Solution of nonlinear fractional differential equations using homotopy analysis method (original) (raw)
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In this paper, we present an algorithm of the homotopy analysis method (HAM) to obtain symbolic approximate solutions for linear and nonlinear differential equations of fractional order. We show that the HAM is different from all analytical methods; it provides us with a simple way to adjust and control the convergence region of the series solution by introducing the auxiliary parameter , the auxiliary function ( ), the initial guess ( ) and the auxiliary linear operator . Three examples, the fractional oscillation equation, the fractional Riccati equation and the fractional Lane-Emden equation, are tested using the modified algorithm. The obtained results show that the Adomain decomposition method, Variational iteration method and homotopy perturbation method are special cases of homotopy analysis method. The modified algorithm can be widely implemented to solve both ordinary and partial differential equations of fractional order.
Analysis of nonlinear fractional partial differential equations with the homotopy analysis method
Communications in Nonlinear Science and Numerical Simulation, 2009
In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2, 2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m, n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Modified Homotopy Analysis Method for Nonlinear Fractional Partial Differential Equations
International Journal of Analysis and Applications, 2017
In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM). The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.
The homotopy analysis method for handling systems of fractional differential equations
Applied Mathematical Modelling, 2010
In this article, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve systems of fractional integro-differential equations. Comparing with the exact solution, the HAM provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter h. Four examples are tested using the proposed technique. It is shown that the solutions obtained by the Adomian decomposition method (ADM) are only special cases of the HAM solutions. The present work shows the validity and great potential of the homotopy analysis method for solving linear and nonlinear systems of fractional integro-differential equations. The basic idea described in this article is expected to be further employed to solve other similar nonlinear problems in fractional calculus.
2014
In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations.Based on the homotopy analysis method, a scheme is developed to obtain the approximate solution of the nonlinearfractional heat conduction, Kaup–Kupershmidt, Fisher and Huxley equations with initial conditions, introduced byreplacing some integer-order time derivatives by fractional derivatives. The solutions of the studied models are calculatedin the form of convergent series with easily computable components. The results of applying this procedure tothe studied cases show the high accuracy and efficiency of the new technique. The fractional derivative is describedin the Caputo sense. Some illustrative examples are presented to observe some computational results.
In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Revised fractional homotopy analysis method for solving nonlinear fractional PDEs
PROCEEDING OF THE 1ST INTERNATIONAL CONFERENCE ON ADVANCED RESEARCH IN PURE AND APPLIED SCIENCE (ICARPAS2021): Third Annual Conference of Al-Muthanna University/College of Science
In paper, A of fractional homotopy method (MFHAM) is and used for solving some p p fractional partial differential equations (NFPDEs) with Caputo fractional derivative (CFD). Examples have presented, to the method the results are compared with homotopy analysis p (FHAM).
An interdisciplinary journal of discontinuity, nonlinearity, and complexity, 2023
We present an efficient approach for solving nonlinear fractional differential equations. The convergence analysis of the approach is studied. To demonstrate the working of the presented approach, we consider three special cases of nonlinear fractional differential equations. The results of theses examples and comparison with different methods provide confirmation for the validity of the proposed approach.
Homotopy Analysis Method for Solving Some Partial Time Fractional Differential Equation
In this paper, the homotopy analysis method (HAM) is applied to solve a time-fractional nonlinear partial differential equation. The fractional derivatives are described by Caputo's sense, and the (HAM) gives a series of solutions which converge rapidly within a few terms with the help of the nonzero convergence control parameter ℏ. After applying this method we reach the conclusion that the HAM is very efficient and accurate. Graphical representations of the solution obtained..
Analysis of a time fractional wave-like equation with the homotopy analysis method
Physics Letters A, 2008
The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, whenh f =h g = −1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.
A New Analytical Method for Solving Linear and Nonlinear Fractional Partial Differential Equations
In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for solving linear and the nonlinear fractional partial differential equation is introduced. The proposed analytical method is an elegant combination of a well-known method, Homotopy Perturbation Method (HPM) and the Natural Transform Method (NTM). In this new analytical method, the fractional derivative is computed in Caputo sense and the nonlinear terms are calculated using He's polynomials. Exact solution of linear and nonlinear fractional partial differential equations are successfully obtained using the new analytical method, and the result is compared with the result of the existing methods.
Solving Fractional Integro Differential Equations by Homotopy Analysis Transform Method
International Journal of Pure and Apllied Mathematics, 2016
In this paper, we introduce an analytical method, which so called the homotopy analysis transform method (HATM) which is a combination of HAM and Laplace decomposition method (LDM). This scheme is simple to apply linear and nonlinear fractional integro-differential equation and having less computational work in comparison of other exiting methods. The fractional derivatives are described in the Caputo sense. The most useful advantage of this method is to solve the fractional integro-differential equation without using Adomian polynomials and He's polynomials for the computation of nonlinear terms.
International Journal of Electrical and Computer Engineering (IJECE), 2022
In recent years nonlinear problems have several methods to be solved and utilize a well-known analytic tools such as homotopy analysis method. In general, homotopy analysis method had gain a wide focus and improvement especially in typical nonlinear problem. The aim of this paper is to use homotopy method of analysis to solve partial differential equation in addition to improve method's efficiency. The method in this paper is to apply approximation to Padé approach to obtain sufficient efficiency. As a result, the improvement has been verified by solving two cases beside a mean value comparison of the homotopy analysis method's squared error with the improved form.
Comparison between Some Methods for Solving Fractional Differential Equations
In this paper, the Homotopy Analysis Method (HAM) is applied to obtain the solution of fractional differential equations. The fractional derivatives are described in the Caputo sense. The solution obtained by this method has been compared with those obtained by Homotopy Perturbation Method (HPM) and the Variational Iteration Method (VIM). Results show that (HPM) and (VIM) are all special cases of the homotopy analysis method (HAM) when the nonzero convergence-control parameter ℏ = −1.
Tamkang Journal of Mathematics , 2018
The reliability of the homotopy analysis method (HAM) and reduction in the size of the computational work give this method a wider applicability. In this paper, HAM has been successfully applied to find the approximate solutions of Caputo fractional Volterra-Fredholm integro-differential equations. Also, the behavior of the solution can be formally determined by analytical approximation. Moreover, the study proves the existence and uniqueness results and the convergence of the solution. This paper concludes with an example to demonstrate the validity and applicability of the proposed technique.
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Mathematical Sciences, 2014
The current work highlights the following issues: a brief survey of the development in the theory of fractional differential equations has been raised. A very recent technique based on the generalized Taylor series called-residual power series (RPS)-is introduced in detailed manner. The time-fractional foam drainage equation is considered as a target model to test the validity of the RPS method. Analysis of the obtained approximate solution of the fractional foam model reveals that RPS is an alternative method to be added for the fractional theory and computations and considered to be a significant method for exploring nonlinear fractional models.
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In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differen...
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