system. This coupled nonlinear system usually arises as a description of waves in multi-component plasmas, traffic flow, electric circuits, electrodynamics and elastic media, shallow water waves etc. The prime purpose of this study is to provide a new class of technique, which need not to use small parameters for finding approximate solution of fractional coupled systems and eliminate linearization and unrealistic factors. Numerical solutions represent that proposed technique is efficient, reliable, and easy to use to large variety of physical systems. This study shows that numerical solutions gained by HPSTM are very accurate and effective for analysis the nonlinear behaviour of system. This study also states that HPSTM is much easier, more convenient and efficient than other available analytical methods.
This work focuses on the heat transfer analysis of the magnetohydrodynamic peristaltic flow of blood through the gap between two coaxial flexible tubes of different wavelengths. In this model, the …
This work focuses on the heat transfer analysis of the magnetohydrodynamic peristaltic flow of blood through the gap between two coaxial flexible tubes of different wavelengths. In this model, the non-Newtonian biviscosity fluid is assumed to be blood, which is flowing through the annulus region between the inclined tubes. The governing equations for the considered problem are simplified under the assumptions of a zero Reynolds number and long wavelength approximations. An analytical expression for the temperature is obtained in its closed form, while a semi-analytical solution for the axial velocity of the moving fluid is determined using the homotopy perturbation method. Expressions for various flow variables such as shear stress, volumetric flow rate, pressure rise, and frictional force at the surface of inner and outer tubes are also obtained. In this work, we discussed the impact of various flow parameters like the Hartmann number, Grashof number, heat source, upper limit apparent viscosity coefficient, amplitude ratios of inner and outer tubes, radius ratio, and wavelength ratio on the above flow variables. The streamline contour plots are also drawn for the realization of the flow pattern of the blood inside the endoscope. The comparison of shear stresses for the peristaltic and rigid endoscopes and the validation of the present result with the previously established results are also discussed. A noteworthy observation drawn from the present model is that the pressure gradient enhances for increasing values of wavelength ratio when the wavelength of the inner tube is less than the wavelength of the outer tube, whereas the pressure gradient gets suppressed for increasing values of wavelength ratio when the wavelength of the inner tube is greater than the wavelength of the outer tube. From the present analysis, it is also found that the shear stress [Formula: see text] of blood is the least for a peristaltic endoscope as compared to the rigid one. Therefore, this study may be applicable to medical practitioners for laparoscopic purposes, as a peristaltic endoscope may be a more appropriate device due to its flexibility than a rigid endoscope.
Bu çalışma, lineer ve lineer olmayan zaman kesirli mertebeli kısmi diferensiyel denklemleri çözmek için ayrık uzak biçimli ayrık homotopi perturbasyon metodunu geliştirmiştir. Kesirli mertebe türevler Caputo anlamında göz önüne alınmıştır. …
Bu çalışma, lineer ve lineer olmayan zaman kesirli mertebeli kısmi diferensiyel denklemleri çözmek için ayrık uzak biçimli ayrık homotopi perturbasyon metodunu geliştirmiştir. Kesirli mertebe türevler Caputo anlamında göz önüne alınmıştır. Bu metodun başarısı ve uygulanabilirliği bazı örnek problemler ile gösterilmiştir. Elde edilen sonuçlar kesirli mertebe bir olduğunda, tam çözümler ile iyi bir uyumluluk göstermiştir. Bu çalışmada gösterilen metodun kesirli mertebe hesabındaki benzer problemleri çözmesi beklenmektedir.
The time evolution of the multispecies Lotka‐Volterra system is investigated by the homotopy analysis method (HAM). The continuous solution for the nonlinear system is given, which provides a convenient and …
The time evolution of the multispecies Lotka‐Volterra system is investigated by the homotopy analysis method (HAM). The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth‐order Runge‐Kutta method. The convergence theorem for the three‐dimensional case is also given.
Direct solution of a class of n th‐order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides …
Direct solution of a class of n th‐order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven‐ and eight‐order Runge‐Kutta method (RK78).
The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a …
The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a limited number of analytical solutions for the number density function corresponding to empirical kernels are available in the literature. This article introduces a semi-analytical approach using the homotopy perturbation method to obtain series solutions for the nonlinear collisional fragmentation equation. The method presented here can be readily adapted to solve both linear and nonlinear integral equations, eliminating the need for domain discretization. To gain deeper insights intothe accuracy of the proposed method, a convergence analysis is conducted. This analysis employs the concept of contractive mapping within the Banach space, a well-established technique universally acknowledged for ensuring convergence. Various collisional kernels (product and polymerization kernels), breakage distribution functions (binary and multiple breakage) and various initial particle distributions are considered to obtain the new series solutions. The obtained results are successfully compared against finite volume method [26] solutions in terms of number density functions and their moments. The error between the exact and obtained series solutions is shown in plots and tables to confirm the applicability and accuracy of the proposed 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 semi-exact methods are introduced for estimating the distribution of tangential displacement and shear stress in non-uniform rotating disks. At high variable angular velocities, the effect of shear …
In this paper semi-exact methods are introduced for estimating the distribution of tangential displacement and shear stress in non-uniform rotating disks. At high variable angular velocities, the effect of shear stress on Von Mises stress is important and must be considered in calculations. Therefore, He’s homotopy perturbation method (HPM) and Adomian’s decomposition method (ADM) is implemented for solving equilibrium equation of rotating disk in tangential direction under variable mechanical loading. The results obtained by these methods are then verified by the exact solution and finite difference method. The comparison among HPM and ADM results shows that although the numerical results are the same approximately but HPM is much easier, straighter and efficient than ADM. Numerical calculations for different ranges of thickness parameters, boundary conditions and angular accelerations are carried out. It is shown that with considering disk profile variable, level of displacement and stress in tangential direction are not always reduced and type of changing the thickness along the radius of disk and boundary condition are an important factor in this case. Finally, the optimum disk profile is selected based on the tangential displacement-shear stress distribution. The presented algorithm is useful for the analysis of rotating disk with any arbitrary function form of thickness and density that it is impossible to find exact solutions.
An analytical effort is made to achieve cognition on the effect of time-dependent mechanical loading on the stress fields of rotating disks with non-uniform thickness and density. At high variable …
An analytical effort is made to achieve cognition on the effect of time-dependent mechanical loading on the stress fields of rotating disks with non-uniform thickness and density. At high variable angular velocities and accelerations, evaluation of the effect of shear stress on the values of von Mises stress is significant and it is excellent to consider shear stress in this equivalent stress calculation alongside the radial and tangential stress. In the proposed analytical model, the Homotopy perturbation method (HPM) solves the general structure of rotating disks equilibrium equations in both radial and tangential directions. HPM is an efficient tool to solve linear and nonlinear equations, providing solutions in quick converging series. The results obtained through this process are then confirmed using the finite difference method and the exact solution in the literature. The effect of parameters in angular velocity and acceleration functions with the parameter in the thickness function and the effect of boundary conditions on the values of elastic limit angular velocity and acceleration are established by performing numerical examples. Furthermore, the effect of shear stress on the maximum values of von Mises stress is discussed. It is shown that shear stress has more influence on the distribution of equivalent von Mises stress in the elastic region. It is shown the introduced analytical model is useful for evaluating rotating disk with any arbitrary shape of thickness and density function, without using the commercial finite element simulation software.
In this paper, we used the homotopy analysis method to ordinary differential equations of type boundary value problems with a parameter representing turning points."To show the high accuracy of the …
In this paper, we used the homotopy analysis method to ordinary differential equations of type boundary value problems with a parameter representing turning points."To show the high accuracy of the solution results, we compare the numerical results applying the standard homotopy analysis method with the integral equation and the numerical solution of the Simpson and Trapezoidal rules."Also, we give the estimated order of convergence (local) and the global estimated order of convergence along the interval.
The current study looks at the peristaltically driven motion of Carreau fluid in a symmetric channel under the influence of an induced and applied magnetic field. Tantalum (Ta) and Gold …
The current study looks at the peristaltically driven motion of Carreau fluid in a symmetric channel under the influence of an induced and applied magnetic field. Tantalum (Ta) and Gold (Au) nanoparticles (NPs) with thermal radiation effects are incorporated in the hybrid nanofluid. The proposed mathematical modeling in two dimensions is finalized by employing lubrication theory. The finalized forms of mathematical modeling are nonlinear, and a perturbation approach is used to solve them. Up to the second-order approximation, solutions for velocity distribution, induction equation, and temperature distribution are reported. For velocity, current density, magnetic force function, induced magnetic field, and temperature profile, graphical and numerical results are presented. For simple and hybrid nanofluids, shear-thinning, Newtonian, and shear-thickening scenarios, numerical data are also presented. The current study aims to be useful in the medical field since Ta-NPs with low cytotoxicity can remove unwanted reactive oxygen species, providing protection for biomedical applications. Magnetic drug targeting is an effective and accurate way for drug delivery to the affected areas in biomedical science. It is accomplished by binding the medicine to biologically suitable magnetic nanoparticles, which are then steered towards the target by carefully placing magnets on the body's external surface.
The variational homotopy perturbation method VHPM is used for solving<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty …
The variational homotopy perturbation method VHPM is used for solving<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy.
The aim of this article is to apply the collocation method for boundary layer in unbounded domain. The solution for velocity and temperature are computed by applying the collocation method. …
The aim of this article is to apply the collocation method for boundary layer in unbounded domain. The solution for velocity and temperature are computed by applying the collocation method. It does not need any perturbation, linearization, or small parameter versus homotopy perturbation method and parameter perturbation method. Also the determination of the auxiliary parameter and auxiliary function versus homotopy analysis method is not necessary. The use of a special technique to obtain solutions that are very close to the exact solution of the equation has been attempted. The idea is to transform the equations and boundary conditions into another set of variables. In comparison with previous studies, the solution shows that the results of the present method are in excellent agreement with those of the numerical methods. As an important result, it is depicted that approximation of the physical quantities f ′′(0) and θ′(0) are more accurate in comparison with those obtained using homotopy perturbation method. Also, the results reveal that this method is more suitable for solution of boundary layer problems with infinite boundary values.
In this paper, we present homotopy analysis method (HAM) for solving system of linear equations and use of different H(x) in this method. The numerical results indicate that this method …
In this paper, we present homotopy analysis method (HAM) for solving system of linear equations and use of different H(x) in this method. The numerical results indicate that this method performs better than the homotopy perturbation method (HPM) for solving linear systems.
The Duffing oscillator represents an important example to describe, mathematically, the nonlinear behavior of several phenomena in physics and engineering. In view of its diverse applications in science and engineering, …
The Duffing oscillator represents an important example to describe, mathematically, the nonlinear behavior of several phenomena in physics and engineering. In view of its diverse applications in science and engineering, the current work investigates the stability analysis of the parametric Duffing oscillator. This equation represents a second-order ordinary differential equation with cubic nonlinearity and periodic coefficient. A coupling of the homotopy perturbation method (HPM) and Laplace transform (LT), an approximate solution is derived. The HPM is adapted to find another accurate approximate solution. The latter analysis reveals the exact solution of the cubic Duffing equation. In addition, an expanded frequency parameter is achieved to find another approximate periodic solution. Therefore, this method is exercised to govern the stability criteria of the problem. Finally, the multiple time scales with the HPM is used to judge the stability criteria. The analyses include the resonance as well as nonresonance cases. Numerical estimations are performed to confirm, graphically, the perturbed solutions together with the stability examination. It is shown that the damped parameter and the cubic stiffness parameter have a destabilizing influence. In contrast, the natural and parametric frequencies are of stabilizing influences.
In this paper, He's homotopy perturbation method is applied to solve two-dimensional linear and nonlinear Volterra integral equations. Then homotopy perturbation method (HPM) is compared with the differential transform method …
In this paper, He's homotopy perturbation method is applied to solve two-dimensional linear and nonlinear Volterra integral equations. Then homotopy perturbation method (HPM) is compared with the differential transform method (DTM) for solving two dimen- sional integral equations. We also give some examples to demonstrate the accuracy of the method. From the computational view point, the homotopy perturbation method is more
The beam deformation equation has very wide applications in structural engineering. As a differential equation, it has its own problem concerning existence, uniqueness and methods of solutions. Often, original forms …
The beam deformation equation has very wide applications in structural engineering. As a differential equation, it has its own problem concerning existence, uniqueness and methods of solutions. Often, original forms of governing differential equations used in engineering problems are simplified, and this process produces noise in the obtained answers. This paper deals with solution of second order of differential equation governing beam deformation using four analytical approximate methods, namely the Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM). The comparisons of the results reveal that these methods are very effective, convenient and quite accurate to systems of non-linear differential equation.
Purpose The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain. Design/methodology/approach …
Purpose The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain. Design/methodology/approach A weighted algorithm based on the homotopy perturbation method is used to minimize the volume of computations. The authors show that this technique yields the analytical solution of the desired problem in the form of a rapidly convergent series with easily computable components. Findings The authors illustrate that the proposed method produces satisfactory results with respect to Adomian decomposition method and standard homotopy perturbation method. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. Originality/value This research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.
Optimal Variational Iteration Method (OVIM) is Variational Iteration Method (VIM) coupled with auxiliary parameter h.In this paper, we have discussed a hydrological problem with pressure distribution phenomenon solved with Optimal …
Optimal Variational Iteration Method (OVIM) is Variational Iteration Method (VIM) coupled with auxiliary parameter h.In this paper, we have discussed a hydrological problem with pressure distribution phenomenon solved with Optimal Variational Iteration Method (OVIM).In the framework of Optimal Variational Iteration Method (OVIM), the auxiliary parameter h, that is convergence controlling parameter is the primary tool which guarantees the convergence of said technique.Moreover, the convergence is obtained by so-called residual error method.Results shows that the reliability of the method with the least error and provide the required solution to the pressure distribution of water in a water reservoir in initial four iteration.
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.
Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. …
Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger's equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.
In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) . The homotopy perturbation method …
In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) . The homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to …
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.
Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities …
Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equations with Variable Coefficients Differential Equations with a Large Parameter Solvability Conditions Index
Article Α Review on Some New Recently Developed Nonlinear Analytical Techniques was published on March 1, 2000 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 1, …
Article Α Review on Some New Recently Developed Nonlinear Analytical Techniques was published on March 1, 2000 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 1, issue 1).