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Weak formulations are an important tool for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". This is equivalent to formulating the problem to require a solution in the sense of a distribution. We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem. ==General concept== Let be a Banach space. We want to find the solution of the equation :, where and , with being the dual of . Calculus of variations tells us that this is equivalent to finding such that for all holds: :. Here, we call a test vector or test function. We bring this into the generic form of a weak formulation, namely, find such that : by defining the bilinear form : Since this is very abstract, let us follow this by some examples. 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「Weak formulation」の詳細全文を読む スポンサード リンク
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