Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation

TL;DR

This paper applies Lie symmetry analysis to reduce a fractional 2D heat equation and introduces a spectral method based on Bernstein's matrices for its numerical solution.

Contribution

It combines Lie symmetry reduction with a spectral method to solve a space-time fractional heat equation, which is a novel approach.

Findings

Derived similarity solutions for the fractional heat equation.

Established a one-dimensional optimal Lie symmetry algebra.

Developed a spectral method using Bernstein's matrices for numerical solutions.

Abstract

In this paper, the Lie symmetry analysis is proposed for a space-time convection-diffusion fractional differential equations with the Riemann-Liouville derivative by (2+1) independent variables and one dependent variable. We find a reduction form of our governed fractional differential equation using the similarity solution of our Lie symmetry. One-dimensional optimal system of Lie symmetry algebras is found. We present a computational method via the spectral method based on Bernstein's operational matrices to solve the two-dimensional fractional heat equation with some initial conditions.