Everything about Weak Solution totally explained
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.
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 don't 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's 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
»
for every smooth function
φ with compact support in
W.
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 doesn't guarantee uniqueness, and it's necessary to supplement it with
entropy conditions or some other selection criterion. In fully nonlinear PDE such as the
Hamilton-Jacobi equation, there's a very different definition of weak solution called
viscosity solution.
Further Information
Get more info on 'Weak Solution'.
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