Author | Rektorys, Karel. author |
---|---|
Title | Variational Methods in Mathematics, Science and Engineering [electronic resource] / by Karel Rektorys |
Imprint | Dordrecht : Springer Netherlands, 1977 |
Connect to | http://dx.doi.org/10.1007/978-94-011-6450-4 |
Descript | 571 p. online resource |
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 -- 5. Orthogonal Systems in L2(G) -- 6. Hilbert Space -- 7. Some Remarks to the Preceding Chapters. Normed Space, Banach Space -- 8. Operators and Functionals, especially in Hilbert Spaces -- 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 Poincarรฉ Inequality -- 19. Boundary Value Problems in Ordinary Differential Equations -- 20. Problem of the Choice of a Base -- 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 Poincarรฉ Inequalities -- 31. Elliptic Differential Operators of Order 2k. Weak Solutions of Elliptic Equations -- 32. The Formulation of Boundary Value Problems -- 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 -- 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 -- 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 -- 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 -- 47. Concluding Remarks, Perspectives of the Presented Theory -- Table for the Construction of Most Current Functionals and of Systems of Ritz Equations -- References