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The dual reciprocity boundary element method. (English) Zbl 0758.65071
Southampton: Computational Mechanics Publications. London: Elsevier Applied Science. xvi, 281 p. (1992).
This interesting book puts together most of the authors’ large experience in the subject title. The dual reciprocity method (DRM) has emerged in the early eighties as a simple and alternative procedure to deal with problems in which the classical boundary integral formulations would require domain discretization for numerical implementation. The great attractiveness of the procedure is the possibility of keeping the “boundary only discretization” obtained by employing a fundamental solution to a simpler equation to treat the domain variables via series expansions and global approximating functions which are taken to the boundary through reciprocity principles.
The book covers a wide number of problems solved by boundary elements; Chapters 1 and 2 introduce the reader to Laplace’s equation and include different alternatives to accomodate domain distributed sources typical of Poisson’s equation. Chapter 3 concentrates on the DRM theory for the known non-homogeneous terms of the Poisson equation and Chapter 4 generalizes the ideas to include potential dependent domain terms. Linear and non-linear problems are discussed, comprising convective terms, the Helmholtz and Burger’s equations among others. Transient problems such as diffusion and wave propagation are left to Chapter 5 and 2-D elasticity, plate bending and transient convection-diffusion are included under the title “Other fundamental solutions” in Chapter 6.
The text is well written and self-consistent, also presenting FORTRAN computer programs to cover the ideas introduced in every Chapter, from 2 to 5.
The book is highly recommended to anyone working with state of art boundary element techniques and profits from the fact that most of the original material covered emerged from the authors’ research group.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65R20 Numerical methods for integral equations
35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35K15 Initial value problems for second-order parabolic equations
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