Generalized 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. 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.
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
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