https://doi.org/10.36719/2663-4619/112/119-125
Shahnaz Qadimova
Azerbaijan State Oil and Industry University
master student
https://orcid.org/0009-0008-4399-6967
kadimovashahnaz01@gmail.com
The Solution of Boundary Problems for Parabolic Type Equations By the
Finite Difference and Grid Methods
Abstract
The article presents an algorithm for solving boundary value problems of parabolic type equations using the finite difference method. The introduction of a coordinate grid, based on the functions involved in the initial and boundary conditions, leads to the formulation of a matrix equation for the grid functions. The key contribution of this work is the formulation of fundamental and diagonal matrices, which facilitate the transition from the matrix equation to the ordinary differential equations related to the grid functions. Formulas for both direct and inverse transitions between the sought and newly introduced functions are provided. The resulting ordinary differential equations can be solved either exactly or approximately. The results are valuable for solving parabolic, elliptic, and hyperbolic type equations under mixed boundary conditions, both in single-domain and multi-domain scenarios.
Keywords: finite difference method, differential equations, boundary conditions, approximation, algorithm, computational experiments