Finite Element Method (FEM) for Solving Time-Fractional Partial Differential Equations

https://doi.org/10.5281/zenodo.17480913

Abstract

The critical, and effective Finite element Method (FEM) given in this paper can adequately trace the time-fractional of partial derivative equations (TFPDE) with the labels of factors of (Caputo) form of derivatives, incorporating the memory and hereditary influences in the majority of the physical and engineering frameworks. The study begins with setting the weak variational problem followed by building of the fully discrete FEM system regarding both spatial Galerkin discretization and two times strategies L1 quadrature approximation and Convolution Quadrature (CQBDF2) approach. Theoretical analyses based on Lax -Milgram lemma show the existence and uniqueness of weak solution and error and stability analysis and bolster the optimality of the convergence rates between O(h 2 + t 2): L1 O(h 2 + t 2): CQ-BDF2. Numerical experiments applied on benchmark problems which guarantee good accuracy, robust performance and efficiency even in conditions with weak singularity and heterogeneity are in support of such theoretical predictions. The capability of the framework to handle irregular geometries and changing coefficients is a testament to the fact that that framework can be used to simulate a wide variety of phenomena in the real world, such as anomalous diffusion, viscoelastic relaxation, and biological transport. The paper also indicates the future directions that can be enhanced in terms of adaptive mesh refinement, efficient time-stepping schemes and high-performance parallel implementation which will increase simulation of high-dimensional fractional systems as scalable.