| Abstract: |
Partial differential equations (PDEs) with time-varying coefficients emerge across numerous scientific and engineering domains, including fluid dynamics, quantum mechanics, and financial modeling. This research develops a comprehensive unified framework integrating exact analytical solutions with advanced numerical methodologies for solving time-varying PDEs. The primary objective investigates the convergence characteristics and computational efficiency of hybrid approaches combining spectral methods, finite element techniques, and symbolic computation. We hypothesize that unified frameworks demonstrate superior accuracy compared to standalone methods when applied to complex time-dependent problems. The methodology employs comparative analysis across benchmark problems using MATLAB and Python implementations. Results indicate that hybrid spectral-finite element methods achieve error reductions of 10^-8 compared to traditional approaches, with computational time improvements of 35-42%. Statistical validation through Richardson extrapolation and L2-norm analysis confirms the framework's robustness. The unified approach demonstrates particular effectiveness for problems with discontinuous coefficients and singular perturbations. This framework provides researchers and practitioners with systematic tools for selecting appropriate solution strategies based on problem characteristics, advancing both theoretical understanding and practical applications in computational mathematics and physics. |
