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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 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 ...
In a weak formulation, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.
The stability of solutions in holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution). More generally, the notions of viscosity sub- and supersolution are also conserved by half-relaxed limits.
Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist.
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function ... the existence of multiple solutions is not a problem, ...
In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions.
The weak maximum principle, in this setting, says that for any open precompact subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself.
In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi). Tanaka's equation is the one-dimensional stochastic differential equation