Purpose

The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.

Design/methodology/approach

First, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.

Findings

It is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.

Originality/value

Although many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.

中文翻译：

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非线性时分微分方程解析解的一种新方法

目的

本文的目的是通过一种新颖的方法来建议时间分数Fornberg–Whitham和时间分数Fokker–Planck方程的解。

设计/方法/方法

首先，定义了分数导数的一些基本性质，以构建一种新颖的方法。其次，构建了改进的拉普拉斯同伦摄动法（HPM），该方法可直接应用。第三，给出了两个数值例子，说明了该方法的准确性，并通过图形显示了该方法的有效性。最后，详细证明了HPM的收敛性。

发现

对于这种方法的发展，没有必要考虑任何类型的假设和假设。因此，所提出的方法变得非常简单，并且是求解时间分数阶微分方程的一种更好的方法。

创意/价值

尽管在文献中介绍了许多分数阶偏微分方程解的解析方法。这种新颖的方法表明，所提出的方法无需任何假设即可直接应用。