On One Approximate Method of a Boundary Value Problem for a One-Dimensional Advection–Diffusion Equation

• Published in: Axioms
• Main Authors: Temirkhan Aleroev, Victor Orlov
• Format: Article
• Language: English
• Published: MDPI AG 2022-10-01
• Subjects: advection–diffusion; eigenvalue; eigenfunction; fractional derivative
• Online Access: https://www.mdpi.com/2075-1680/11/10/541

This article discusses the author’s version of the technology for solving a one-dimensional boundary value problem for a one-dimensional advection–diffusion equation based on the method of separation of variables, as well as the theory of eigenvalues and eigenfunctions when constructing a solution to a differential equation. This problem is solved in two stages. Firstly, we illustrate the technology of separating variables for equations with fractional derivatives, and then apply the theory of eigenvalues and eigenfunctions to obtain an exact solution in the form of an infinite series. Since this series converges very quickly, it is natural to replace it with the sum of the first few terms. The approximate solution obtained in this way is quite suitable for numerical calculations in practice. The article provides a listing of the program for performing calculations, as well as the results of calculations themselves.