Variational methods in mathematics, science, and engineering

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. 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