Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

Abstract

This study presents numerical solutions for initial boundary value problems of homogeneous and non-homogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes' robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.