|
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions. Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions. Weak solutions are important because a great many differential equations encountered in modelling real world phenomena do not admit sufficiently smooth solutions and then the only way of solving such equations is using the weak formulation. Even in situations where an equation does have differentiable solutions, it is often convenient to first prove the existence of weak solutions and only later show that those solutions are in fact smooth enough. ==A concrete example== As an illustration of the concept, consider the first-order wave equation : (see partial derivative for the notation) where ''u'' = ''u''(''t'', ''x'') is a function of two real variables. Assume that ''u'' is continuously differentiable on the Euclidean space R2, multiply this equation (1) by a smooth function of compact support, and integrate. One obtains : Using Fubini's theorem which allows one to interchange the order of integration, as well as integration by parts (in ''t'' for the first term and in ''x'' for the second term) this equation becomes : (Notice that while the integrals go from −∞ to ∞, the integrals are essentially over a finite box because has compact support, and it is this observation which also allows for integration by parts without the introduction of boundary terms.) We have shown that equation (1) implies equation (2) as long as ''u'' is continuously differentiable. The key to the concept of weak solution is that there exist functions ''u'' which satisfy equation (2) for any , and such ''u'' may not be differentiable and thus, they do not satisfy equation (1). A simple example of such function is ''u''(''t'', ''x'') = |''t'' − ''x''| for all ''t'' and ''x''. (That ''u'' defined in this way satisfies equation (2) is easy enough to check, one needs to integrate separately on the regions above and below the line ''x'' = ''t'' and use integration by parts.) A solution ''u'' of equation (2) is called a weak solution of equation (1). 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 solution」の詳細全文を読む スポンサード リンク
|