Fundamental solution

In mathematics, a fundamental solution for a linear partial differential operator Template:Mvar 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" Template:Math, a fundamental solution Template:Mvar is a solution of the inhomogeneous equation Template:Block indent Here Template:Mvar 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, and a proof is available in Joel Smoller (1994). In the context of functional analysis, fundamental solutions are usually developed via the Fredholm alternative and explored in Fredholm theory.

Example

Consider the following differential equation Template:Math with $$L=\frac{d^2}{d{x}^2}.$$

The fundamental solutions can be obtained by solving Template:Math, explicitly, $$\frac{d^2}{d{x}^2}F(x)=\delta (x).$$

Since for the unit step function (also known as the Heaviside function) Template:Mvar we have $$\frac{d}{dx}H(x)=\delta (x),$$ there is a solution $$\frac{d}{dx}F(x)=H(x)+C.$$ Here Template:Mvar is an arbitrary constant introduced by the integration. For convenience, set Template:Math.

After integrating $\frac{dF}{dx}$ and choosing the new integration constant as zero, one has $$F(x)=xH(x)-\frac{1}{2}x=\frac{1}{2}|x|.$$

Motivation

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.

Application to the example

Consider the operator Template:Mvar and the differential equation mentioned in the example, $$\frac{d^2}{d{x}^2}f(x)=\sin (x).$$

We can find the solution $f(x)$ of the original equation by convolution (denoted by an asterisk) of the right-hand side $\sin (x)$ with the fundamental solution $F(x)=\frac{1}{2}|x|$: $$f(x)=(F*\sin )(x):=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{2}|x-y|\sin (y)dy.$$

This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, L¹ integrability) since, we know that the desired solution is Template:Math, while the above integral diverges for all Template:Mvar. The two expressions for Template:Mvar are, however, equal as distributions.

An example that more clearly works

$$\frac{d^2}{d{x}^2}f(x)=I(x),$$ where Template:Mvar is the characteristic (indicator) function of the unit interval Template:Closed-closed. In that case, it can be verified that the convolution of Template:Math with Template:Math is $$(I*F)(x)=\{\begin{array}{cc}\frac{1}{2}{x}^2-\frac{1}{2}x+\frac{1}{4},& 0\le x\le 1\\ |\frac{1}{2}x-\frac{1}{4}|,& \text{otherwise}\end{array}$$ which is a solution, i.e., has second derivative equal to Template:Mvar.

Proof that the convolution is a solution

Denote the convolution of functions Template:Mvar and Template:Mvar as Template:Math. Say we are trying to find the solution of Template:Math. We want to prove that Template:Math is a solution of the previous equation, i.e. we want to prove that Template:Math. When applying the differential operator, Template:Mvar, to the convolution, it is known that $$L(F*g)=(LF)*g,$$ provided Template:Mvar has constant coefficients.

If Template:Mvar is the fundamental solution, the right side of the equation reduces to $$\delta *g.$$

But since the delta function is an identity element for convolution, this is simply Template:Math. Summing up, $$L(F*g)=(LF)*g=\delta (x)*g(x)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x-y)g(y)dy=g(x).$$

Therefore, if Template:Mvar is the fundamental solution, the convolution Template:Math is one solution of Template:Math. This does not mean that it is the only solution. Several solutions for different initial conditions can be found.

Fundamental solutions for some partial differential equations

The following can be obtained by means of Fourier transform:

Laplace equation

For the Laplace equation, $$[-\mathrm{\Delta }]\mathrm{\Phi }(𝐱,{𝐱}^{\prime })=\delta (𝐱-{𝐱}^{\prime })$$ the fundamental solutions in two and three dimensions, respectively, are $${\mathrm{\Phi }}_{\text{2D}}(𝐱,{𝐱}^{\prime })=-\frac{1}{2\pi }\ln |𝐱-{𝐱}^{\prime }|,{\mathrm{\Phi }}_{\text{3D}}(𝐱,{𝐱}^{\prime })=\frac{1}{4\pi |𝐱-{𝐱}^{\prime }|}.$$

Screened Poisson equation

For the screened Poisson equation, $$[-\mathrm{\Delta }+k^2]\mathrm{\Phi }(𝐱,{𝐱}^{\prime })=\delta (𝐱-{𝐱}^{\prime }),k\in ℝ,$$ the fundamental solutions are $${\mathrm{\Phi }}_{\text{2D}}(𝐱,{𝐱}^{\prime })=\frac{1}{2\pi }{K}_0(k|𝐱-{𝐱}^{\prime }|),{\mathrm{\Phi }}_{\text{3D}}(𝐱,{𝐱}^{\prime })=\frac{\exp (-k|𝐱-{𝐱}^{\prime }|)}{4\pi |𝐱-{𝐱}^{\prime }|},$$ where $K_0$ is a modified Bessel function of the second kind.

In higher dimensions the fundamental solution of the screened Poisson equation is given by the Bessel potential.

Biharmonic equation

For the Biharmonic equation, $$[-{\mathrm{\Delta }}^2]\mathrm{\Phi }(𝐱,{𝐱}^{\prime })=\delta (𝐱-{𝐱}^{\prime })$$ the biharmonic equation has the fundamental solutions $${\mathrm{\Phi }}_{\text{2D}}(𝐱,{𝐱}^{\prime })=-\frac{|𝐱-{𝐱}^{\prime }{|}^2}{8\pi }\ln |𝐱-{𝐱}^{\prime }|,{\mathrm{\Phi }}_{\text{3D}}(𝐱,{𝐱}^{\prime })=\frac{|𝐱-{𝐱}^{\prime }|}{8\pi }.$$

Signal processing

Template:Main In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter.

See also

• Green's function
• Impulse response
• Parametrix

References

• Template:Springer
• For adjustment to Green's function on the boundary see Shijue Wu notes.