Weak solution
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.Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of sufficiently smooth solutions, and the only way of solving such equations is using the weak formulation.As an illustration of the concept, consider the first-order wave equation: where u = u(t, x) is a function of two real variables.To indirectly probe the properties of a possible solution u, one integrates it against an arbitrary smooth functionThus, assume a solution u is continuously differentiable on the Euclidean space R2, multiply the equation (1) by a test functionThe key to the concept of weak solution is that there exist functions u that satisfy equation (2) for anyThe general idea that follows from this example is that, when solving a differential equation in u, one can rewrite it using a test function, such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts toIndeed, consider a linear differential operator in an open set W in Rn: where the multi-index (α1, α2, ..., αn) varies over some finite set in Nn and the coefficientsThe differential equation P(x, ∂)u(x) = 0 can, after being multiplied by a smooth test functionwith compact support in W and integrated by parts, be written as where the differential operator Q(x, ∂) is given by the formula The number shows up because one needs α1 + α2 + ⋯ + αn integrations by parts to transfer all the partial derivatives from u toin each term of the differential equation, and each integration by parts entails a multiplication by −1.with compact support in W. The notion of weak solution based on distributions is sometimes inadequate.In the case of hyperbolic systems, the notion of weak solution based on distributions does not guarantee uniqueness, and it is necessary to supplement it with entropy conditions or some other selection criterion.