A computational approach based on the fractional Euler functions and Chebyshev cardinal functions for distributed-order time fractional 2D diffusion equation

Abstract

In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonality, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative operational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss–Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.

Keywords

• Fractional Euler functions
• Chebyshev cardinal functions
• Distributed-order fractional derivative
• Diffusion equation