A new approach to solving multi-order fractional equations using BEM and Chebyshev matrix

TL;DR

This paper introduces a hybrid numerical scheme combining boundary element method and Chebyshev operational matrix to efficiently solve multi-order two-dimensional time-fractional PDEs, demonstrating high accuracy and convergence.

Contribution

The paper presents a novel hybrid approach integrating BEM and Chebyshev matrices for solving complex multi-order fractional PDEs with improved accuracy.

Findings

High accuracy demonstrated through numerical experiments

Convergence rates analyzed for different mesh sizes

Efficient solution of multi-order fractional PDEs achieved

Abstract

In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multi-order time-fractional partial differential equations; nonlinear and linear in respect to spatial and temporal variables, respectively. Fractional derivatives are estimated by Caputo sense. Boundary element method is used to convert the main problem into a system of a multi-order fractional ordinary differential equation. Then, the produced system is approximated by Chebyshev operational matrix technique, ans its condition number is analyzed. Accuracy and efficiency of the proposed hybrid scheme are demonstrated by solving three different types of two-dimensional time fractional convection-diffusion equations numerically. The convergent rates are calculated for different meshing within the boundary element technique. Numerical results are given by graphs and tables for solutions and different type of error norms.