DOI: https://doi.org/10.36719/2663-4619/125/199-205
Nazrin Imamverdiyeva
Azərbaijan State Oil and İndustry University
Master’s student
https://orcid.org/0009-0001-9854-5409
inezrin2003@gmail.com
Numerical Solution of a Boundary Value Problem for a Second-Order Linear Ordinary Differential Equation
Abstract
The most widespread and universal numerical method for solving differential equations is the finite difference method. The main idea of the method is as follows. The continuous domain of variation of the argument (for example, an interval) is replaced by a set of discrete points called nodes. These nodes form a difference grid. The sought function of a continuous argument is approximated by a discrete-argument function defined on the given grid. This function is called a grid function. The original differential equation is replaced by a difference equation with respect to the grid function. In this process, appropriate finite difference relations are used to approximate the derivatives appearing in the equation. The replacement of a differential equation by a difference equation is called its approximation on the grid (or finite difference approximation). Solving the differential equation is thus reduced to finding the values of the grid function at the grid nodes.
The justification of replacing the differential equation with a difference equation, the accuracy of the obtained solutions, and the stability of the method are among the most important issues that require careful study. The finite difference method is widely used, especially in the numerical solution of boundary and initial value problems. With the help of this method, differential equations that do not have a complex analytical solution can be solved in a practical manner. The choice of the grid step directly affects the accuracy of the calculations and the computational process. Therefore, determining the optimal step size is considered an important practical issue.
Keywords: finite difference method, differential equations, difference equation, approximation, stability and accuracy, continuity, grid point