MODERN APPROACHES TO SOLVING LINEAR PARABOLIC FILTRATION PROBLEMS

Abstract

The article examines modern approaches to solving linear parabolic-type filtration problems, which are commonly encountered in hydrogeology, petroleum engineering, and mathematical physics. Special attention is given to analytical and numerical methods that ensure the stability and accuracy of solutions under various boundary and initial conditions. Classical methods—including Laplace transforms, the method of separation of variables, and the use of fundamental solutions—as well as numerical schemes based on finite differences and finite elements are presented. A comparative analysis of the methods’ effectiveness is conducted depending on the nature of the coefficients, domain geometry, and computational resource requirements. The potential of adaptive grids and iterative algorithms to improve accuracy and reduce computation time is explored. Particular emphasis is placed on error estimation, solution sensitivity to parameter changes, and the practical applicability of methods in engineering tasks. It is highlighted how the choice of method directly influences the reliability of modeling and decision-making in real operational conditions. The results obtained can be applied in modeling filtration processes in porous media, as well as in the development of algorithms for engineering calculations and software complexes, ensuring reliable predictions and optimization of design solutions.