Efficient delannoy polynomials method for nonlinear fractional diffusion and liénard equations

Abstract

Fractional ordinary differential equations modeling diffusion-like processes are important tools for describing systems where transport is unusual or depends on past behavior, such as in porous media, biological systems, or viscoelastic materials. Since exact solutions are rarely available, we rely on numerical methods to solve them. In this work, we introduce a spectral collocation method that uses Delannoy polynomials to approximate solutions of these equations involving the Caputo derivative. By building operational matrices for the Caputo derivative, the method directly constructs the algebraic system, which is then solved efficiently using Newton’s method. Tests show that the method is highly accurate, converges quickly, and is computationally efficient. These results suggest that Delannoy polynomials are a strong alternative to classical polynomial bases for spectral solutions of fractional differential problems.