DOI: https://doi.org/10.36719/2663-4619/125/194-198
Minara Aliyeva
Azərbaijan State Oil and İndustry University
Master’s student
https://orcid.org/0009-0000-5833-217X
minareeliyeva2202@gmail.com
Application and Analysis of the Finite Difference Method for the
Numerical Solution of the Helmholtz Equation
Abstract
In this article, we will study the application and analysis of the finite difference method for the numerical solution of the Helmholtz equation. The Helmholtz equation is one of the fundamental equations of mathematical physics. The Helmholtz equation was formulated by the German physicist and mathematician Hermann von Helmholtz to simplify the mathematical description of wave processes. This equation is used in acoustics, wave propagation, electromagnetic waves, thermal dissipation, and other processes. The analytical solution of the Helmholtz equation can be calculated only for domains with simple geometric shapes and with selected boundary conditions. In complex processes, inhomogeneous environments, and for boundary conditions reflecting real physical processes, it is difficult or impossible to find an analytical solution. Therefore, approximate solutions of the equation are investigated, and numerical methods are used.
In this article, we will consider the problems of reducing the differential equation to a system of algebraic equations using the finite difference method and solving the resulting system of equations.
Keywords: Helmholtz equation, finite difference method, electromagnetic waves, numerical solution, mathematical physics, differential equation