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.

中文翻译：

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利用拉普拉斯同伦方法求解变系数分数阶类热方程和类波动方程

目的

本文旨在提出具有可变系数的时间分数热似和波动（TFH-L和WL）方程的近似解。所提出的方案表明结果非常接近于精确解。

设计/方法/方法

首先借助分数导数的一些基本性质，构造了具有求解分数阶偏微分方程能力的方案。然后，通过该方案求解变系数的TFH-L和WL方程，其结果非常接近精确解。得出的结果表明，该方案非常有效。最后，讨论了该方法的收敛性。

发现

将传统方法与Laplace变换相结合以构造此方案。为了分解非线性项，本文介绍了具有He多项式的同伦摄动方法，因此，该解以一系列形式提供，可以很快收敛到精确解。

创意/价值

所提出的方法是原始且非常有效的，因为据作者所知，该方法首次非常成功地用于解决分数阶偏微分方程，这引起了人们的极大兴趣。