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Concept
Weak solution
Summary
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 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. 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.
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 function of compact support, known as a test function, taking
For example, if is a smooth probability distribution concentrated near a point , the integral is approximately . Notice that while the integrals go from to , they are essentially over a finite box where is non-zero.
Thus, assume a solution u is continuously differentiable on the Euclidean space R2, multiply the equation () by a test function (smooth of compact support), and integrate:
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:
(Boundary terms vanish since is zero outside a finite box.
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