METHOD OF LINES FOR A LOADED PARABOLIC EQUATION

Loaded parabolic equations belong to a complex yet important class of differential equations and are widely applied in various scientific and engineering problems, as well as in ecology, epidemic propagation modeling, and biological systems. Special analytical and numerical methods are used to solve these equations, taking into account the influence of integral and functional loads. This article examines a two-point boundary value problem for loaded parabolic equations, defined in a closed domain. The solution is approached using the method of lines with respect to the variable x. As a result of this method, a discretized problem is formulated. The obtained discretized problem is represented in a vector-matrix form and is reduced to a two-point boundary value problem for a loaded system of differential equations. The parameterization method proposed by Professor Dzhumabaev is used to solve the boundary value problem. The efficiency of this method lies in the high accuracy of the numerical-analytical solution compared to the exact solution, as well as in the possibility of formulating the solvability conditions of the problem. As a theoretical justification of the method, an additional theorem is proven, based on which the solvability conditions of the problem are determined. The study explores the relationship between the original boundary value problem and its discretized form for the loaded parabolic equation. This relationship is substantiated using an additional theorem derived from the parameterization method.