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..