Fundamental solutions of stationary equations and the method of separation of variables in linear problems of mathematical physics

Abstract

The paper considers two key methods for solving linear differential equations of the second order:

• construction of a fundamental solution (Green function) for stationary inhomogeneous equations, and
• method of separation of variables in solving boundary value problems for parabolic and hyperbolic equations.

The basic definitions, formulas and algorithms are given, and examples of application of each method are given. Particular attention is paid to the practical significance of fundamental solutions in constructing particular solutions and the application of the method of separation of variables in problems of heat transfer and string vibrations