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 The fundamental solutions can be obtained by solving , explicitly, Since for the Heaviside function we have there is a solution Here is an arbitrary constant introduced by the integration. For convenience, set . After integrating and choosing the new integration constant as zero, one has 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, We can find the solution of the original equation by convolution (denoted by an asterisk) of the right-hand side with the fundamental solution : 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 , while the above integral diverges for all . The two expressions for are, however, equal as distributions. An example that more clearly works where is the characteristic (indicator) function of the unit interval . In that case, it can be verified that the convolution of with is 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