Purpose

The purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.

Design/methodology/approach

The author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.

Findings

The author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.

Originality/value

Many scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

中文翻译：

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一种求解Caputo-Hadamard分数微分方程的小波方法

目的

本工作的目的是提出一种小波方法，用于任意区间上 Caputo-Hadamard 分数微分方程的数值解。

设计/方法/方法

作者修改了 CAS 小波 (mCAS) 并将其用于解决 Caputo-Hadamard 分数线性/非线性初始和边值问题。作者推导并构造了 mCAS 小波的新运算矩阵。此外，作者还提出了一种结合mCAS小波和拟线性化技术的方法来求解非线性Caputo-Hadamard分数微分方程。

发现

作者证明了mCAS小波的正交性。针对Caputo-Hadamard分数边值问题，作者构建了mCAS小波矩阵、任意阶Hadamard分数积分的mCAS小波运算矩阵和Hadamard分数积分的mCAS小波运算矩阵。这些运算矩阵用于快速计算。此外，作者对该方法进行了误差分析。作者介绍了 Caputo-Hadamard 分数初始和边值问题的实现过程。提供数值模拟来说明该方法的可靠性和准确性。

原创性/价值

许多科学家、医生和工程师可以利用所提出的方法来模拟他们的线性/非线性 Caputo-Hadamard 分数微分模型。据作者所知，目前的工作从未针对线性/非线性 Caputo-Hadamard 分数微分方程提出和实施。