User:Cesiumfrog/weak solution

estWeak solution theory is a branch of mathematics which generalises the concept of differential equations to admit some solutions which are distributions other than differentiable functions. It underpins the widely used finite element method.

Introduction
An example is the partial differential equation for the first order wave equation. The usual formulation is termed the "strong" form:

$$\partial_x u + \partial_t u = 0$$

The scalar-field solution $$u(x,t)$$ must clearly be differentiable, in order for the strong form of the differential equation to be satisfied at every point in the domain. Indeed, the most general formula for the solution to this equation is $$u = f(x-t)$$, where $$f:\mathbb R \rightarrow \mathbb R$$ is any function for which a first-order derivative exists everywhere.

In contrast, the scalar field $$u=|x-t|$$ does not completely solve the strong form of the wave equation, because there are some points where the absolute value function does not have any defined derivative (i.e. where $$x-t = 0 $$). However, this might otherwise be considered a reasonable solution (e.g. it is obtainable as the limit of a sequence of valid solutions, and it satisfies the equation almost everywhere). It may be desirable to be able to admit this as a solution, and no solution will exist otherwise under certain boundary conditions (such as if the problem presupposes that $$u(x,0) = |x| $$).

The differential equation can be expressed differently by multiplying by an arbitrary smooth test-function $$w(x,t) $$ with compact support, integrating over the domain, and using integration by parts to transfer differential operators from the solution onto the test-function (letting the boundary terms vanish because of the compact support).

$$\begin{align} 0 &= \int \left(\partial_x u + \partial_t u\right)w\ d(x,t) \qquad \forall w \\ \therefore 0 &= \int u \left(\partial_x w + \partial_t w\right)\ d(x,t) \qquad \forall w \end{align} $$

This is termed the weak formulation. Although the weak formulation is still satisfied by the solutions to the strong formulation, it is also satisfied by some additional solutions (termed "weak solutions"). These additional solutions include $$u=|x-t| $$ (as can be verified by treating the integrals of each half-plane separately).