Weak Solution - A Concrete Example

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).