Two-dimensional Müntz–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 Müntz–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) Müntz–Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann–Liouville operational matrix of 2D Müntz–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.