(BQ) Part 2 book "Numerical analysis" has contents: Boundary value problems, partial differential equations, random numbers and applications, trigonometric interpolation and the FFT, compression, optimization, eigenvalues and singular values. | C H A P T E R 7 Boundary Value Problems Underground and undersea pipelines must be designed to withstand pressure from the outside environment. The deeper the pipe, the more expensive a failure due to collapse will be. The oil pipelines connecting North Sea platforms to the coast lie at a 70-meter depth. The increasing importance of natural gas, and the danger and expense of transportation by ship, may lead to the construction of intercontinental gas pipelines. Mid-Atlantic depths exceed 5 kilometers, where the hydrostatic pressure of 7000 psi will require C innovation in pipe materials and construction to avoid buckling. The theory of pipe buckling is central to a wide array of applications, from architectural supports to coronary stents. Numerical models of buckling are valuable when direct experimentation is expensive and difficult. Reality Check 7 on page 355 represents a cross-sectional slice of a pipe as a circular ring and examines when and how buckling occurs. hapter 6 described methods for calculating the solution to an initial value problem (IVP), a differential equation together with initial data, specified at the left end of the solution interval. The methods we proposed were all “marching’’ techniques—the approximate solution began at the left end and progressed forward in the independent variable t. An equally important set of problems arises when a differential equation is presented along with boundary data, specified at both ends of the solution interval. Chapter 7 describes methods for approximating solutions of a boundary value problem (BVP). The methods are of three types. First, shooting methods are presented, a combination of the IVP solvers from Chapter 6 and equation solvers from Chapter 1. Then, finite difference methods are explored, which convert the differential equation and boundary conditions into a system of linear or nonlinear equations to be solved. The final section is focused on collocation methods and the Finite Element Method, .
