A.A. Samarskii's Books

Difference Schemes with Operator Factors

By: A.A. Samarskii,P.P. Matus,P.N. Vabishchevich

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Difference Schemes with Operator Factors

By: A.A. Samarskii,P.P. Matus,P.N. Vabishchevich

Two-and three-level difference schemes for discretisation in time, in conjunction with finite difference or finite element approximations with respect to the space variables, are often used to solve numerically non stationary problems of mathematical physics. In the theoretical analysis of difference schemes our basic attention is paid to the problem of sta bility of a difference solution (or well posedness of a difference scheme) with respect to small perturbations of the initial conditions and the right hand side. The theory of stability of difference schemes develops in various di rections. The most important results on this subject can be found in the book by A.A. Samarskii and A.V. Goolin [Samarskii and Goolin, 1973]. The survey papers of V. Thomee [Thomee, 1969, Thomee, 1990], A.V. Goolin and A.A. Samarskii [Goolin and Samarskii, 1976], E. Tad more [Tadmor, 1987] should also be mentioned here. The stability theory is a basis for the analysis of the convergence of an approximative solu tion to the exact solution, provided that the mesh width tends to zero. In this case the required estimate for the truncation error follows from consideration of the corresponding problem for it and from a priori es timates of stability with respect to the initial data and the right hand side. Putting it briefly, this means the known result that consistency and stability imply convergence.

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390

Publisher

Springer Science & Business Media

Published Date

ISBN 10

9401598746

ISBN 13

9789401598743

A Collection of Problems on Mathematical Physics

International Series of Monographs in Pure and Applied Mathematics

By: B. M. Budak,A. A. Samarskii,A. N. Tikhonov

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A Collection of Problems on Mathematical Physics

International Series of Monographs in Pure and Applied Mathematics

By: B. M. Budak,A. A. Samarskii,A. N. Tikhonov

A Collection of Problems on Mathematical Physics is a translation from the Russian and deals with problems and equations of mathematical physics. The book contains problems and solutions. The book discusses problems on the derivation of equations and boundary condition. These Problems are arranged on the type and reduction to canonical form of equations in two or more independent variables. The equations of hyperbolic type concerns derive from problems on vibrations of continuous media and on electromagnetic oscillations. The book considers the statement and solutions of boundary value problems pertaining to equations of parabolic types when the physical processes are described by functions of two, three or four independent variables such as spatial coordinates or time. The book then discusses dynamic problems pertaining to the mechanics of continuous media and problems on electrodynamics. The text also discusses hyperbolic and elliptic types of equations. The book is intended for students in advanced mathematics and physics, as well as, for engineers and workers in research institutions.

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783

Publisher

Elsevier

Published Date

ISBN 10

1483184862

ISBN 13

9781483184869

Equations of Mathematical Physics

By: A. N. Tikhonov,A. A. Samarskii

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Equations of Mathematical Physics

By: A. N. Tikhonov,A. A. Samarskii

Mathematical physics plays an important role in the study of many physical processes — hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced undergraduate- or graduate-level text considers only those problems leading to partial differential equations. Contents: I. Classification of Partial Differential Equations II. Evaluations of the Hyperbolic Type III. Equations of the Parabolic Type IV. Equations of Elliptic Type V. Wave Propagation in Space VI. Heat Conduction in Space VII. Equations of Elliptic Type (Continuation) The authors — two well-known Russian mathematicians — have focused on typical physical processes and the principal types of equations dealing with them. Special attention is paid throughout to mathematical formulation, rigorous solutions, and physical interpretation of the results obtained. Carefully chosen problems designed to promote technical skills are contained in each chapter, along with extremely useful appendixes that supply applications of solution methods described in the main text. At the end of the book, a helpful supplement discusses special functions, including spherical and cylindrical functions.

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802

Publisher

Courier Corporation

Published Date

ISBN 10

0486173364

ISBN 13

9780486173368

Numerical Methods for Solving Inverse Problems of Mathematical Physics

By: A. A. Samarskii,Petr N. Vabishchevich

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Numerical Methods for Solving Inverse Problems of Mathematical Physics

By: A. A. Samarskii,Petr N. Vabishchevich

The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in applied mathematics, computational mathematics, and mathematical modelling.

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453

Publisher

Walter de Gruyter

Published Date

ISBN 10

3110205793

ISBN 13

9783110205794

Numerical Methods for Grid Equations

Volume I Direct Methods

By: A.A. Samarskij,E.S. Nikolaev

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Numerical Methods for Grid Equations

Volume I Direct Methods

By: A.A. Samarskij,E.S. Nikolaev

The finite-difference solution of mathematical-physics differential equations is carried out in two stages: 1) the writing of the difference scheme (a differ ence approximation to the differential equation on a grid), 2) the computer solution of the difference equations, which are written in the form of a high order system of linear algebraic equations of special form (ill-conditioned, band-structured). Application of general linear algebra methods is not always appropriate for such systems because of the need to store a large volume of information, as well as because of the large amount of work required by these methods. For the solution of difference equations, special methods have been developed which, in one way or another, take into account special features of the problem, and which allow the solution to be found using less work than via the general methods. This work is an extension of the book Difference M ethod3 for the Solution of Elliptic Equation3 by A. A. Samarskii and V. B. Andreev which considered a whole set of questions connected with difference approximations, the con struction of difference operators, and estimation of the ~onvergence rate of difference schemes for typical elliptic boundary-value problems. Here we consider only solution methods for difference equations. The book in fact consists of two volumes.

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273

Publisher

Birkhäuser

Published Date

ISBN 10

3034892721

ISBN 13

9783034892728

Blow-Up in Quasilinear Parabolic Equations

By: A. A. Samarskii,Victor a. Galaktionov,Sergey p. Kurdyumov,A. P. Mikhailov

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Blow-Up in Quasilinear Parabolic Equations

By: A. A. Samarskii,Victor a. Galaktionov,Sergey p. Kurdyumov,A. P. Mikhailov

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

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561

Publisher

Walter de Gruyter

Published Date

ISBN 10

3110889862

ISBN 13

9783110889864

Blow-Up in Quasilinear Parabolic Equations

By: A. A. Samarskii,Victor a. Galaktionov,Sergey p. Kurdyumov,A. P. Mikhailov

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Blow-Up in Quasilinear Parabolic Equations

By: A. A. Samarskii,Victor a. Galaktionov,Sergey p. Kurdyumov,A. P. Mikhailov

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561

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Walter de Gruyter

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ISBN 10

3110889862

ISBN 13

9783110889864

Qualitative Methods in Nonlinear Dynamics

Novel Approaches to Liapunov's Matrix Functions

By: A.A. Martynyuk

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Qualitative Methods in Nonlinear Dynamics

Novel Approaches to Liapunov's Matrix Functions

By: A.A. Martynyuk

Presents new approaches to qualitative analysis of continuous, discreteptime, and impulsive nonlinear systems via Liapunov matrix-valued functions that introduce more effective tests for solving problems of estimating the domains of asymptotic stability.

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320

Publisher

CRC Press

Published Date

ISBN 10

1482294745

ISBN 13

9781482294743

Theory and Applications of Fractional Differential Equations

By: A.A. Kilbas,H. M. Srivastava,J.J. Trujillo

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Theory and Applications of Fractional Differential Equations

By: A.A. Kilbas,H. M. Srivastava,J.J. Trujillo

This work aims to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy Type and Cauchy problems involving nonlinear ordinary fractional differential equations.

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550

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Elsevier

Published Date

ISBN 10

0444518320

ISBN 13

9780444518323

Applied Integral Transforms

By: M. Ya. Antimirov,A. A. Kolyshkin,Remi Vaillancourt

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Applied Integral Transforms

By: M. Ya. Antimirov,A. A. Kolyshkin,Remi Vaillancourt

This book constructs the kernels of integral transforms by solving the generalized Sturm-Liouville problems associated with the partial differential equations at hand. In the first part of the book, the authors construct the kernels and use them to solve elementary problems of mathematical physics. This part requires little mathematical background and provides an introduction to the subject of integral transforms as it proceeds mainly by examples and includes a variety of exercises. In the second part of the book, the method of integral transforms is used to solve modern applied problems in convective stability, temperature fields in oil strata, and eddy-current testing. The choice of topics reflects the authors' research experience and involvement in industrial applications. The first part of the book is accessible to undergraduates, while the second part is aimed at graduate students and researchers. Because of the applications, the book will interest engineers (especially petroleum engineers) and physicists.

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288

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821843141

ISBN 13

9780821843147

Advances in Nonlinear Dynamics

By: S. Sivasundaram,A.A. Martynyuk

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Advances in Nonlinear Dynamics

By: S. Sivasundaram,A.A. Martynyuk

Dedicated to Professor S. Leela in recognition of her significant contribution to the field of nonlinear dynamics and differential equations, this text consists of 38 papers contributed by experts from 15 countries, together with a survey of Professor Leela's work. The first group of papers examines stability, the second process controls, and the third section contains papers on various topics, including solutions for new classes of systems of equations and boundary problems, and proofs of basic theorems. Many of the featured problems are associated with the ideas and methods proposed and developed by Professor Leela.

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406

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CRC Press

Published Date

ISBN 10

9056990306

ISBN 13

9789056990305

Applied Methods of the Theory of Random Functions

By: A. A. Sveshnikov

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Applied Methods of the Theory of Random Functions

By: A. A. Sveshnikov

International Series of Monographs in Pure and Applied Mathematics, Volume 89: Applied Methods of the Theory of Random Functions presents methods of random functions analysis with their applications in various branches of technology, such as in the theory of ships, automatic regulation and control, and radio engineering. This book discusses the general properties of random functions, spectral theory of stationary random functions, and determination of optimal dynamical systems. The experimental methods for the determination of characteristics of random functions, method of envelopes, and some supplementary problems of the theory of random functions are also deliberated. This publication is intended for engineers and scientists who use the methods of the theory of probability in various branches of technology.

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332

Publisher

Elsevier

Published Date

ISBN 10

1483222632

ISBN 13

9781483222639

G-Convergence and Homogenization of Nonlinear Partial Differential Operators

By: A.A. Pankov

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G-Convergence and Homogenization of Nonlinear Partial Differential Operators

By: A.A. Pankov

Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space.

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