NUMERICAL SOLUTION OF FRACTIONAL ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION

This article develops a finite element method (FEM) for solving nonlinear fractional-order partial differential equations (PDEs) using the Caputo fractional derivative. The scheme achieves second-order convergence in <i>L</i>² and <i>L</i>^∞ norms and is supported by a rigorous qualitative analysis, including existence, uniqueness, and Ulam-Hyers stability, established via nonlinear functional analysis and fixed-point theorems. Applied to a fractionalorder Burgers' equation modeling longitudinal dispersion in homogeneous porous media, the method demonstrates superior accuracy compared to finite difference and B-spline Galerkin methods. Numerical results validate robustness across fractional orders <i>α</i> ∈ {0.5, 0.75, 1.0}, with applications in environmental engineering and fluid dynamics.