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# Variational Methods in Mathematics, Science and Engineering

## byKarel Rektorys

### Paperback | January 22, 2012

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The impulse which led to the writing of the present book has emerged from my many years of lecturing in special courses for selected students at the College of Civil Engineering of the Tech nical University in Prague, from experience gained as supervisor and consultant to graduate students-engineers in the field of applied mathematics, and - last but not least - from frequent consultations with technicians as well as with physicists who have asked for advice in overcoming difficulties encountered in solving theoretical problems. Even though a varied combination of problems of the most diverse nature was often in question, the problems discussed in this book stood forth as the most essential to this category of specialists. The many discussions I have had gave rise to considerations on writing a book which should fill the rather unfortunate gap in our literature. The book is designed, in the first place, for specialists in the fields of theoretical engineering and science. However, it was my aim that the book should be of interest to mathematicians as well. I have been well aware what an ungrateful task it may be to write a book of the present type, and what problems such an effort can bring: Technicians and physicists on the one side, and mathematicians on the other, are often of diametrically opposing opinions as far as books con ceived for both these categories are concerned.

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Title:Variational Methods in Mathematics, Science and EngineeringFormat:PaperbackDimensions:11.02 × 8.27 × 0.07 inPublished:January 22, 2012Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401164525

ISBN - 13:9789401164528

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Table of Contents

Preface.- Notation Frequently Used.- 1. Introduction.- I. Hilbert Space.- 2. Inner Product of Functions. Norm, Metric.- 3. The Space L2.- 4. Convergence in the Space L2(G) (Convergence in the Mean). Complete Space. Separable Space.- a) Convergence in the space L2(G).- b) Completeness.- c) Density. Separability.- 5. Orthogonal Systems in L2(G).- a) Linear dependence and independence in L2(G).- b) Orthogonal and orthonormal systems in L2(G).- c) Fourier series. Complete systems. The Schmidt orthonormalization.- d) Decomposition of L2(G) into orthogonal subspaces.- e) Some properties of the inner product.- 6. Hilbert Space.- a) Pre-Hilbert space. Hilbert space.- b) Linear dependence and independence in a Hilbert space. Orthogonal systems, Fourier series.- c) Orthogonal subspaces. Some properties of the inner product.- d) The complex Hilbert space.- 7. Some Remarks to the Preceding Chapters. Normed Space, Banach Space.- 8. Operators and Functionals, especially in Hilbert Spaces.- a) Operators in Hilbert spaces.- b) Symmetric, positive, and positive definite operators. Theorems on density.- c) Functionals. The Riesz theorem.- II. Variational Methods.- 9. Theorem on the Minimum of a Quadratic Functional and its Consequences.- 10. The Space HA.- 11. Existence of the Minimum of the Functional F in the Space HA. Generalized Solutions.- 12. The Method of Orthonormal Series. Example.- 13. The Ritz Method.- 14. The Galerkin Method.- 15. The Least Squares Method. The Courant Method.- 16. The Method of Steepest Descent. Example.- 17. Summary of Chapters 9 to 16.- III. Application of Variational Methods to the Solution of Boundary Value Problems in Ordinary and Partial Differential Equations.- 18. The Friedrichs Inequality. The Poincaré Inequality.- 19. Boundary Value Problems in Ordinary Differential Equations.- a) Second order equations.- b) Higher order equations.- 20. Problem of the Choice of a Base.- a) General principles.- b) Choice of the base for ordinary differential equations.- 21. Numerical Examples: Ordinary Differential Equations.- 22. Boundary Value Problems in Second Order Partial Differential Equations.- 23. The Biharmonic Operator. (Equations of Plates and Wall-beams.).- 24. Operators of the Mathematical Theory of Elasticity.- 25. The Choice of a Base for Boundary Value Problems in Partial Differential Equations.- 26. Numerical Examples: Partial Differential Equations.- 27. Summary of Chapters 18 to 26.- IV. Theory of Boundary Value Problems in Differential Equations Based on the Concept of a Weak Solution and on the Lax-Milgram Theorem.- 28. The Lebesgue Integral. Domains with the Lipschitz Boundary.- 29. The Space W2(k)(G).- 30. Traces of Functions from the Space W2(k)(G). The Space W?2(k)(G). The Generalized Friedrichs and Poincaré Inequalities.- 31. Elliptic Differential Operators of Order 2k. Weak Solutions of Elliptic Equations.- 32. The Formulation of Boundary Value Problems.- a) Stable and unstable boundary conditions.- b) The weak solution of a boundary value problem. Special cases.- c) Definition of the weak solution of a boundary value problem. The general case.- 33. Existence of the Weak Solution of a Boundary Value Problem. V-ellipticity. The Lax-Milgram Theorem.- 34. Application of Direct Variational Methods to the Construction of an Approximation of the Weak Solution.- a) Homogeneous boundary conditions.- b) Nonhomogeneous boundary conditions.- c) The method of least squares.- 35. The Neumann Problem for Equations of Order 2k (the Case when the Form ((v, u)) is not V-elliptic).- 36. Summary and Some Comments to Chapters 28 to 35.- V. The Eigenvalue Problem.- 37. Introduction.- 38. Completely Continuous Operators.- 39. The Eigenvalue Problem for Differential Operators.- 40. The Ritz Method in the Eigenvalue Problem.- a) The Ritz method.- b) The problem of the error estimate.- 41. Numerical Examples.- VI. Some Special Methods. Regularity of the Weak solution.- 42. The Finite Element Method.- 43. The Method of Least Squares on the Boundary for the Biharmonic Equation (for the Problem of Wall-beams). The Trefftz Method of the Solution of the Dirichlet Problem for the Laplace Equation.- a) First boundary value problem for the biharmonic equation (the problem of wall-beams).- b) The idea of the method of least squares on the boundary.- c) Convergence of the method.- d) The Trefftz method.- 44. The Method of Orthogonal Projections.- 45. Application of the Ritz Method to the Solution of Parabolic Boundary Value Problems.- 46. Regularity of the Weak Solution, Fulfilment of the Given Equation and of the Boundary Conditions in the Classical Sense. Existence of the Function w ? W2(k)(G) satisfying the Given Boundary Conditions.- a) Smoothness of the weak solution.- b) Existence of the function w ? W2(k)(G) satisfying the given boundary conditions.- 47. Concluding Remarks, Perspectives of the Presented Theory.- Table for the Construction of Most Current Functionals and of Systems of Ritz Equations.- References.