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Foundations and Applications of Variational and Perturbation Methods
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Authors: S. Raj Vatsya (Senior Research Officer, National Research Council of Canada) 
Book Description:
Variational and perturbation methods, together with their hybrids, are the most widely used techniques employed by applied mathematicians, physical scientists and engineers. This book presents the topic as a unified and coherent discipline encompassing the problem specific procedures. Usable results are deduced, rigorously increasing their clarity, reliability and scope of their applications. The concepts are developed from the premise assuming that the background of the reader is one expected of an advanced undergraduate student in scientific and applied mathematics discipline and is presumed to increase the accessibility and appeal to students and researchers in a broad range of areas. In this text, material scattered in literature is collected together and a number of results available so far only in the journal papers and specialized monographs included. The concepts and their contents are illustrated with comments and examples to facilitate their comprehension. The results and techniques to exploit them are illustrated by a copious use of worked out examples from a broad range of disciplines, constituting about half of the volume. The examples are selected to illustrate the applications of the results, each to a class of problems. A discerning reader can use suitable results and techniques illustrated in this text to solve a specific problem encountered. The applications part is aimed at filling the void between the foundational and the computational literature.

Table of Contents:
Preface
I. Foundations

1. Integration and vector spaces
1.i. Preliminaries
1.ii. Integration
1.ii.1. Basic Concepts
1.ii.2. Integration over trajectories
1.iii. Vector spaces

2. Operators in vector spaces
2.i. Operators in Banach Spaces
2.ii. Operators in Hilbert spaces
2.iii. Forms in Hilbert spaces
2.iv. Integral transforms
2.v. Differential operators

3. Variational methods
3.i. Formulation
3.ii. Convergence
3.ii.1. Basic results
3.ii.2. Compact operators
3.ii.3. Bounded operators
3.ii.4. Semi-bounded operators
3.ii.5. Unbounded operators
3.iii. Padé approximants
3.iii.1. Formulation
3.iii.2. Representations
3.iii.3. Convergence and applications
3.iv. Monotonic convergence
3.iv.1. Diagonal forms
3.iv.2. Eigenvalues

4. Perturbation methods
4.i. Perturbed operator
4.ii. Spectral perturbation
4.ii.1. Resolvent and point spectrum
4.ii.2. Continuous spectrum
4.iii. Spectral differentiation
4.iv. Iteration

II. Applications

5. Matrices
5.i. Tridiagonal matrices
5.ii. Structured matrices
5.iii. Conjugate residual-like methods

6. Atomic systems
6.i. Preliminaries
6.ii. Eigenvalues and critical points
6.ii.1. Helium atom
6.ii.2. Short range potentials
6.iii. Scattering
6.iii.1. Formulation
6.iii.2. Born series
6.iii.3. Schwinger’s method
6.iii.4. Hulthén-Kohn methods
6.iii.5. Rotated Hamiltonians

7. Supplementary examples
7.i. Ray tomography
7.ii. Maxwell’s equations
7.iii. Positivity lemma for the elliptic operators
7.iii.1. Basic results
7.iii.2. Applications
7.iv. Transport and propagation
7.iv.1. Reaction-diffusion
7.iv.2. Heat transfer
7.iv.3. Radiative transfer
7.iv.4. Turbulent diffusion
7.iv.5. Optical beam propagation
7.v. Quantum theory

References

Index

   Binding: Hardcover
   Pub. Date: 2009
   Pages: 7 x 10, 337 pp.
   ISBN: 978-1-60692-591-1
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Foundations and Applications of Variational and Perturbation Methods