- Fundamental solution
In

mathematics , a**fundamental solution**for a linearpartial differential operator "L" is a formulation in the language ofdistribution theory of the older idea of aGreen's function . In terms of theDirac delta function δ("x"), a fundamental solution "f" is the solution of theinhomogeneous 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 byMarcel Riesz . The existence of a fundamental solution for any operator withconstant coefficients — the most important case, directly linked to the possibility of usingconvolution to solve an arbitrary right hand side — was shown byMalgrange andEhrenpreis .**Example**Consider the following differential equation "Lf" = sin(x) with L as,

:$L=frac\{partial^2\}\{partial\; x^2\}$

The fundamental solutions can be obtained by solving "Lf" = δ("x"), explicitly,

:$frac\{partial^2\}\{partial\; x^2\}\; F(x)\; =\; delta(x)$

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

:$F(x)=frac\{1\}\{2\}\; |x|$

**Fundamental solutions for some**partial differential equation sLaplace equation :$[-\; abla^2]\; Phi(mathbf\{x\},mathbf\{x\}\text{'})\; =\; delta(mathbf\{x\}-mathbf\{x\}\text{'})$

The fundamental solutions in two and three dimensions are

:$Phi\_\{2D\}(mathbf\{x\},mathbf\{x\}\text{'})=-frac\{1\}\{2pi\}ln|mathbf\{x\}-mathbf\{x\}\text{'}|,quad\; Phi\_\{3D\}(mathbf\{x\},mathbf\{x\}\text{'})=frac\{1\}\{4pi|mathbf\{x\}-mathbf\{x\}\text{'}$

Helmholtz equation where the parameter "k" is real and the fundamental solution a modifiedBessel function .:$[-\; abla^2+k^2]\; Phi(mathbf\{x\},mathbf\{x\}\text{'})\; =\; delta(mathbf\{x\}-mathbf\{x\}\text{'})$

The two and three dimensional Helmholtz equations have the fundamental solutions :$Phi\_\{2D\}(mathbf\{x\},mathbf\{x\}\text{'})=frac\{1\}\{2pi\}K\_0(k|mathbf\{x\}-mathbf\{x\}\text{'}|),quadPhi\_\{3D\}(mathbf\{x\},mathbf\{x\}\text{'})=frac\{1\}\{4pi|mathbf\{x\}-mathbf\{x\}\text{'}exp(-k|mathbf\{x\}-mathbf\{x\}\text{'}|)$

Biharmonic equation :$[-\; abla^4]\; Phi(mathbf\{x\},mathbf\{x\}\text{'})\; =\; delta(mathbf\{x\}-mathbf\{x\}\text{'})$

The biharmonic equation has the fundamental solutions

:$Phi\_\{2D\}(mathbf\{x\},mathbf\{x\}\text{'})=-frac\{8pi\}$

**Motivation**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.

:$frac\{partial^2\}\{partial\; x^2\}\; f(x)\; =\; sin(x)$

Since we have found the fundamental solution, we can easily find the solution of the original equation by convolution,

:$int\_\{-infty\}^\{infty\}\; frac\{1\}\{2\}|x\; -\; y|sin(y)dy$

**Proof that the convolution is the desired solution**Denote the

convolution operation as:"f"*"g".

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

:"L"("f"*"g")=("Lf")*"g",

provided "L" has constant coefficients.

If "f" is the fundamental solution, the RHS reduces to

:δ*"g".

It is straightforward to verify that this is in fact "g"("x") (in other words the delta function acts as

identity element for convolution). Summing up,:$L(F*g)=(LF)*g=delta(x)*g(x)=int\_\{-infty\}^\{infty\}\; delta\; (x-y)\; g(y)\; dy=g(x)$

Therefore, if "F" is the fundamental solution, the convolution "F"*"g" is the solution of "Lf" = "g"("x").

**See also***

parametrix

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