Purpose

The main aim of the paper is to introduce the shifted fractional-order Gegenbauer wavelets (SFGWs) and the development of a method for solving fractional nonlinear initial and boundary value problems on a semi-infinite domain.

Design/methodology/approach

The proposed method is the combination of SFGWs and parametric iteration method. We have derived and constructed the new operational matrices for the SFGWs, which are utilized for the solutions of nonlinear fractional differential equations.

Findings

We have constructed the weight function and normalizing factor for SFGWs. The operational matrices for the SFGWs are derived and constructed, to make the calculations fast. Furthermore, we work out an error analysis for the method. The procedure of implementation for both fractional nonlinear initial and boundary value problems is presented. Numerical simulation is provided to illustrate the reliability and accuracy of the method.

Originality/value

Many engineers can utilize the presented method for solving their nonlinear fractional differential models. To the best of the author’s knowledge, the shifted fractional-order Gegenbauer wavelets have never been introduced and implemented for nonlinear fractional differential equations.

中文翻译：

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求解分数阶微分方程的分数Gegenbauer小波运算矩阵方法

目的

本文的主要目的是介绍平移分数阶Gegenbauer小波（SFGWs）以及开发在半无限域上求解分数阶非线性初值和边值问题的方法。

设计/方法/方法

所提出的方法是SFGW和参数迭代方法的组合。我们已经导出并构造了SFGW的新运算矩阵，并将其用于非线性分数阶微分方程的解。

发现

我们已经构造了SFGW的权重函数和归一化因子。推导并构造了SFGW的运算矩阵，以加快计算速度。此外，我们对该方法进行了误差分析。给出了分数阶非线性初值和边值问题的实现过程。数值仿真表明了该方法的可靠性和准确性。

创意/价值

许多工程师可以利用提出的方法来求解其非线性分数差分模型。据作者所知，从未引入移位分数阶Gegenbauer小波并将其用于非线性分数阶微分方程。