Solutions of Nonlinear Parabolic PDEs: A Novel Technique Based on Galerkin-Finite Difference Residual Corrections

Numerical solutions for second-order parabolic partial differential equations (PDEs), specifically the nonlinear heat equation, are investigated with a focus on analyzing residual corrections. Initially, the Galerkin weighted residual method is employed to rigorously formulate the heat equation and derive numerical solutions using third-degree Bernstein polynomials as basis functions. Subsequently, a proposed residual correction scheme is applied, utilizing the finite difference method to solve the error equations while adhering to the associated error boundary and initial conditions. Enhanced approximations are achieved by incorporating the computed error values derived from the error equations into the original weighted residual results. The stability and convergence of the residual correction scheme are also analyzed. Numerical results and absolute errors are compared against exact solutions and published literature for various time and space step sizes, demonstrating the effectiveness and precision of the proposed scheme in achieving high accuracy.