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.
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).
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.
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
Thus, assume a solution u is continuously differentiable on the Euclidean space R2, multiply the equation (1) by a test function
The key to the concept of weak solution is that there exist functions u that satisfy equation (2) for any
The general idea that follows from this example is that, when solving a differential equation in u, one can rewrite it using a test function
, such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to
Indeed, consider a linear differential operator in an open set W in Rn: where the multi-index (α1, α2, ..., αn) varies over some finite set in Nn and the coefficients
The differential equation P(x, ∂)u(x) = 0 can, after being multiplied by a smooth test function
with compact support in W and integrated by parts, be written as where the differential operator Q(x, ∂) is given by the formula The number shows up because one needs α1 + α2 + ⋯ + αn integrations by parts to transfer all 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 Q(x, ∂) is the formal adjoint of P(x, ∂) (cf.
In summary, if the original (strong) problem was to find an |α|-times differentiable function u defined on the open set W such that (a so-called strong solution), then an integrable function u would be said to be a weak solution if for every smooth function
with compact support in W. 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.