Fundamental solution

Consider the following differential equation {{math|1=Lf = sin(x)}} with

$$L\; =\; \backslash frac\{d^2\}\{d\; x^2\}\; .$$

The fundamental solutions can be obtained by solving {{math|1=LF = δ(x)}}, explicitly,

$$\backslash frac\{d^2\}\{d\; x^2\}\; F(x)\; =\; \backslash delta(x)\; \backslash ,.$$

Since for the unit step function (also known as the Heaviside function) {{mvar|H}} we have

$$\backslash frac\{d\}\{d\; x\}\; H(x)\; =\; \backslash delta(x)\; \backslash ,,$$

there is a solution

$$\backslash frac\{d\}\{d\; x\}\; F(x)\; =\; H(x)\; +\; C\; \backslash ,.$$

Here {{mvar|C}} is an arbitrary constant introduced by the integration. For convenience, set {{math|1=C = −1/2}}.

After integrating $\backslash frac\{dF\}\{dx\}$ and choosing the new integration constant as zero, one has

$$F(x)\; =\; x\; H(x)\; -\; \backslash frac\{1\}\{2\}x\; =\; \backslash frac\{1\}\{2\}\; |x|\; ~.$$

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.

Consider the operator {{mvar|L}} and the differential equation mentioned in the example,

$$\backslash frac\{d^2\}\{d\; x^2\}\; f(x)\; =\; \backslash sin(x)\; \backslash ,.$$

We can find the solution $f(x)$ of the original equation by convolution (denoted by an asterisk) of the right-hand side $\backslash sin(x)$ with the fundamental solution $F(x)\; =\; \backslash frac\{1\}\{2\}|x|$:

$$f(x)\; =\; (F\; *\; \backslash sin)(x)\; :=\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash frac\{1\}\{2\}|x\; -\; y|\backslash sin(y)\; \backslash ,\; dy\; \backslash ,.$$

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 {{math|1=f(x) = −sin(x)}}, while the above integral diverges for all {{mvar|x}}. The two expressions for {{mvar|f}} are, however, equal as distributions.

$$\backslash frac\{d^2\}\{d\; x^2\}\; f(x)\; =\; I(x)\; \backslash ,,$$

where {{mvar|I}} is the characteristic (indicator) function of the unit interval {{closed-closed|0,1}}. In that case, it can be verified that the convolution of {{math|I}} with {{math|1=F(x) = {{!}}x{{!}}/2}} is

$$(I\; *\; F)(x)\; =\; \backslash begin\{cases\}$$

\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{cases}

which is a solution, i.e., has second derivative equal to {{mvar|I}}.

Denote the convolution of functions {{mvar|F}} and {{mvar|g}} as {{math|F ∗ g}}. Say we are trying to find the solution of {{math|1=Lf = g(x)}}. We want to prove that {{math|F ∗ g}} is a solution of the previous equation, i.e. we want to prove that {{math|1=L(F ∗ g) = g}}. When applying the differential operator with constant coefficients, {{mvar|L}}, to the convolution, it is known that

$$L(F*g)\; =\; (LF)*g\; \backslash ,,$$

provided {{mvar|L}} has constant coefficients.

If {{mvar|F}} is the fundamental solution, the right side of the equation reduces to

$$\backslash delta\; *\; g~.$$

But since the delta function is an identity element for convolution, this is simply {{math|g(x)}}. Summing up,

$$L(F*g)\; =\; (LF)*g\; =\; \backslash delta(x)*g(x)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash delta\; (x-y)\; g(y)\; \backslash ,\; dy\; =\; g(x)\; \backslash ,.$$

Therefore, if {{mvar|F}} is the fundamental solution, the convolution {{math|F ∗ g}} is one solution of {{math|1=Lf = g(x)}}. This does not mean that it is the only solution. Several solutions for different initial conditions can be found.

The following can be obtained by means of Fourier transform:

For the Laplace equation,

$$[-\backslash Delta]\; \backslash Phi(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; \backslash delta(\backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'})$$

the fundamental solutions in two and three dimensions, respectively, are

$$\backslash Phi\_\backslash textrm\{2D\}(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; -\backslash frac\{1\}\{2\backslash pi\}\backslash ln|\backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'}|,\backslash qquad$$

\Phi_\textrm{3D}(\mathbf{x},\mathbf{x}') = \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} ~.

For the screened Poisson equation,

$$[-\backslash Delta+k^2]\; \backslash Phi(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; \backslash delta(\backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'}),\; \backslash quad\; k\; \backslash in\; \backslash R,$$

the fundamental solutions are

$$\backslash Phi\_\backslash textrm\{2D\}(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; \backslash frac\{1\}\{2\backslash pi\}K\_0(k|\backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'}|),\backslash qquad$$

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

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.

For the Biharmonic equation,

$$[-\backslash Delta^2]\; \backslash Phi(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; \backslash delta(\backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'})$$

the biharmonic equation has the fundamental solutions

$$\backslash Phi\_\backslash textrm\{2D\}(\backslash mathbf\{x\},\backslash mathbf\{x\}\text{'})\; =\; -\backslash frac$$

{{Main|Impulse response}}

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

• Green's function
• Impulse response
• Parametrix

{{reflist}}

• {{Springer|id=f/f042250|title=Fundamental solution}}
• For adjustment to Green's function on the boundary see [http://www.math.lsa.umich.edu/~sijue/greensfunction.pdf Shijue Wu notes].

{{DEFAULTSORT:Fundamental Solution}}

Category:Partial differential equations

Category:Generalized functions

Category:Schwartz distributions

Category:Fredholm theory