Fundamental solution

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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 RHS — was shown by Malgrange and Ehrenpreis.

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[edit] 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|

[edit] Fundamental solutions for some partial differential equations

Laplace equation

[-\nabla^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')

The fundamental solutions in two and three dimensions are

\Phi_{2D}(\mathbf{x},\mathbf{x}')= -\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}

Helmholtz equation where the parameter k is real and the fundamental solution a modified Bessel function.

[-\nabla^2+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')

The two and three dimensional Helmholtz equations have the fundamental solutions

\Phi_{2D}(\mathbf{x},\mathbf{x}')= \frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}\exp(-k|\mathbf{x}-\mathbf{x}'|)

Biharmonic equation

[-\nabla^4] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')

The biharmonic equation has the fundamental solutions

\Phi_{2D}(\mathbf{x},\mathbf{x}')= -\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}(\ln|\mathbf{x}-\mathbf{x}'| - 1),\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{|\mathbf{x}-\mathbf{x}'|}{8\pi}

[edit] 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.

[edit] 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

[edit] 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).

[edit] See also

  • parametrix
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