A novel numerical method for solution of fractional partial differential equations involving the $\psi$-Caputo fractional derivative

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.