A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative
A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative
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DOI：
10.3934/math.2023110
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发表时间：
2022
期刊：
影响因子：
2.2
通讯作者：
Amjid Ali;Teruya Minamoto;Rasool Shah;K. Nonlaopon
中科院分区：
文献类型：
--
作者：
Amjid Ali;Teruya Minamoto;Rasool Shah;K. Nonlaopon
本文推导了积分的$\psi$-Haar小波运算矩阵，并将其用于解线性$\psi$-分数阶偏微分方程组($\psi$-FPDEs)，其中分数阶导数由$\psi$-Caputo算子定义。我们用Haar小波逼近线性$\psi$-FPDE解的最高阶分数偏导数。通过将运算矩阵与$\psi$-分数次积分相结合，我们逼近了解及其其他$\psi$-分数阶偏导数。然后将这些近似代入给定的$\psi$-FPDE中，得到一个线性代数方程组。最后通过对该系统的求解，得到了该系统的近似解。作为一种求解分数阶偏微分方程组的数学工具，该方法的简单性和有效性是其主要优点之一。运算矩阵的稀疏性提高了该方法以较小的计算复杂度执行的能力。数值算例表明了该方法的有效性和有效性。
In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.