Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations

In this paper, we consider a new fractional function based on Legendre and Laguerre polynomials for solving a class of linear and nonlinear time-space fractional partial differential equations with variable coefficients. The concept of the fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on operational and pseudo-operational matrices of integer and fractional order integration with collocation method. We convert the problem to a system of algebraic equations by applying the operational matrices, pseudo-operational matrices and collocation method. Also, we calculate the upper bound for the error of integral operational matrix of the fractional order. We illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph. We also describe the physical application of some examples.