A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation

A class of partial differential equations with fractional derivatives in the spatial variables are called space-fractional diffusion equations. They can be applied to simulate anomalous diffusion, in which the classical diffusion equation does not accurately describe how a plume of particles disperses. Analytically solving fractional diffusion equations can be problematic due to the typically complex structures of fractional derivative models. Hence, this study proposes the utilisation of a satisfier function in combination with the Ritz method to effectively address fractional diffusion equations in the Caputo sense. By employing this approach, the equations are transformed into an algebraic system, so facilitating their solution and providing a numerical result. This method can achieve a high level of accuracy in solving the Caputo fractional diffusion equations by utilising only a small number of terms from the shifted Legendre polynomials in two variables. The precision and effectiveness of our approach may be evaluated, as it yielded dependable approximations of the solutions.