The main advantage of this proposed method is its capability of combining two powerful methods for obtaining rapid convergent series for fractional partial differential equations. HAM was first proposed and applied by Liao 10 — 13 based on homotopy, a fundamental concept in topology and differential geometry. The HAM has been successfully applied by many researchers for solving linear and nonlinear partial differential equations.
Among these, we may mention the following: Laplace decomposition methods 23 , 24 and homotopy perturbation transform method. To illustrate the basic idea of the HATM for the fractional partial differential equation, we consider the following fractional partial differential equation as. For simplicity, we ignore all initial and boundary conditions, which can be treated in a similar way. Now, the methodology consists of applying Laplace transform first on both sides of equation 3 , and we get. By means of generalizing the traditional homotopy method, Liao 10 — 13 constructs the zero-order deformation equation.
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It is important that one has great freedom to choose auxiliary thing in HATM. Thus, as q increases from 0 and 1 , the solution varies from the initial guess u 0 x , t to the solution u r , t. Combining equations 17 and 20 , we get. We combine the new hybrid iterative algorithm for nonlinear fractional partial differential equations arising in science and engineering. It should be stated that the selection of optimal value of the auxiliary parameter in the HATM and HAM has an approximately similar procedures or more details, see Cheng et al.
In this section, two examples on time-fractional wave equations are solved to demonstrate the performance and efficiency of the HATM. We consider the one-dimensional wave-like equation with variable coefficients 7 — 9 as. Operating the inverse Laplace transform on both sides in equation 29 , we get. Consequently, we obtain the solution of equation 24 in a series form.
However, mostly, the results given by the Laplace decomposition method and homotopy perturbation transform method converge to the corresponding numerical solutions in a rather small region. From Figure 2 , the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. In this example, we consider the following two-dimensional wave-like equation with variable coefficients 7 — 9 as. Applying the Laplace transform on both sides in equation 35 and after using the differentiation property of Laplace transform, we get.
Operating the inverse Laplace transform on both sides in equation 43 , we get. Hence, the solution of the given problem is given as. From Figure 4 , the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. In this article, we carefully proposed a reliable modification of the HAM which introduces a promising tool for solving wave equations of fractional order.
Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter. This work illustrates the validity and great potential of the HATM for fractional differential equations. An excellent agreement is achieved. The solution is very rapidly convergent by utilizing the modified HAM by modification of Laplace operator. It may be concluded that the modified HATM methodology is very powerful and efficient in finding approximate solutions as well as analytical solutions of many fractional physical models.
The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper. The second author is highly grateful to the Department of Mathematics, National Institute of Technology, Jamshedpur, India, for the provision of some excellent facilities and research environment. Skip to main content. Advances in Mechanical Engineering. A modified homotopy analysis method for solution of fractional wave equations.
Download Citation If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Via Email All fields are required. Send me a copy Cancel. Article first published online: December 7, ; Issue published: December 1, Received: September 14, ; Accepted: This article is part of the following special collection s: Advances in Fractional Dynamics in Mechanical Engineering ,.
This article is distributed under the terms of the Creative Commons Attribution 3. Keywords Fractional wave equation , fractional derivatives , analytical solution , Mittag-Leffler function , Laplace transform , homotopy analysis transform method.
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A numerical scheme for the solution of multi-order fractional differential equations
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