Hybrid Fourier Series and Weighted Residual Function Method for Caputo-Type Fractional PDEs With Variable Coefficients

Abstract

This study presents a novel computational framework for approximating solutions to time-fractional partial differential equations (TFPDEs) with variable coefficients, employing the Caputo definition of fractional derivatives. TFPDEs, distinguished by their fractional-order time derivatives, inherently capture the non-local and memory-dependent dynamics observed in a wide range of physical and engineering systems. The proposed method reformulates the TFPDE into a set of decoupled fractional-order ordinary differential equations (FODEs) via Fourier expansion strategy. This decomposition facilitates analytical tractability while preserving the essential features of the original system. The initial conditions of each resulting FODE are systematically obtained from the governing equation's initial data. Auxiliary initial value problems are formulated for each FODE to facilitate the construction of explicit particular solutions. These solutions are then synthesized through a carefully designed linear superposition, optimized to minimize the residual error across the domain of interest. This residual minimization ensures that the composite solution closely approximates the behavior of the original TFPDE, offering both accuracy and computational efficiency. Theoretical analysis demonstrates that the method is convergent. A FLOP-based analysis confirms that the proposed method is computationally efficient. The validity and effectiveness of the proposed scheme are demonstrated through a set of benchmark problems. Empirical convergence rates are compared with those from existing numerical methods in each case. The findings confirm that the proposed approach consistently achieves superior accuracy and demonstrates robust performance under a wide range of scenarios. These findings highlight the method's potential as a powerful and versatile tool for solving complex TFPDEs in mathematical modeling and applied sciences.