Methods of Numerical Mathematics

Häftad, Engelska, 2011

Del 2 i serien Stochastic Modelling and Applied Probability

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Beskrivning

The present volume is an adaptation of a series of lectures on numerical mathematics which the author has been giving to students of mathematics at the Novosibirsk State University during the span of several years. In dealing with problems of applied and numerical mathematics the author sought to focus his attention on those complicated problems of mathe matical physics which, in the course of their solution, can be reduced to simpler and theoretically better developed problems allowing effective algorithmic realization on modern computers. It is usually these kinds of problems that a young practicing scientist runs into after finishing his university studies. Therefore this book is pri marily intended for the benefit of those encountering truly complicated problems of mathematical physics for the first time, who may seek help regarding rational approaches to their solution. In writing this book the author has also tried to take into account the needs of scientists and engineers who already have a solid background in practical problems but who lack a systematic knowledge in areas of numerical mathematics and its more general theoretical framework.

Produktinformation

Utgivningsdatum:2011-11-08
Mått:155 x 235 x 29 mm
Vikt:791 g
Format:Häftad
Språk:Engelska
Serie:Stochastic Modelling and Applied Probability
Antal sidor:510
Upplaga:2
Förlag:Springer-Verlag New York Inc.
ISBN:9781461381525
Översättare:A.A. Brown, A. a. Brown, A. A. Brown

Utforska kategorier

Beräkning och matematisk analys inom Naturvetenskap och teknik

Innehållsförteckning

1 Fundamentals of the Theory of Difference Schemes.- 1.1. Basic Equations and Their Adjoints.- 1.2. Approximation.- 1.3. Countable Stability.- 1.4. The Convergence Theorem.- 2 Methods of Constructing Difference Schemes for Differential Equations.- 2.1. Variational Methods in Mathematical Physics.- 2.2. The Method of Integral Identities.- 2.3. Difference Schemes for Equations with Discontinuous Coefficients Based on Variational Principles.- 2.4. Principles for the Construction of Subspaces for the Solution of One-Dimensional Problems by Variational Methods.- 2.5. Variational-Difference Schemes for Two-Dimensional Equations of Elliptic Type.- 2.6. Variational Methods for Multi-Dimensional Problems.- 2.7. The Method of Fictive Domains.- 3 Interpolation of Net Functions.- 3.1. Interpolation of Functions of One Variable.- 3.2. Interpolation of Functions of Two or More Variables.- 3.3. An r-Smooth Approximation to a Function of Several Variables.- 3.4. Elements of the General Theory of Splines.- 4 Methods for Solving Stationary Problems of Mathematical Physics.- 4.1. General Concepts of Iteration Theory.- 4.2. Some Iterative Methods and Their Optimization.- 4.3. Nonstationary Iteration Methods.- 4.4. The Splitting-Up Method.- 4.5. Iteration Methods for Systems with Singular Matrices.- 4.6. Iterative Methods for Inaccurate Input Data.- 4.7. Direct Methods for Solving Finite-Difference Systems.- 5 Methods for Solving Nonstationary Problems.- 5.1. Second-Order Approximation Difference Schemes with Time-Varying Operators.- 5.2. Nonhomogeneous Equations of the Evolution Type.- 5.3. Splitting-Up Methods for Nonstationary Problems.- 5.4. Multi-Component Splitting.- 5.5. General Approach to Component-by-Component Splitting.- 5.6. Methods of Solving Equations of the Hyperbolic Type.-6 Richardson’s Method for Increasing the Accuracy of Approximate Solutions.- 6.1 Ordinary First-Order Differential Equations.- 6.2. General Results.- 6.3. Simple Integral Equations.- 6.4. The One-Dimensional Diffusion Equation.- 6.5. Nonstationary Problems.- 6.6. Richardson’s Extrapolation for Multi-Dimensional Problems.- 7 Numerical Methods for Some Inverse Problems.- 7.1. Fundamental Definitions and Examples.- 7.2. Solution of the Inverse Evolution Problem with a Constant Operator.- 7.3. Inverse Evolution Problems with Time-Varying Operators.- 7.4. Methods of Perturbation Theory for Inverse Problems.- 7.5. Perturbation Theory for Complex Nonlinear Models.- 8 Methods of Optimization.- 8.1. Convex Programming.- 8.2. Linear Programming.- 8.3. Quadratic Programming.- 8.4. Numerical Methods in Convex Programming Problems.- 8.5. Dynamic Programming.- 8.6. Pontrjagin’s Maximum Principle.- 8.7. Extremal Problems with Constraints and Variational Inequalities.- 9 Some Problems of Mathematical Physics.- 9.1. The Poisson Equation.- 9.2. The Heat Equation.- 9.3. The Wave Equation.- 9.4. The Equation of Motion.- 9.5. The Neutron Transport Equation.- 10 A Review of the Methods of Numerical Mathematics.- 10.1. The Theory of Approximation, Stability, and Convergence of Difference Schemes.- 10.2. Numerical Methods for Problems of Mathematical Physics.- 10.3. Conditionally Well-Posed Problems.- 10.4. Numerical Methods in Linear Algebra.- 10.5. Optimization Problems in Numerical Methods.- 10.6. Optimization Methods.- 10.7. Some Trends in Numerical Mathematics.- References.- Index of Notation.