Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation

This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples.