Solving an inverse problem for a time-fractional advection-diffusion equation with variable coefficients by rationalized Haar wavelet method

Abstract

In this research work, we use of rationalized Haar wavelet (RHW) method and Crank–Nicolson finite-difference scheme to propose a new idea for solving the inverse time-fractional advection-diffusion equation with variable coefficients. Based on this method, the expansion of the solution is a series of RHW functions with variable coefficients. One of the advantages of the proposed method is the use of operational matrices for single and double integration of basic functions without the need for direct integration. In addition, the numerical estimation of the order of convergence and extrapolation of the results are provided. To reveal the efficiency of our method, the numerical solutions of two examples have been compared with Cubic B-spline function (CBS) and Radial basis function (RBF) methods. Finally, the diagrams of error for the Haar wavelet and RBF methods are shown in the figures.