Variational Methods in Mathematics, Science and Engineering

Front Matter....Pages 1-9
Preface....Pages 11-14
Notation Frequently Used....Pages 15-16
Introduction....Pages 17-20
Inner Product of Functions. Norm, Metric....Pages 21-31
The Space L 2 ....Pages 32-37
Convergence in the Space L 2 ( G ) (Convergence in the Mean). Complete Space. Separable Space....Pages 38-46
Orthogonal Systems in L 2 ( G )....Pages 47-65
Hilbert Space....Pages 66-80
Some Remarks to the Preceding Chapters. Normed Space, Banach Space....Pages 81-85
Operators and Functionals, especially in Hilbert Spaces....Pages 86-112
Theorem on the Minimum of a Quadratic Functional and its Consequences....Pages 113-120
The Space H A ....Pages 121-132
Existence of the Minimum of the Functional F in the Space H A . Generalized Solutions....Pages 133-145
The Method of Orthonormal Series. Example....Pages 146-152
The Ritz Method....Pages 153-160
The Galerkin Method....Pages 161-165
The Least Squares Method. The Courant Method....Pages 166-171
The Method of Steepest Descent. Example....Pages 172-177
Summary of Chapters 9 to 16....Pages 178-185
Front Matter....Pages 187-187
The Friedrichs Inequality. The Poincaré Inequality....Pages 188-198
Front Matter....Pages 187-187
Boundary Value Problems in Ordinary Differential Equations....Pages 199-222
Problem of the Choice of a Base....Pages 223-237
Numerical Examples: Ordinary Differential Equations....Pages 238-253
Boundary Value Problems in Second Order Partial Differential Equations....Pages 254-265
The Biharmonic Operator. (Equations of Plates and Wall-beams.)....Pages 266-275
Operators of the Mathematical Theory of Elasticity....Pages 276-284
The Choice of a Base for Boundary Value Problems in Partial Differential Equations....Pages 285-292
Numerical Examples: Partial Differential Equations....Pages 293-306
Summary of Chapters 18 to 26....Pages 307-311
Front Matter....Pages 313-313
The Lebesgue Integral. Domains with the Lipschitz Boundary....Pages 314-327
Elliptic Differential Operators of Order 2 k . Weak Solutions of Elliptic Equations....Pages 328-336
The Formulation of Boundary Value Problems....Pages 337-343
Existence of the Weak Solution of a Boundary Value Problem. V-Ellipticity. The Lax-Milgram Theorem....Pages 344-354
Application of Direct Variational Methods to the Construction of an Approximation of the Weak Solution....Pages 355-382
The Neumann Problem for Equations of Order 2 k (The Case when the Form (( v , u )) is Not V -Elliptic)....Pages 383-398
Summary and Some Comments to Chapters 28 to 35....Pages 399-416
Introduction....Pages 417-433
Completely Continuous Operators....Pages 434-440
The Eigenvalue Problem for Differential Operators....Pages 441-444
The Ritz Method in the Eigenvalue Problem....Pages 445-461
Numerical Examples....Pages 462-476
The Finite Element Method....Pages 477-494
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....Pages 495-501
The Method of Orthogonal Projections....Pages 503-510
Application of the Ritz Method to the Solution of Parabolic Boundary Value Problems....Pages 511-523
Concluding Remarks, Perspectives of the Presented Theory....Pages 524-532
Table for the Construction of Most Current Functionals and of Systems of Ritz Equations....Pages 533-544
Back Matter....Pages 545-550
....Pages 551-553