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In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives appearing in the equation 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. Euclid, detail from The School of Athens by Raphael. ...
In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
Partial plot of a function f. ...
In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability 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). This results in the somewhat surprising fact that a differential equation may have solutions which are not differentiable, and the weak formulation allows one to find such solutions.
Weak solutions are extremely important because a great many differential equations encountered in modelling real world phenomena do not admit smooth enough 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.
A concrete example As an illustration of the concept, consider the first-order wave equation The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ...
 (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 by a smooth function  of compact support, and integrate. One obtains In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
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 In mathematical analysis, Fubinis theorem, named in honor of Guido Fubini, states that if the integral being taken with respect to a product measure on the space over , then the first two integrals being iterated integrals, and the third being an integral with respect to a product measure. ...
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...
(Notice that while the integrals go from  to  , the integrals are essentially over a finite box due to the fact that has compact support, and it is this observation which also allows for integration by parts.)
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).
General case The general idea which follows from this example is that, when solving a differential equation in u, one can rewrite it using a so-called test function such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to . In this way one obtains solutions to the original equation which are not necessarily differentiable.
The approach illustrated above works for equations more general than the wave equation. Indeed, consider a linear differential operator in an open set W in Rn In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
where varies over some finite set in Nn and the coefficients  are smooth enough functions of x.
The differential equation can, after being multiplied by a test function and integrated by parts, be written as
where the differential operator is given by the formula The number shows up because one needs integrations by parts to transfer the partial derivatives from u to in each term of the differential equation, and each integration by parts entails a multiplication by − 1.
The differential operator is the formal adjoint of (see also adjoint of an operator for the concept of adjoint). In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...
In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...
Other kinds of weak solution 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. In fully nonlinear PDE such as the Hamilton-Jacobi equation, there is a very different definition of weak solution called viscosity solution. A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ...
The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ...
The Viscosity Solution concept was introduced in the early 1980s by Lions and Crandall as a generalization of the classical concept of what is meant by a solution to a partial differential equation. ...
References - L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
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