The Technique of Analyzing a Non-Homogeneous Partial Differential Equation by Splitting The Problem into The Inverse and Direct Problems

Abstract

Background: In this study, we decompose a non-homogeneous second-order partial differential equation into a homogeneous part and a non-homogeneous part. Previous research on the existence and uniqueness of solutions for both sections provide a clear mathematical foundation. The main goal of this research is to divide the partial differential equation into two separate parts: one homogeneous and one non-homogeneous. These parts are then addressed using initial conditions along with various boundary and overdetermination conditions.

Materials and Methods: We use the finite difference method to numerically solve both the homogeneous and non-homogeneous components. Various boundary conditions are considered, including Dirichlet, Neumann, and mixed conditions, each requiring specific supplementary information. To assess the stability of the numerical solutions, we introduce different noise levels and apply Tikhonov regularization to stabilize the results.

Results: Several numerical examples are presented to evaluate the effectiveness of the proposed approach. The findings indicate that the finite difference method yields accurate solutions under a range of boundary conditions, and that Tikhonov regularization effectively enhances stability in noisy environments.

Conclusion: The numerical results validate the robustness and reliability of the proposed method. By combining Tikhonov regularization with finite difference discretization, this study establishes an efficient framework for solving non-homogeneous second-order partial differential equations under various boundary conditions. The research results indicate that, under various boundary, extra, and initial conditions, we can solve the problem and find convergent solutions for both parts.