A Legendre spectral finite difference method for the solution of non-linear space-time fractional Burger’s–Huxley and reaction-diffusion equation with Atangana–Baleanu derivative

Abstract

In this research, we have solved non-linear reaction-diffusion equation and non-linear Burger's–Huxley equation with Atangana Baleanu Caputo derivative. We developed a numerical approximation for the ABC derivative of Legendre polynomial. A difference scheme is applied to deal with fractional differential term in the time direction of differential equation. We applied Legendre spectral method to deal with unknown function and spatial ABC derivatives. A formulation to deal with Dirichlet boundary condition is also included. After applying this spectral method our problem reduces to a system of fractional partial differential equation. To solve this system we developed finite difference scheme by which our FPDEs system reduces to a system of algebraic equations. Taking the help of initial conditions we solve this algebraic system and find the value of unknowns, To demonstrate the effectiveness and validity of our proposed method some numerical examples are also presented. We compare our obtained numerical results with the analytical results.