Fractional Partial Differential Equations and Their Applications

Fractional partial differential equations (FPDEs) extend classical PDEs by allowing differentiation of non-integer (fractional) order. Over the past two decades, FPDEs have been recognised as powerful tools for modelling physical, biological, and financial systems that exhibit anomalous diffusion, long-range dependence, and memory effects. This paper reviews the theoretical foundations of fractional calculus, examines the formulation and properties of FPDEs, and discusses contemporary analytical and numerical solution methods. Applications spanning anomalous diffusion, viscoelastic materials, control systems, and image processing are examined to demonstrate the versatility of FPDEs. Finally, the paper highlights challenges and frontiers in the theory and applications of FPDEs, including stochastic fractional models and high-dimensional problems.