By Goong Chen; Jianxin Zhou
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Extra info for Boundary element methods with applications to nonlinear problems
And its residue at λ = k ∈ N is φ (k−1) (0)/(k − 1)!. Since φ (k−1) (0) = (−1)(k−1) δ (k−1) (x), φ , λ we may say that the functional x− + itself has simple poles at λ = k, and the residues there are (−1)k−1 (k−1) δ (x), k = 1, 2, . . (k − 1)! 1. For λ λ 0, it is easy to define x− + . 25)). 40 Boundary Element Methods with Applications to Nonlinear Problems The distribution corresponding to |x|−λ , x < 0, λ x− − = 0, x 0, λ can be treated in the same way as x− + , by making a reflection x → −x: ∞ λ x− − ,φ = = 1 0 = 0 x−λ φ (−x) dx = − xλ+ , φ (−x) x−λ φ (−x) − φ (0) + xφ (0) − · · · − + ∞ 0 ∞ 1 (−x)n−1 (n−1) φ (0) dx (n − 1)!
40) ⎡ 2η1 ⎤ ⎢ ⎥ ⎢ 2η2 ⎥ ⎢ ⎥ Z = ⎢ . ⎥, ⎢ .. 39) as (I − K) W = Z . 41) This linear system is noninvertible, since it can be easily verified that m ∑ θi j = π , i = 1, 2, . . 42) j=1 implying W 0 ≡ [1, 1, . . , 1]Tr ∈ N (I − K). 41) is solvable if and only if Z ⊥ N (I − K)∗ . 29) of the original problem (NBVP) thus its occurrence is rather natural. 41) still lacks uniqueness, as W + α W 0 is a solution for any α ∈ R, whenever W itself is. This is the case for the original (NBVP), since w + α is a solution whenever w is, for any α ∈ R.
The Dirac delta function δ (x) concentrated at the origin is a generalized function whose defining property is RN f (x)δ (x) dx = f (0) for any sufficiently smooth function f . According to this property, δ (ξ ) = F (δ )(ξ ) = RN 0 δ (x)e−2π i x,ξ dx = e−2π i 0,ξ = e = 1. (Thus the Fourier transform of the delta function is the constant 1. Taking the inverse Fourier transform of 1, F (1) = RN 1 · e2π i x,ξ d ξ , we realize that the integral does not exist in the classical sense. Nevertheless, we can define F (1) = δ (x) so that formally the Fourier inversion formula is made to hold: F (F (δ )) = δ .
Boundary element methods with applications to nonlinear problems by Goong Chen; Jianxin Zhou