Boundary element methods with applications to nonlinear by Goong Chen; Jianxin Zhou

By Goong Chen; Jianxin Zhou

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Example text

1) to denote a mixed partial derivative of f of order |α | ≡ α1 + α2 + · · · + αN . 1), we actually want to take the derivatives of f in the sense of distributions (see Chapter 3). 1) as the classical derivative: the one about which we have learned in advanced calculus. Let 1 The f k,p p ∞, and k ∈ Z+ . norm of f is defined by ⎧ 1/p ⎪ ⎪ ⎪ |Dα f | p dx , 1 p < ∞, ⎨ ∑ |α | k Ω f k,p = ⎪ α ⎪ ⎪ p = ∞, ⎩ ess. 2) x∈Ω |α | k if the RHS is finite. 2). 2) are Lebesgue integrals (as are those throughout the rest of the book), but for all practical purposes in this book they may be regarded as Riemann integrals.

V) For engineering applications, an important motivation for using BEM is its amenability for “singular geometry” such as cracks. The formulation of BIE from given boundary value problems depends on the availability of an explicit fundamental solution E(x, ξ ). For linear PDE with variable coefficients or for nonlinear PDE, such fundamental solutions are usually not available in explicit form. Therefore, BEM are in general not applicable. Research is under way to partially overcome some of these difficulties by coupling the BEM with various iteration methods.

The Dirac delta function δ is now well defined as a distribution because δ , φ ≡ φ (0) is a continuous linear functional on D(RN ). We may add distributions or multiply them by C∞ functions to form new distributions. However, the product of two distributions is not well defined in general. Every L1loc (RN ) function f defines a distribution via f,φ ≡ RN f (x)φ (x) dx, ∀ φ ∈ D. 2) for a continuous linear functional T even when T is not an L1loc function. Distributions such as L1loc (RN ) functions can not be assigned values at isolated points.

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