Comparative Study of Numerical Approaches for Fractional Order Partial Differential Equations
Keywords:
numerical methods, Dirichlet problem, weak solution, finite difference, spectral methodAbstract
Fractional order partial differential equations (FPDEs) have emerged as a powerful tool for modeling complex phenomena characterized by memory and hereditary properties in physics, engineering, biology, and finance. The numerical solution of FPDEs presents significant challenges due to their nonlocal nature and the complexity of fractional derivatives. This paper provides a comparative study of prominent numerical approaches for solving FPDEs, including finite difference methods, finite element methods, spectral techniques, and Ritz-Galerkin schemes. We assess each method in terms of accuracy, computational efficiency, stability, and suitability for various classes of problems. The study also discusses the theoretical underpinnings of weak solutions, minimization principles, and best approximation results, with a particular focus on the Dirichlet problem. Our findings offer valuable insights for researchers and practitioners seeking reliable and effective computational strategies for FPDEs.
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