A Galerkin algorithm leveraging Bernoulli polynomials for accurate solutions of time-fractional diffusion-wave equations

Abstract

This study presents a modified Galerkin technique utilizing Bernoulli polynomials for time-fractional diffusion-wave equations (TFDWEs). The proposed approach combines fractional calculus, namely Caputo derivatives, with a semi-discrete approach to achieve a high numerical accuracy. By utilizing Bernoulli polynomials as an efficient basis to approximate the solution, the algorithm transforms the governing equations into very sparse linear systems that can be solved computationally efficiently. Detailed numerical investigations, including applications to fractional wave equations and fourth-order diffusion-wave equations, demonstrate the method’s ability to achieve reduced errors and better computational efficiency. The results underline the stability and accuracy of the proposed technique, which turns out to be particularly suitable for simulating complex physical systems characterized by memory effects and anomalous diffusion.