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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.
Weak form and strong form may refer to: Weaker and stronger versions of a hypothesis, theorem or physical law; Weak formulations and strong formulations of differential equations in mathematics; Differing pronunciations of words depending on emphasis; see Weak and strong forms in English; Weak and strong pronouns
Weakened weak form (or W2 form) [1] is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems.
This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal ( X 2 + 1) in R [ X ] {\displaystyle \mathbb {R} [X]} do ...
In this way, the strong form represents a determining pronoun (such as in "what is that?"), while the weak form is a subordinating word (as in "I think that it's a mistake"). [32] The pronunciation of the voiced dental fricative /ð/ may vary, such as being stopped in Cameroonian English, resulting in a pronunciation of [dat]. [33]
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation [20] + + = multiply it for a test function , defined in a suitable space , and integrate both members with respect to the domain : [20] + + = Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem ...
Particular confusion was caused by the 1986 book The Anthropic Cosmological Principle by John D. Barrow and Frank Tipler, [15] which distinguished between a "weak" and "strong" anthropic principle in a way very different from Carter's, as discussed in the next section. Carter was not the first to invoke some form of the anthropic principle.
The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form. Consider the Laplace problem Δ u = f {\displaystyle \Delta u=f} . Multiply each side of the equation by a "test function" φ {\displaystyle \varphi } and integrate by parts using Green's theorem to obtain