Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator 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" , a fundamental solution is a solution of the inhomogeneous equation Here 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 with formula_1 The fundamental solutions can be obtained by solving , explicitly, formula_2 Since for the Heaviside function we have formula_3 there is a solution formula_4 Here is an arbitrary constant introduced by the integration. For convenience, set . After integrating  formula_5 and choosing the new integration constant as zero, one has formula_6 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 and the differential equation mentioned in the example, formula_7 We can find the solution formula_8 of the original equation by convolution (denoted by an asterisk) of the right-hand side formula_9  with the fundamental solution formula_10: formula_11 This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, "L" integrability) since, we know that the desired solution is , while the above integral diverges for all . The two expressions for are, however, equal as distributions. An example that more clearly works. formula_12 where is the characteristic (indicator) function of the unit interval . In that case, it can be verified that the convolution of with is formula_13 which is a solution, i.e., has second derivative equal to . Proof that the convolution is a  solution. Denote the convolution of functions and as . Say we are trying to find the solution of . We want to prove that is a solution of the previous equation, i.e. we want to prove that . When applying the differential operator, , to the convolution, it is known that formula_14 provided has constant coefficients. If is the fundamental solution, the right side of the equation reduces to formula_15 But since the delta function is an identity element for convolution, this is simply . Summing up, formula_16 Therefore, if is the fundamental solution, the convolution is one solution of . 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, formula_17 the fundamental solutions in two and three dimensions, respectively, are formula_18 Screened Poisson equation. For the screened Poisson equation, formula_19 the fundamental solutions are formula_20 where formula_21 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, formula_22 the biharmonic  equation has the fundamental solutions formula_23 Signal processing. In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter.