Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. 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 Lax–Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.
Weakformulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems...
continuous operator problem, such as a differential equation, commonly in a weakformulation, to a discrete problem by applying linear constraints determined by...
computer. The first step is to convert P1 and P2 into their equivalent weakformulations. If u {\displaystyle u} solves P1, then for any smooth function v...
equation show up (the new form is called the weakformulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential...
Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions...
Weak form and strong form may refer to: Weaker and stronger versions of a hypothesis, theorem or physical law Weakformulations and strong formulations...
methods, where the so-called weakened weak (W2) formulation based on the G space theory were developed. The W2 formulation offers possibilities for formulate...
condition of extremum (functional derivative equal zero) appears in a weakformulation (variational form) integrated with an arbitrary function δf. The fundamental...
consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weakformulations involve the class of Sobolev functions...
{\displaystyle X} and X ∗ {\displaystyle X^{*}} are endowed with strong topology and weak* topology respectively. There exists r > 0 {\displaystyle r>0} such that...
optimization, the fundamental theorem of linear programming states, in a weakformulation, that the maxima and minima of a linear function over a convex polygonal...
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations...
plausible: weakformulations (which hold that precaution in the face of uncertain harms is permissible) are trivial, while strong formulations (which hold...
associated a bilinear form B on the Sobolev space Hk, so that the weakformulation of the equation Lu = f is B [ u , v ] = ( f , v ) {\displaystyle B[u...
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces...
to be large relative to these bounds. Evans's book has a slightly weakerformulation, in which there is assumed to be a positive number λ which is a lower...
both lepton number and baryon number. Standard Model (mathematical formulation) weak charge Donoghue, J.F.; Golowich, E.; Holstein, B.R. (1994). Dynamics...
mathematical problem (the partial differential equation) to be cast in a weakformulation. This is typically done by multiplying the differential equation by...
equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall...
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