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Universal Publishers, 2007. — 256 p. — ISBN-10: 15811297424; ISBN-13: 978-1581129748.This is a course in boundary element methods for the absolute beginners. Basic concepts are carefully explained through the use of progressively more complicated boundary value problems in engineering and physical sciences. The readers are assumed to have prior basic knowledge of vector calculus (covering topics such as line, surface and volume integrals and the various integral theorems), ordinary and partial differential equations, complex variables, and computer programming.**Contents with Author's remarks:**

**Two-dimensional Laplace's Equation**

*Essential for a better of understanding of all other chapters.*

This chapter introduces the boundary element method through solving a relatively simple boundary value problem governed by the two-dimensional Laplace's equation. A derivation of the boundary integral equation needed for solving the boundary value problem is given. The boundary integral equation is discretized using constant elements. That is, the boundary of the solution domain is approximated using straight line segments and the required solution and its normal derivative on the boundary are assumed to be constants over each line segment.

**Discontinuous Linear Elements**

*Best read after going through Chapter 1.*

In Chapter 1, the boundary integral equation for the two-dimensional Laplace's equation is discretized using constant elements. This chapter shows how the approximations over the line segments can be improved through the use of discontinuous linear elements. It may be necessary to use higher order elements when there is a need to obtain more accurate results using fewer elements.

**Two-dimensional Helmholtz Type Equation**

*Best read after going through Chapter 1. Sections 3.2 and 3.3 may be read independently of each other.*

In the first part of this chapter, the analysis in Chapter 1 is extended to the two-dimensional homogeneous Helmholtz equation. The fundamental solution of the Helmholtz equation is given by a special function in the form of a zeroth order Bessel function of the second kind. With this fundamental solution, a boundary integral solution is obtained and discretized in order to solve boundary value problems governed by the Helmholtz equation. A generalized version of the Helmholtz equation with variable coefficients is considered in the second part of this chapter. For the generalized equation, the fundamental solution may be difficult (if not impossible) to obtain in analytical form. Hence, it may not be possible to derive a boundary integral solution for developing a boundary element procedure. An approach that may be used to overcome this difficulty is to "borrow" the fundamental solution of the Laplace's equation (in Chapter 1) to obtain an integral solution for the generalized Helmholtz equation. The integral solution contains not only the usual boundary integral but also a double integral over the entire solution domain, however. The so called dual-reciprocity method may be applied to convert the double integral approximately into a line integral. This gives rise to the dual-reciprocity boundary element method which allows the boundary element approach to be used for solving a wider range of engineering problems.

**Two-dimensional Diffusion Equation**

*Best read after going through Chapters 1, 2 and Section 3.3 of Chapter 3.*

This chapter shows how the dual-reciprocity boundary element approach in Chapter 3 can be extended to solve numerically initial-boundary value problems governed by the two-dimensional diffusion equation. The fundamental solution for the Laplace's equation is applied to obtain an integro-differential formulation for the diffusion equation. For a more accurate spatial approximation, discontinuous linear elements are used to discretize the boundary integral in the integro-differential formulation. This gives rise to a system of linear algebraic-differential equations containing unknown functions of time. The time derivatives of the unknown functions are approximated using a finite-difference formula. A time-stepping dual-reciprocity boundary element method is thus derived for the numerical solution of the diffusion equation.

*Green's Functions for Potential Problems*

*Best read after going through Chapter 1.*

In this chapter, the basic idea of using Green's functions in boundary element formulations is explained in the context of two-dimensional potential problems. Special Green's functions for a half plane, an infinitely long strip and the region exterior to a circle, which satisfy certain boundary conditions, are given with examples of applications.

**Three-dimensional Problems**

*Best read after going through Chapters 1 and 3.*

This chapter shows how the analyses and boundary element procedures in Chapters 1 and 3 for Laplace's and Helmholtz type equations can be extended to include three-dimensional problems.

This chapter introduces the boundary element method through solving a relatively simple boundary value problem governed by the two-dimensional Laplace's equation. A derivation of the boundary integral equation needed for solving the boundary value problem is given. The boundary integral equation is discretized using constant elements. That is, the boundary of the solution domain is approximated using straight line segments and the required solution and its normal derivative on the boundary are assumed to be constants over each line segment.

In Chapter 1, the boundary integral equation for the two-dimensional Laplace's equation is discretized using constant elements. This chapter shows how the approximations over the line segments can be improved through the use of discontinuous linear elements. It may be necessary to use higher order elements when there is a need to obtain more accurate results using fewer elements.

In the first part of this chapter, the analysis in Chapter 1 is extended to the two-dimensional homogeneous Helmholtz equation. The fundamental solution of the Helmholtz equation is given by a special function in the form of a zeroth order Bessel function of the second kind. With this fundamental solution, a boundary integral solution is obtained and discretized in order to solve boundary value problems governed by the Helmholtz equation. A generalized version of the Helmholtz equation with variable coefficients is considered in the second part of this chapter. For the generalized equation, the fundamental solution may be difficult (if not impossible) to obtain in analytical form. Hence, it may not be possible to derive a boundary integral solution for developing a boundary element procedure. An approach that may be used to overcome this difficulty is to "borrow" the fundamental solution of the Laplace's equation (in Chapter 1) to obtain an integral solution for the generalized Helmholtz equation. The integral solution contains not only the usual boundary integral but also a double integral over the entire solution domain, however. The so called dual-reciprocity method may be applied to convert the double integral approximately into a line integral. This gives rise to the dual-reciprocity boundary element method which allows the boundary element approach to be used for solving a wider range of engineering problems.

This chapter shows how the dual-reciprocity boundary element approach in Chapter 3 can be extended to solve numerically initial-boundary value problems governed by the two-dimensional diffusion equation. The fundamental solution for the Laplace's equation is applied to obtain an integro-differential formulation for the diffusion equation. For a more accurate spatial approximation, discontinuous linear elements are used to discretize the boundary integral in the integro-differential formulation. This gives rise to a system of linear algebraic-differential equations containing unknown functions of time. The time derivatives of the unknown functions are approximated using a finite-difference formula. A time-stepping dual-reciprocity boundary element method is thus derived for the numerical solution of the diffusion equation.

In this chapter, the basic idea of using Green's functions in boundary element formulations is explained in the context of two-dimensional potential problems. Special Green's functions for a half plane, an infinitely long strip and the region exterior to a circle, which satisfy certain boundary conditions, are given with examples of applications.

This chapter shows how the analyses and boundary element procedures in Chapters 1 and 3 for Laplace's and Helmholtz type equations can be extended to include three-dimensional problems.

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