scholarly journals Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

2020 ◽  
Vol 34 (2) ◽  
pp. 203-221
Author(s):  
Ali Khalouta ◽  
Abdelouahab Kadem
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Variable Coefficients ◽  
Numerical Comparison ◽  
Fractional Partial Differential Equations ◽  
Efficient Tool ◽  
Partial Differential

AbstractThis work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.

scholarly journals Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Convergent Series ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Shao-Wen Yao
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type ◽  
The Laplace Transform ◽  
First Time

Purpose This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution. Design/methodology/approach First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed. Findings A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He’s polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly. Originality/value The proposed approach is original and very effective because this approach is, to the authors’ knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest.

Solving Fractional Partial Differential Equations with Variable Coefficients by the Reconstruction of Variational Iteration Method

2015 ◽  
Vol 70 (5) ◽  
pp. 375-382 ◽  
Author(s):  
Esmail Hesameddini ◽  
Azam Rahimi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Closed Form Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Form Solution ◽  
Successive Approximations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

AbstractIn this article, we propose a new approach for solving fractional partial differential equations with variable coefficients, which is very effective and can also be applied to other types of differential equations. The main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations. The fractional derivatives are described based on the Caputo sense. Our method contains an iterative formula that can provide rapidly convergent successive approximations of the exact solution if such a closed form solution exists. Several examples are given, and the numerical results are shown to demonstrate the efficiency of the newly proposed method.

scholarly journals Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Efficient Tool ◽  
Partial Differential ◽  
Fractional Boundary Value Problems

We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sinc-Galerkin method is a very efficient tool in solving space-fractional partial differential equations.

The Invariant Subspace Method for Solving Fractional Partial Differential Equations

2020 ◽  
pp. 267-282
Author(s):  
Mohamed Soror Abdel Latif ◽  
Abass Hassan Abdel Kader
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Balance Principle ◽  
New Exact Solutions

In this chapter, the authors discuss the effectiveness of the invariant subspace method (ISM) for solving fractional partial differential equations. For this purpose, they have chosen a nonlinear time fractional partial differential equation (PDE) with variable coefficients to be investigated through this method. One-, two-, and three-dimensional invariant subspace classifications have been performed for this equation. Some new exact solutions have been obtained using the ISM. Also, the authors give a comparison between this method and the homogeneous balance principle (HBP).

scholarly journals On some nonlinear fractional PDEs in physics

BIBECHANA ◽  
2014 ◽  
Vol 12 ◽  
pp. 59-69
Author(s):  
Jamshad Ahmad ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Variational Iteration Method ◽  
Mathematical Physics ◽  
Inverse Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Pdes ◽  
The Given ◽  
Nonlinear Nature

In this paper, we applied relatively new fractional complex transform (FCT) to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and Variational Iteration Method (VIM) is to find approximate solution of time- fractional Fornberg-Whitham and time-fractional Wu-Zhang equations. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs arising in mathematical physics and hence can be extended to other problems of diversified nonlinear nature. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm.  DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11687BIBECHANA 12 (2015) 59-69 

scholarly journals Solitary and compacton solutions of fractional KdV-like equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 328-336 ◽  
Author(s):  
Bo Tang ◽  
Yingzhe Fan ◽  
Jianping Zhao ◽  
Xuemin Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Reliable Tool ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Liouville Derivative

AbstractIn this paper, based on Jumarie’s modified Riemann-Liouville derivative, we apply the fractional variational iteration method using He’s polynomials to obtain solitary and compacton solutions of fractional KdV-like equations. The results show that the proposed method provides a very effective and reliable tool for solving fractional KdV-like equations, and the method can also be extended to many other fractional partial differential equations.

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