Сызықтық дербес дифференциалдық теңдеулердің негізгі шешімі

Concept in the solution of linear partial differential equations

In mathematics, a fundamental solution for a linear partial differential operator L 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" δ(x), a fundamental solution F is a solution of the inhomogeneous equation

Here F 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.

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.

where I is the characteristic (indicator) function of the unit interval In that case, it can be verified that the convolution of I with is

which is a solution, i. e., has second derivative equal to I.

In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter.