Fundamental solution

A fundamental solution is a tool in the theory of partial differential equations. With the help of a fundamental solution one can construct special solutions to these equations.

Definition

If a linear differential operator with constant coefficients, then a fundamental solution is defined as a distributional solution of

Where the Dirac delta function is.

Application

If a fundamental solution is known for a linear differential operator, we obtain a solution of the equation

By convolution of the fundamental solution with the right side:

Theory

For many differential equations, a fundamental solution is known as the Poisson equation, the heat equation, the wave equation and the Helmholtz equation.

In general, the set of Malgrange - Ehrenpreis, after which each partial differential equation with constant coefficients has a fundamental solution applies.

Examples

• The fundamental solution of the total differential is the Heaviside function, see also delta function # Ableitung_der_Heaviside distribution.
• The function is ( in the distributional sense) the fundamental solution of the Cauchy -Riemann operator.
• For the Laplace operator, the Green's function is the fundamental solution.