- Fundamental solution
mathematics, a fundamental solution for a linear partial differential operator"L" is a formulation in the language of distribution theoryof the older idea of a Green's function. In terms of the Dirac delta functionδ("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
This concept was long known for the
Laplacianin 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 convolutionto solve an arbitrary right hand side — was shown by Malgrangeand Ehrenpreis.
Consider the following differential equation "Lf" = sin(x) with L as,
The fundamental solutions can be obtained by solving "Lf" = δ("x"), explicitly,
Since for the
Heaviside function"H" we have
:"H"′("x") = δ("x")
there is a solution
:"F"′("x") = "H"(x) + "C".
Here "C" is an arbitrary constant. For convenience, set
:"C" = − 1/2.
After integrating and taking the integration constant as zero, we get
Fundamental solutions for some
partial differential equations Laplace equation
The fundamental solutions in two and three dimensions are
Helmholtz equationwhere the parameter "k" is real and the fundamental solution a modified Bessel function.
The two and three dimensional Helmholtz equations have the fundamental solutions :
The biharmonic equation has the fundamental solutions
The motivation to find the fundamental solution is because once one finds the fundamental solution, it is easy to find the desired solution of the original equation. In fact, this process is achieved by convolution.
Fundamental solutions also play an important role in the numerical solution of partial differential equations by the
boundary element method.
Application to the example
Consider the operator L, mentioned in the example.
Since we have found the fundamental solution, we can easily find the solution of the original equation by convolution,
Proof that the convolution is the desired solution
Say we are trying to find the solution of
:"Lf" = "g"("x").
When applying the differential operator, "L", to the convolution it is known that
provided "L" has constant coefficients.
If "f" is the fundamental solution, the RHS reduces to
It is straightforward to verify that this is in fact "g"("x") (in other words the delta function acts as
identity elementfor convolution). Summing up,
Therefore, if "F" is the fundamental solution, the convolution "F"*"g" is the solution of "Lf" = "g"("x").
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