This book presents and explains a general, efficient, and elegant method for solving the Dirichlet, Neumann, and Robin boundary value problems for the extensional deformation of a thin plate on an elastic foundation. The solutions of these problems are obtained both analytically―by means of direct and indirect boundary integral equation methods (BIEMs)―and numerically, through the application of a boundary element technique. The text discusses the methodology for constructing a BIEM, deriving all the attending mathematical properties with full rigor. The model investigated in the book can serve as a template for the study of any linear elliptic two-dimensional problem with constant coefficients. The representation of the solution in terms of single-layer and double-layer potentials is pivotal in the development of a BIEM, which, in turn, forms the basis for the second part of the book, where approximate solutions are computed with a high degree of accuracy.

The book is intended for graduate students and researchers in the fields of boundary integral equation methods, computational mechanics and, more generally, scientists working in the areas of applied mathematics and engineering. Given its detailed presentation of the material, the book can also be used as a text in a specialized graduate course on the applications of the boundary element method to the numerical computation of solutions in a wide variety of problems.

Chapter 3 Existence of Solutions In this chapter, we describe and apply two boundary integral equation methods to solve the fundamental boundary value problems formulated in Sect. 1.2. 3.1 The Classical Indirect Method For the interior and exterior Dirichlet, Neumann, and Robin problems (D± ), (N± ), and (R± ), the solutions are sought in the form u = W +ϕ u = W −ϕ u = V +ψ u = V −ψ u = V +ϕ u = V −ϕ for for for for for for (D+ ), (D− ), (N+ ), (N− ), (R+ ), (R− ). From the properties of

160 2(1 + x1)3 − 3(1 + x1)2 x2 − 6(1 + x1)x22 + x23 3 (1 + x1)2 + x22 Hypergeometric0F1Regularized 1, 20 + 200 3(1 + x1)3 + 3(1 + x1)2 x2 − 9(1 + x1)x22 − x23 Hypergeometric0F1Regularized 1, 38 (1 + x1)2 + x22 − 16 − 20(1 + x1)3 + 3(1 + x1)2 11 + x1(2 + x1) x2 + 6(1 + x1) 11 + x1(2 + x1) x22 + − 7 + 3 x1(2 + x1) x23 + 6(1 + x1)x24 3 Hypergeometric0F1Regularized 2, 20 (1 + x1)2 + x22 − 25 3(1 + x1)3 11 + 3 x1(2 + x1) + 3(1 + x1)2 9 + x1(2 + x1) x2 − 72(1 + x1)x22 − 8 x23 − 9(1 + x1)x24 − 3 x25

. . . . . . . . . . . . 170 5.12.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.12.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.12.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.12.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Robin Problem in a Square Revisited: Piecewise Cubic Spline .

1 2 3 –100 Fig. 5.81 The components of ϕ [x[t]] (parametric form). 5.8 Dirichlet Problem in a Domain with Corners: Piecewise Quintic Spline 145 Numerical approximation. We compute ϕ˜ by the collocation method with a B-spline basis of elements bi, j . Then the approximate density is sought in the form ⎛ ⎞ ∑ ∑ c1,i, j bi, j [t] ⎠, ϕ˜ [x[t]] = ⎝ i j ∑ ∑ c2,i, j bi, j [t] (5.32) i j where the numerical coefficients cα ,i, j are determined by substituting (5.32) in (5.30). Since ϕ is

Boundary Operators 23 (ii) (K0 ) and (K∗0 ) have the same finite number of linearly independent solutions {ϕ1 , ϕ2 , · · · , ϕn } and {ψ1 , ψ2 , · · · , ψn }, respectively, and (K) and (K∗ ) are solvable if and only if ( f , ψi ) = 0 and (g, ϕi ) = 0, i = 1, 2, · · · , n. 2.11 Corollary. The Fredholm alternative holds for the pairs of equations W0 − 12 I ϕ = f , W0∗ − 12 I ψ = g, W0 + 12 I ϕ = f , W0∗ + 12 I ψ = g in the dual system (C0,α (∂ S),C0,α (∂ S)), α ∈ (0, 1), with the