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| In [[mathematics]], a '''fundamental solution''' for a linear [[partial differential operator]] ''L'' is a formulation in the language of [[Distribution (mathematics)|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]]
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| :''LF'' = δ(''x'').
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| Here ''F'' is ''a priori'' only assumed to be a [[Schwartz distribution]].
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| 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]] [[Sides of an equation|right hand side]] — was shown by [[Bernard Malgrange]] and [[Leon Ehrenpreis]].
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| ==Example==
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| Consider the following differential equation ''Lf'' = sin(x) with
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| :<math> L=\frac{\partial^2}{\partial x^2} </math>.
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| The fundamental solutions can be obtained by solving ''LF'' = δ(''x''), explicitly,
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| :<math> \frac{\partial^2}{\partial x^2} F(x) = \delta(x) </math>.
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| Since for the [[Heaviside function]] ''H'' we have
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| :<math> \frac{\partial}{\partial x} H(x) = \delta(x) </math>.
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| there is a solution
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| :<math> \frac{\partial}{\partial x} F(x) = H(x) + C.</math>
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| Here ''C'' is an arbitrary constant introduced by the integration. For convenience, set ''C'' = − 1/2.
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| After integrating <math>\frac{\partial}{\partial x}F(x)</math> and taking the new integration constant as zero, we get
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| :<math> F(x) = x H(x) - \frac{1}{2}x = \frac{1}{2} |x| </math>
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| ==Motivation==
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| Once the fundamental solution is found, it is easy to find the desired solution of the original equation. In fact, this process is achieved by convolution.
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| Fundamental solutions also play an important role in the numerical solution of partial differential equations by the [[boundary element method]].
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| ===Application to the example===
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| Consider the operator L and the differential equation mentioned in the example.
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| :<math> \frac{\partial^2}{\partial x^2} f(x) = \sin(x) </math> | |
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| We can find the solution of the original equation by convolving the right-hand side <math>\sin(x)</math> with the fundamental solution <math>F(x) = \frac{1}{2}|x|</math>:
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| :<math> f(x) = \int_{-\infty}^{\infty} \frac{1}{2}|x - y|\sin(y)dy</math>
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| This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, <math>L^1</math> integrability) since, we know that the desired solution is <math>f(x) = -\sin x</math>, while the above integral diverges for all x. The two expressions for f are, however, equal as distributions.
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| ===An example that more clearly works===
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| :<math> \frac{\partial^2}{\partial x^2} f(x) = I(x) </math>
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| where ''I'' is the characteristic (indicator) function of the unit interval ''[0,1]''. In that case, it can be readily verified that the convolution ''I*F'' with ''F(x)=|x|/2'' is a solution, i.e., has second derivative equal to ''I''.
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| ===Proof that the convolution is a solution===
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| Denote the [[convolution]] of functions ''F'' and ''g'' as
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| :''F''*''g''.
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| Say we are trying to find the solution of
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| :''Lf'' = ''g''(''x'').
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| We want to prove that ''F''*''g'' is a solution of the previous equation, i.e. we want to prove that ''L(''F''*''g'')'' = ''g''(''x''). When applying the differential operator, ''L'', to the convolution it is known that
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| :''L''(''F''*''g'')=(''LF'')*''g'',
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| provided ''L'' has constant coefficients.
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| If ''F'' is the fundamental solution, the right side of the equation reduces to
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| :δ*''g''.
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| But since the delta function is an [[identity element]] for convolution, this is simply ''g''(''x''). Summing up,
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| :<math> L(F*g)=(LF)*g=\delta(x)*g(x)=\int_{-\infty}^{\infty} \delta (x-y) g(y) dy=g(x) </math>
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| Therefore, if ''F'' is the fundamental solution, the convolution ''F''*''g'' is one solution of ''Lf'' = ''g''(''x''). This does not mean that it is the only solution. Several solutions for different initial conditions can be found.
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| ==Fundamental solutions for some partial differential equations==
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| ===Laplace equation===
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| For the [[Laplace equation]],
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| :<math> [-\nabla^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>
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| the fundamental solutions in two and three dimensions are
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| :<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=
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| -\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')=
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| \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} </math>
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| ===Screened Poisson equation===
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| For the [[Screened Poisson equation]], where the parameter ''k'' is real and the fundamental solution a modified [[Bessel function]],
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| :<math> [-\nabla^2+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>
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| the two and three dimensional [[Helmholtz equation]]s have the fundamental solutions
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|
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| :<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=
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| \frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\quad
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| \Phi_{3D}(\mathbf{x},\mathbf{x}')=
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| \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}\exp(-k|\mathbf{x}-\mathbf{x}'|)
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| </math>
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| ===Biharmonic equation===
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| For the [[Biharmonic equation]],
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| :<math> [-\nabla^4] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>
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| the biharmonic equation has the fundamental solutions
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| :<math>\Phi_{2D}(\mathbf{x},\mathbf{x}')=
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| -\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}(\ln|\mathbf{x}-\mathbf{x}'| - 1),\quad
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| \Phi_{3D}(\mathbf{x},\mathbf{x}')=
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| \frac{|\mathbf{x}-\mathbf{x}'|}{8\pi}</math>
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| ==Signal processing==
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| {{Main|Impulse response}}
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| In [[signal processing]], the analog of the fundamental solution of a differential equation is called the [[impulse response]] of a filter.
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| ==See also==
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| * [[Green's function]]
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| * [[Impulse response]]
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| * [[Parametrix]]
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| ==References==
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| *{{Springer|id=f/f042250|title=Fundamental solution}}
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| {{DEFAULTSORT:Fundamental Solution}}
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| [[Category:Partial differential equations]]
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| [[Category:Generalized functions]]
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Greetings! I am Myrtle Shroyer. To gather cash is a thing that I'm totally addicted to. Years in the past we moved to Puerto Rico and my family members enjoys it. Hiring is her working day occupation now but she's usually needed her own business.
Stop by my web site ... std home test