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Concept
Fundamental solution
Summary
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).
In terms of the Dirac delta "function" δ(x), a fundamental solution F is a solution of the inhomogeneous equation
Here F is a priori only assumed to be a distribution.
This concept has long been utilized for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz.
The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis. In the context of functional analysis, fundamental solutions are usually developed via the Fredholm alternative and explored in Fredholm theory.
Consider the following differential equation Lf = sin(x) with
The fundamental solutions can be obtained by solving LF = δ(x), explicitly,
Since for the Heaviside function H we have
there is a solution
Here C is an arbitrary constant introduced by the integration. For convenience, set C = −1/2.
After integrating and choosing the new integration constant as zero, one has
Once the fundamental solution is found, it is straightforward to find a solution of the original equation, through convolution of the fundamental solution and the desired right hand side.
Fundamental solutions also play an important role in the numerical solution of partial differential equations by the boundary element method.
Consider the operator L and the differential equation mentioned in the example,
We can find the solution of the original equation by convolution (denoted by an asterisk) of the right-hand side with the fundamental solution :
This shows that some care must be taken when working with functions which do not have enough regularity (e.g.
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