Английская Википедия:Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator Шаблон: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" Шаблон:Math, a fundamental solution Шаблон:Mvar is a solution of the inhomogeneous equation Шаблон:Block indent Here Шаблон: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. 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 Шаблон:Math with <math display="block"> L = \frac{d^2}{d x^2} .</math>
The fundamental solutions can be obtained by solving Шаблон:Math, explicitly, <math display="block"> \frac{d^2}{d x^2} F(x) = \delta(x) \,.</math>
Since for the unit step function (also known as the Heaviside function) Шаблон:Mvar we have <math display="block"> \frac{d}{d x} H(x) = \delta(x) \,,</math> there is a solution <math display="block"> \frac{d}{d x} F(x) = H(x) + C \,.</math> Here Шаблон:Mvar is an arbitrary constant introduced by the integration. For convenience, set Шаблон:Math.
After integrating <math>\frac{dF}{dx}</math> and choosing the new integration constant as zero, one has <math display="block"> F(x) = x H(x) - \frac{1}{2}x = \frac{1}{2} |x| ~.</math>
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 Шаблон:Mvar and the differential equation mentioned in the example, <math display="block"> \frac{d^2}{d x^2} f(x) = \sin(x) \,.</math>
We can find the solution <math>f(x)</math> of the original equation by convolution (denoted by an asterisk) of the right-hand side <math>\sin(x)</math> with the fundamental solution <math display="inline">F(x) = \frac{1}{2}|x|</math>: <math display="block"> f(x) = (F * \sin)(x) := \int_{-\infty}^{\infty} \frac{1}{2}|x - y|\sin(y) \, dy \,.</math>
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 Шаблон:Math, while the above integral diverges for all Шаблон:Mvar. The two expressions for Шаблон:Mvar are, however, equal as distributions.
An example that more clearly works
<math display="block"> \frac{d^2}{d x^2} f(x) = I(x) \,,</math> where Шаблон:Mvar is the characteristic (indicator) function of the unit interval Шаблон:Closed-closed. In that case, it can be verified that the convolution of Шаблон:Math with Шаблон:Math is <math display="block">(I * F)(x) = \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} </math> which is a solution, i.e., has second derivative equal to Шаблон:Mvar.
Proof that the convolution is a solution
Denote the convolution of functions Шаблон:Mvar and Шаблон:Mvar as Шаблон:Math. Say we are trying to find the solution of Шаблон:Math. We want to prove that Шаблон:Math is a solution of the previous equation, i.e. we want to prove that Шаблон:Math. When applying the differential operator, Шаблон:Mvar, to the convolution, it is known that <math display="block">L(F*g) = (LF)*g \,,</math> provided Шаблон:Mvar has constant coefficients.
If Шаблон:Mvar is the fundamental solution, the right side of the equation reduces to <math display="block">\delta * g~.</math>
But since the delta function is an identity element for convolution, this is simply Шаблон:Math. Summing up, <math display="block"> L(F*g) = (LF)*g = \delta(x)*g(x) = \int_{-\infty}^{\infty} \delta (x-y) g(y) \, dy = g(x) \,.</math>
Therefore, if Шаблон:Mvar is the fundamental solution, the convolution Шаблон:Math is one solution of Шаблон: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, <math display="block"> [-\Delta] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math> the fundamental solutions in two and three dimensions, respectively, are <math display="block"> \Phi_\textrm{2D}(\mathbf{x},\mathbf{x}') = -\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\qquad \Phi_\textrm{3D}(\mathbf{x},\mathbf{x}') = \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} ~. </math>
Screened Poisson equation
For the screened Poisson equation, <math display="block"> [-\Delta+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}'), \quad k \in \R,</math> the fundamental solutions are <math display="block"> \Phi_\textrm{2D}(\mathbf{x},\mathbf{x}') = \frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\qquad \Phi_\textrm{3D}(\mathbf{x},\mathbf{x}') = \frac{\exp(-k|\mathbf{x}-\mathbf{x}'|)}{4\pi|\mathbf{x}-\mathbf{x}'|},</math> where <math>K_0</math> 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, <math display="block"> [-\Delta^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math> the biharmonic equation has the fundamental solutions <math display="block">\Phi_\textrm{2D}(\mathbf{x},\mathbf{x}') = -\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}\ln|\mathbf{x}-\mathbf{x}'|,\qquad \Phi_\textrm{3D}(\mathbf{x},\mathbf{x}') = \frac{|\mathbf{x}-\mathbf{x}'|}{8\pi} ~.</math>
Signal processing
Шаблон:Main In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter.
See also
References
- Шаблон:Springer
- For adjustment to Green's function on the boundary see Shijue Wu notes.
