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In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator. A parametrix for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good. It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it. ==Overview and informal definition== It is useful to start reviewing what a fundamental solution for a differential operator ''P''(''D'') with constant coefficients is: it is a distribution ''u'' on ℝ''n'' such that : in the weak sense, where δ is the Dirac delta distribution. In a similar way, a parametrix for a variable coefficient differential operator ''P''(''x,D'') is a distribution ''u'' such that : where ω is some C∞ function with compact support. The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable coefficient, since for such operators over appropriate domains a parametrix can be shown to exist, can be somewhat easily constructed〔By using known facts about the fundamental solution of constant coefficient differential operators.〕 and be a smooth function away from the origin. Having found the analytic expression of the parametrix, it is possible to compute the solution of the associated fairly general elliptic partial differential equation by solving an associated Fredholm integral equation: also, the structure itself of the parametrix reveals properties of the solution of the problem without even calculating it, like its smoothness〔See the entry about the regularity problem for partial differential operators.〕 and other qualitative properties 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「Parametrix」の詳細全文を読む スポンサード リンク
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