Marchuk Differential Equations and Numerical Mathematics

Selected Papers Presented to a National Conference Held in Novosibirsk, September 1978
1. Auflage 2014
ISBN: 978-1-4831-5454-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 164 Seiten, Web PDF

Differential Equations and Numerical Mathematics contains selected papers presented in a national conference held in Novosibirsk on September 1978. This book, as the conference, is organized into three sections. Section A describes the modern theory of efficient cubature formulas; embedding theorems; and problems of spectral analysis. Section B considers the theoretical questions of partial differential equations, with emphasis on hyperbolic equations and systems, formulations, and methods for nonclassical problems of mathematical physics. Section C addresses the various problems of numerical mathematics, with focus on the optimum and asymptotically optimum algorithms for solving the problems of numerical mathematics.

Autoren/Hrsg.

Marchuk, G. I.

Inhaltsverzeichnis

1;Front Cover;1
2;Differential Equations and Numerical Mathematics;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;6
6;SECTION A: Cubature Formulae and Functional Analysis;10
6.1;CHAPTER 1. On an analogue of Plancherel's theorem and on the qualitative character of the spectrum of a self-adjoint operator
;12
6.2;References;15
6.3;CHAPTER 2. Self-adjoint operators in spaces of functions of an infinite number of variables;18
6.3.1;References;15
6.4;CHAPTER 3. Multidimensional non-linear spectral boundary value problems and soliton superposition of their asymptotic solutions;28
6.4.1;1. A non-linear spectral boundary value problem of Steklov type;28
6.4.2;2. Asymptotic complex-valued solutions, concentrated in the neighbourhood of closed geodesies;29
6.4.3;3. "Non-linear superposition" of asymptotic solutions, multidimensional Dirichlet series and real-valued asymptotic solutions;30
6.4.4;4. Example;32
6.4.5;5. Problem of reflection from a boundary and finite-gap almost periodic solutions;33
6.4.6;References;35
6.5;CHAPTER 4. Réduction de la dimension dans un probléme de contrôle optimal;36
6.5.1;Introduction;36
6.5.2;1. Position du probléme;36
6.5.3;2. Enonce du résultat;37
6.5.4;3. Bornes supérieures;39
6.5.5;4. Dualité;40
6.5.6;5. Bornes inférieures;41
6.5.7;Références;42
6.6;CHAPTER 5. Embedding theorems for a class of weight spaces and applications;44
6.6.1;References;47
6.7;CHAPTER 6. Theory of multipliers in spaces of differentiable functions and applications;48
6.7.1;1. Notation;48
6.7.2;2. The description and properties of multiplier spaces;49
6.7.3;3. Traces and extensions of multipliers on Wl p;50
6.7.4;4. The space M(Mmp. Wl q);51
6.7.5;5. Diffeomorphisms, manifolds and differential operators, connected with MWip
;52
6.7.6;6. Continuity of the convolution transformation in L2 with a weight;54
6.7.7;References;55
7;SECTION B: Differential Equations;56
7.1;CHAPTER 7. On the roots of Euler polynomials;58
7.1.1;1. Introduction;58
7.1.2;2. The second asymptotic formula;59
7.1.3;3. The domain of validity of formula II;62
7.1.4;4. The magnitude of qlIj{k) and the error for large values of q;65
7.1.5;5. The first asymptotic formula;66
7.1.6;6. Estimation of .;69
7.1.7;7. The estimation of S1
;71
7.1.8;8. The estimation of S2;71
7.1.9;9. The estimation of the error of formula I;72
7.1.10;10. The estimation of the length of productive intervals for q = 0(k-1/2);74
7.1.11;References;77
7.2;CHAPTER 8. On certain mathematical problems in hydrodynamics;78
7.2.1;1. On the approximation of solenoidal vector fields;78
7.2.2;2. The second problem we should like to consider is the investigation of the decay and rise of vorticity in a moving continuous medium;81
7.2.3;References;88
7.3;CHAPTER 9. On the solvability of the Sturm–Liouville inverse problem on the entire line
;90
7.3.1;1. Solution of the inverse problem on the entire line by a spectral matrix function;90
7.3.2;2. Application to the Korteveg–de Vries equation
;93
7.3.3;References;95
7.4;CHAPTER 10. Asymptotic properties of solutions of partial differential equations;96
7.4.1;References;104
7.5;CHAPTER 11. Boundary value problems for weakly elliptic systems of differential equations;106
7.5.1;References;111
8;SECTION C: Numerical Mathematics;114
8.1;CHAPTER 12. A generalization of the finite element method for solution of hyperbolic equations;116
8.1.1;1. Introduction;116
8.1.2;2. Variational formulation;116
8.1.3;3. A finite element solution and error estimates;118
8.1.4;References;120
8.2;CHAPTER 13. An asymptotic minimization of computational costs for solving strongly elliptic boundary value problems;122
8.2.1;References;128
8.3;CHAPTER 14. On optimal algorithms for solving the problems of numerical mathematics;130
8.3.1;1. General theory;130
8.3.2;2. Optimum algorithms in the algebra of polynomials and in linear algebra;131
8.3.3;3. Optimum algorithms for approximation, solution of equations, and minimization of functions;132
8.3.4;References;133
8.4;CHAPTER 15. Game theory and optimality of iterative methods;136
8.4.1;References;141
8.5;CHAPTER 16. The block-relaxation method for solution of the Dirichlet problem;142
8.5.1;References;147
8.6;CHAPTER 17. On the asymptotic behaviour of solutions of the homogeneous transport equation;148
8.6.1;1. Introduction;148
8.6.2;2. Definitions and notations;149
8.6.3;3. Main results;150
8.6.4;4. Application to the transport equation;151
8.6.5;References;152
8.7;CHAPTER 18. The method of inner boundary conditions and its applications. A new approach to the numerical solution of boundary integral equations;154
8.7.1;References;159
8.8;CHAPTER 19. Inverse problems and energy inequalities;160
8.8.1;References;165