A Novel Hybrid Approach Combining Fractional Calculus and Metaheuristic Optimization for Solving High-Dimensional Non-Linear Partial Differential Equations

Abstract

This paper presents a novel hybrid approach for solving high-dimensional non-linear partial differential equations (PDEs) by integrating fractional calculus concepts with metaheuristic optimization algorithms. Specifically, we employ the Caputo fractional derivative to model the PDE and then utilize a modified Particle Swarm Optimization (PSO) algorithm to minimize the error functional associated with the fractional PDE. The proposed method addresses the challenges posed by high dimensionality and non-linearity, which often render traditional numerical techniques computationally infeasible or inaccurate. We demonstrate the efficacy and accuracy of our approach through several benchmark problems, comparing our results with those obtained by existing methods. The convergence analysis and computational efficiency of the hybrid algorithm are also investigated. The results demonstrate that the proposed method offers a promising alternative for solving complex fractional PDEs in various scientific and engineering applications.