By Goong Chen; Jianxin Zhou
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Additional resources for Boundary element methods with applications to nonlinear problems
9) where θ = θ (x) depends on x and 0 < θ < 1, and ψ (x) ≡ φ (θ x). Thus φ (ε ) φ (0) = λ + ε 1−λ φ (θ ε ) ελ ε and ε 1−λ φ (θ ε ) −→ 0 as ε ↓ 0. 8) becomes −λ (x+ ) , φ = lim ε ↓0 φ (0) −λ ελ ∞ ε φ (x) dx . 9) is C∞ everywhere except perhaps at x = 0. At x = 0, we can easily define ψ (0) so that ψ (x) is continuous at x = 0. Let R > 0 be sufficiently large that φ (x) = 0 for x > R. 9) we have lim λ ε ↓0 R ε R φ (0) + xψ (x) φ (x) dx = lim λ dx ε ↓0 xλ +1 xλ +1 ε R ψ (x) φ (0) φ (0) = lim − + λ dx .
And residues δ (k−1) (x)/(k − 1)! at λ = k. 19), but without Re λ being restricted to (0, 1): λ −λ |x|−λ = x− + + x− , λ −λ |x|−λ sgn x = x− + − x− . , pseudofunctions). 29) above that at λ = k both x− + and x− have poles, with respective residues (−1)(k−1) (k−1) 1 δ (x) and δ (k−1) (x). (k − 1)! (k − 1)! Thus |x|−λ has poles only at λ = 1, 3, 5, . , with residues 2 δ (2m) (x) at λ = 2m + 1, m ∈ Z+ . (2m)! At λ = 2m, m ∈ Z+ , the distribution |x|−λ is well defined, and for these values of λ we shall naturally write x−2m instead of |x|−2m .
10 Boundary Element Methods with Applications to Nonlinear Problems (Q8) What are effective ways to discretize the BIE and to obtain numerical solutions? (Q9) How can we utilize the BIE and BEM to solve nonlinear BVPs? We will try to answer some of these questions in this book. 23). Consider, respectively, the Dirichlet and Neumann BVP on a two-dimensional bounded smooth open domain Ω: Δw(x) = 0 (DBVP) on Ω, w(x) = g1 (x), x ∈ ∂ Ω, ⎧ ⎨ Δw(x) = 0 (NBVP) ⎩ ∂ w(x) = g2 (x), ∂n We discuss them separately below.
Boundary element methods with applications to nonlinear problems by Goong Chen; Jianxin Zhou