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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. In terms of the Dirac delta "function" , a fundamental solution is the 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. ==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 = − 1/2. After integrating and choosing the new integration constant as zero, one has : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「fundamental solution」の詳細全文を読む スポンサード リンク
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