Variational Methods in Mathematics, Science and Engineering

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.