. 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 provided has constant coefficients. If is the fundamental solution, the right side of the equation reduces to But since the delta function is an identity element for convolution, this is simply . Summing up, 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, the fundamental solutions in two and three dimensions, respectively, are Screened Poisson equation For the screened Poisson equation, the fundamental solutions are where 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, the biharmonic equation has the fundamental solutions Signal processing 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 For adjustment to Green's function on the boundary see Shijue Wu notes. Partial differential equations Generalized functions