Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection–diffusion equation

• The operational matrix of fractional-order Legendre functions method are considered.
• The fractional order convection–diffusion problem is solved.
• The problem can be used extensively in science and engineering as in oil reservoir simulations.
• Problems which, an initially discontinuous profile is propagated by diffusion and convection.

In this paper, the application of the operational matrix of fractional-order Legendre functions (FLFs) to solve the time-fractional convection–diffusion equation has been investigated. Fractional calculus has been applied to model the engineering and physical processes which are best described with other mathematical tools. The time variable of the time-fractional convection–diffusion equation and its space variable have been approximated by FLFs and shifted Legendre polynomials, respectively. The fractional derivatives together with product matrices of FLFs are employed to convert the solution of this problem to the solution of a system of algebraic equations.