Two-dimensional Muntz-Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations

In this manuscript, we present a new numerical technique based on two-dimensional Muntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Muntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Muntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works.