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. In terms of the Dirac delta function δ(x), a fundamental solution F is the solution of the inhomogeneous equation
- LF = δ(x).
Here F is a priori only assumed to be a Schwartz distribution.
This concept was long known 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 Malgrange and Leon Ehrenpreis.
Read more about Fundamental Solution: Example, Motivation, Signal Processing
Famous quotes containing the words fundamental and/or solution:
“Much of what contrives to create critical moments in parenting stems from a fundamental misunderstanding as to what the child is capable of at any given age. If a parent misjudges a child’s limitations as well as his own abilities, the potential exists for unreasonable expectations, frustration, disappointment and an unrealistic belief that what the child really needs is to be punished.”
—Lawrence Balter (20th century)
“There is a lot of talk now about metal detectors and gun control. Both are good things. But they are no more a solution than forks and spoons are a solution to world hunger.”
—Anna Quindlen (b. 1953)