Tamtam Proceedings - lamsin

472 Ben Abda et al.4. Applications4.1. The Robin inverse problemLet u the solution of the problem⎧⎨ ∆u = 0, in Ω∂u⎩∂n = g, on Γ c∂u∂n + ϕu = 0 on Γ iThe inverse Robin problem is to find a function ϕ such that the solution of the previousproblem satisfies u = f, on M, where M is a part of accessible boundary Γ c ; for theexistence and uniqueness of the solution see [3, 4, 6, 7]. In this work we suppose thatM = Γ c , our objective is to approach the function ψ on Γ i . The field we simulate is givenby u(x, y) = f 2 (x, y) = exp(10y) ∗ sin(10x) + exp(3y) cos(3x) + 10(y 3 − 3yx 2 ), thenwe use both of the algorithm to approach u and ∂u∂n , respectively, on Γ i. We give in thefigure 3 the approximation of ϕ for a = b = 5, 10 and 15 and we take for the degree ofthe polynomial of approximation n = 5, 8 and 10 respectively.(11)241012500−1−20−2−4−5−3−6−10−4−8−5−10−15−60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 3. Curves of ϕ with u = f 1 and its approximation for a = b = 5, 10 and a = b = 15and n = 5, 8, 104.2. Cracks problemIn this section, we give an algorithm to determine cracks lying on a priori knowninternal line. We want to identify the crack on Γ c = Ω 1 ∩Ω 2 by considering the followingCauchy problem⎧⎪ ⎨⎪ ⎩∆u = 0, in Ω \ σu = f, on Γ c∂u∂n = g, on Γ cu = 0 on σWhere Ω and Γ c are defined in section 3; σ is a set in Γ c . For numerical tests we considerthe function f 3 = (r) 1/2 cos( θ 2), the crack is on the x-axis and is specified by the partwhere f has a jump, it coincides with σ = [0, 0.5]. By using the algorithm defined insection 2 we can complete the solution of (1) − (2) with f = f 3 and φ = ∂f3∂n on Γ c.(12)TAMTAM –Tunis– 2005