Weak Form Solution Sample Clauses

Weak Form Solution. Although familiar, the differential (strong) formulation of a partial differential equa- tion is often unsuitable for describing real problems. This is due to the the concept that the objective relies upon continuity in the problem, which we do not always have (for example, a traffic jam problem)[10]. A more natural formulation is therefore the so called variational one. A complete understanding of this is based on functional analysis concepts, beyond the purpose of this work. We refer to [9] for a complete description. Let u be the solution to our problem: ∂ u ∂2u ∂t − µ∂x2 = f (3.2) u(a, t) = u(b, t) = 0  (3.3) u(x, 0) = u0(x) Let u belong to a set or, more precisely, to a functional space V . The variational formulation can be formally obtained by equation (3.2) with a weighted average approach and after an integration by parts. We say that we look for u ∈ V such that ∫ b ∂u v + µ a ∂t b ∂ u ∂ v ∫ = a ∂x ∂x b ∫ fv ∀v ∈ V (3.4) a This is the basis for well posedness theory and the numerical discretization with the finite element method. For the well posedness, a classical result of functional analysis called Lax-Milgram Lemma guarantees that for a problem in the form of equation (3.4) the solution exists, is unique and depends continuously on the data, under usual assumptions on the initial condition function. The same result can be extended to a more general case: ∂ u ∂ u ∂ u ∂ µ(x) ∂x ∂t − + β(x) + σ(x)u = f (x) (3.5) ∂x ∂x with u(a, t) = φa(t) u(b, t) = φb(t) u(x, 0) = u0(x)