## Alexander Andreevich Samarskii's books

A A Samarskii was the author of a remarkable number of books, both advanced textbooks and monographs. Many are co-authored with one or more colleagues. Some, but not all, are Russian books with English translations. Some of these books have gone through a number of volumes and often the English translation has been made from one of the later Russian volumes. We present the books in roughly chronological order of the original edition.

**1. Equations of mathematical physics, by A N Tikhonov and A A Samarskii.**(Russian edition 1951, English edition 1963).

**1.1. Review of Russian edition by: Murray H Protter.**

*Mathematical Reviews*, MR0058827

**(15,430b).**

This book consists of three parts: (i) the theory of the equations of mathematical physics; (ii) applications to physical problems; and (iii) special functions. The first part comprises the body of the text. Each of the chapters (except the first) has an appendix which discusses applications to physical problems of the material just presented. The theory of special functions is taken up separately in a special lengthy (about 100 pages) appendix. ... This book has a wealth of applications, many to topics not usually treated. For example, two applications which the reviewer has not seen elsewhere concern the diffusion of clouds and the influence of radioactivity on the temperature of the earth's core.

**1.2. Review of English edition by: David A Conrad.**

*American Scientist*

**52**(4) (1964), 456A.

The contents of this book are a necessary part of the mathematical equipment of the physicist and engineer. The subject is restricted largely to partial differential equations of the second order, with brief mention of higher order equations arising in elasticity. The book was written as a textbook, and is there fore, more expository and requires less mathematical sophistication than Courant-Hilbert (Volume II), which covers much of the same material. The treatment here is classical, with an initial chapter on the classification of second order equations, followed by three chapters on hyperbolic, parabolic and elliptic equations, illustrated by detailed treatments of the wave, heat and Laplace equations. The remaining three chapters are devoted to wave propagation and heat conduction in three dimensions and to some further aspects of elliptic equations.

**2. A Collection of Problems on Mathematical Physics, by B M Budak, A A Samarskii and A N Tikhonov.**(Russian edition 1956, English edition 1964).

**2.1. Review of Russian edition by: Jenny E Rosenthal.**

*Mathematical Reviews*, MR0083655

**(18,740f).**

The present book is intended to supplement the usual text on partial differential equations of physics so as to enable a student (or a graduate engineer faced with a theoretical problem) to acquire the facility to formulate and solve boundary value problems of classical physics. The first part of the book is devoted to the listing of a large number of problems. The chapter divisions are made according to the type of differential equation considered, i.e., hyperbolic, parabolic, or elliptic. A brief statement is given in each chapter as to the branches of classical physics where these particular equations are likely to arise. The second part of the book gives the answers together with the methods needed to find them. The treatment of the first problems of a given classification starts with the selection of the variables and the formulation of the problem as a partial differential equation. The methods of solution discussed primarily are the separation of variables and integral representations. The amount of detail given appears to be adequate, typical problems being worked out step by step. After going through a detailed example, a student should find sufficient the indications on procedure given for similar problems.

**2.2. Review of English edition by: Ian P Grant.**

*The Mathematical Gazette*

**51**(376) (1967), 184-185.

This is the translation of a Russian book based on practical work with the partial differential equations of mathematical physics in the Physics Faculty and external section of Moscow State University. The authors' aim is "to give students the opportunity, by means of independent work, of gaining elementary technique in solving problems in the principal classes" of these equations. ... The editor of this translation suggests that a student should attempt only a few problems from each section for himself, but will have solutions of the remaining problems for reference. Those following this advice would gain the technical skill the authors wish to impart ...

**2.3. Review of English edition by: Fritz Steinhardt.**

*Amer. Math. Monthly*

**75**(1) (1968), 99.

This volume contains 835 problems, most of them stated in physical terms, but easily set up as boundary-value problems in second order partial differential equations. There are references to the textbook of Tikhonov and Samarskii (also recently brought out by Pergamon Press in an English translation), but the problems should make a most valuable adjunct to any text or course on second order partial differential equations.

**3. Introduction to the theory of difference schemes, by A A Samarskii.**(Russian edition 1971).

**3.1. Review of Russian edition by: N N Janenko.**

*Mathematical Reviews*, MR0347102

**(49 #11822).**

In the already extensive literature on the application of difference methods in problems of mathematical physics, the book under review is outstanding with respect to the fundamentality of the problems as well as with respect to the completeness of the results obtained. ... In the first part the author presents the basic concepts of the theory of difference schemes by means of examples of the simplest problems of mathematical physics: the boundary value problem for a second order ordinary differential equation, the mixed Cauchy problem for a one-dimensional parabolic equation, the Cauchy problem for the one-dimensional equation of oscillations, the boundary value problem for a second order elliptic equation. In the second part he presents the general theory of difference schemes, where he treats the difference scheme as an operator equation in a Banach space of network functions. Such an approach permits him to solve, in a unified way, the problems of constructing difference schemes with prescribed properties: accuracy, stability, simplicity of realization. He applies the constructed theory to obtain economical difference schemes and iteration processes in multidimensional problems of mathematical physics.

**4. Difference schemes for gas dynamics, by A A Samarskii and Yu P Popov.**(Russian edition 1975).

**4.1. Authors' summary:**

This book gives a systematic presentation of methods for constructing, studying, and implementing difference schemes for the numerical solution of time-dependent problems of gas dynamics and magnetohydrodynamics. The second edition is significantly expanded by the inclusion of a new chapter devoted totally to conservative difference schemes for the two-dimensional case, as well as by studies of the convergence of iteration processes for nonlinear gas-dynamics schemes. The section on applications has been expanded. The book is intended for advanced undergraduates and for graduates in applied mathematics. It may also prove interesting to scientists in the fields of applied mathematics, physics, and mechanics.

**4.2. Review of Russian edition by: Jakub Siemek.**

*Mathematical Reviews*, MR0462194

**(57 #2169).**

The monograph contains a systematic presentation of methods of construction and investigation of difference schemes oriented for the numerical solution of one dimensional, unsteady problems of gas dynamics and magnetohydrodynamics. Results obtained during the recent years are presented. Particular attention is paid to conservative schemes which have convincing physical motivation. The algorithms of numerical execution of difference schemes are shown, and numerous examples which prove the effectiveness of the algorithms and schemes are specified. ... The book is based on lectures given by the authors at Moscow University.

**5. Difference methods for elliptic equations, by A A Samarskii and V B Andreev.**(Russian edition 1976, French edition 1978).

**Review of Russian and French editions by:**Georges A Lebaud.

*Mathematical Reviews*, MR0502017

**(58 #19209a); MR0502018**

**(58 #19209b).**

This book develops the finite difference method for the numerical solution of elliptic partial differential equations, with particular emphasis on equations of order 2 or 4 of immediate practical value. Specifically, the authors address the issues related to the construction and study of discrete schemes. Their resolution is not addressed in this book.

**6. Theory of difference schemes, by A A Samarskii.**(Russian edition 1977, English edition 2001).

**6.1. Review of Russian edition by: Milos A Zlamal.**

*Mathematical Reviews*, MR0483271

**(58 #3288).**

The material of this book and of the first book of the author on difference schemes [

*Introduction to the theory of difference schemes*(Russian)] is largely the same. The main difference is that the earlier book was written as a monograph whereas this book is primarily a textbook. Also new results published 1970-75 were incorporated into this book. The author gives a systematic exposition of the foundations of the theory of difference schemes and applications of this theory to the solution of simple typical problems of mathematical physics. ... The material of the book is extensive and illustrated by many examples. The book is well written and will be useful not only to students but also to those interested in the theory of difference schemes as well as in numerical computations.

**6.2. Review of English edition by: Raytcho Lazarov.**

*Mathematical Reviews*, MR1818323

**(2002c:65003).**

In Russia this book has had numerous editions and has been used in the education and training of two generations of numerical analysts. The book is well written, transparent, self-contained in notation and mathematical tools, and delivers more than one may expect. It covers both a detailed construction of finite difference schemes for basic linear partial differential equations and a rigorous analysis of their stability, error analysis, and implementation. The book is unique in the English literature. It introduces the concept of difference schemes as operators in finite-dimensional spaces of grid functions and uses both the mathematical language and the tools of operator theory. In order to achieve this compact and transparent exposition the author gradually builds the necessary mathematical concepts and tools in the first three chapters. Stability in various norms, local truncation error, convergence rate and accuracy, and the concept of operator-difference scheme are introduced and illustrated on various one-dimensional test problems. ... Overall, this is an excellent book, with a wealth of mathematical material and techniques. It will fill an existing gap in the English language literature in the area of finite difference methods for differential equations.

**7. Methods for the solution of difference equations, by A A Samarskii and E S Nikolaev.**(Russian edition 1978).

**7.1. Publisher's description:**

We consider methods for solving algebraic systems of high order which arise in applying the difference method to problems of mathematical physics. In addition to iteration methods, which have been most widely used in practice to compute the solution of such problems, we discuss direct methods. This book has been written for students of applied mathematics as well as for engineers and specialists in the area of numerical mathematics.

**8. An introduction to numerical methods, by A A Samarskii.**(Russian edition 1982).

**8.1. From the introduction:**

This book introduces the theory of numerical methods using a minimum of information from analysis, linear algebra, and the theory of differential equations. The book is a polished version of lectures given by the author over a period of several years to second-year students in the department of numerical mathematics and cybernetics of Moscow State University. The book is traditional in scope, covering interpolation and approximation, numerical integration, the solution of nonlinear equations, direct and iterative methods for solving systems of linear algebraic equations, difference methods for solving the Cauchy problem, and boundary value problems for ordinary differential equations. The author has attempted to make his presentation understandable on the first reading, paying attention to the basic concepts of the theory of numerical methods and illustrating them by very simple examples.

**9. What mathematical physics is, by A A Arsen'ev and A A Samarskii.**(Russian edition 1983).

**9.1. From the summary:**

Contemporary mathematical physics is the science that deals with the study of fundamental laws of nature by mathematical methods. In this booklet the authors discuss, in popular form, its place in the system of natural sciences, and its significance in forming a scientific picture of the world. They give a description of the methods of mathematical physics and talk about its role in solving scientific and technological problems.

**10. Blow-up in Quasilinear Parabolic Equations, by A A Samarskii, V A Galaktionov, S P Kurdyumov and A P Mikhailov.**(Russian edition 1987, English edition 1995).

**10.1. Review of Russian edition by: Pawel Szeptycki.**

*Mathematical Reviews*, MR0919951

**(89a:35002).**

This monograph is devoted to an exposition of results mostly due to the authors and concerned with the behaviour of solutions of boundary and initial value problems for quasilinear parabolic equations [of a particular form]. The physical terminology is mostly motivated by heat transfer consideration, but the discussion is also relevant for various other models in physics, engineering, biology, biophysics, ecology, etc. The study undertaken in the monograph centres around the phenomenon of blowing-up of solutions in finite time and of propagation of disturbances with finite speed.

**10.2. Review of English edition by: Howard Levine.**

*SIAM Review*

**38**(4) (1996), 692-694.

This book is primarily concerned with the properties of nonnegative solutions of the quasi-linear partial differential equation ...[which] arises naturally as the equation of motion in all sorts of physical situations such as heat transfer, flows in porous media, propagation of magnetic fields in media with finite conductivities, and in chemical kinetics to name just a few.

**11. Numerical Methods for Grid Equations - Vol. I: Direct Methods, Vol. II: Iterative Methods, by A A Samarskii and E S Nikolaev.**(English edition 1989).

**11.1. Review of English edition by: Gerhard Maess.**

*Mathematical Reviews*, MR1004468

**(90m:65003a); MR1004469**

**(90m:65003b).**

Volume I is devoted to direct methods for solving large sparse linear systems of special structure (tridiagonal, five-point discretization, block systems). ... Volume II contains iterative methods, (among others) Gauss-Seidel, SOR, Chebyshev semi-iterative methods, triangular and alternate-triangular methods, and the method of alternating directions. Both volumes include many examples that apply the methods described to the solution of special equations from the field of mathematical physics.

**11.2. Review by: Zbigniew Leyk.**

*Mathematics of Computation*

**55**(192) (1990), 867-868.

These volumes are devoted to the solution of systems of equations that arise in applying the finite difference method to problems of mathematical physics, mainly to boundary value problems for second-order elliptic equations. They are focused on iterative methods, although direct methods are also discussed. The aim is to gather in one place information on iterative methods for solving difference equations. The book has primarily been written for students of applied mathematics at the Moscow State University. ... The mathematical level of the text is very high. Apart from some introductory lemmas and theorems in Chapters 1 and 5, every theorem and lemma has a proof. The book is directed at advanced readers. However, it contains many remarks and examples illustrating the methods discussed. ... It is recommended for those acquainted with fundamentals of functional analysis and finite difference methods. The book is a classic, and should be a valuable addition for practitioners as well as students in the field.

**12. Nonstationary structures and diffusion chaos, by T S Akhromeeva, S P Kurdyumov, G G Malinetskii and A A Samarskii.**(Russian edition 1992).

**12.1. Review of Russian edition by:**Alexander Loskutov.

*Mathematical Reviews*, MR1261183

**(95f:58050).**

The book is devoted to the problems of nonlinear dynamics and chaos in open systems; it has an interdisciplinary character. Mathematical modelling of complex phenomena and processes appearing in plasma physics, the theory of combustion, chemical kinetics, and quantum field theory have a dominant role in the book. In addition, the authors give detailed insight into the recent ideas of synergetics, novel economics models, the theory of neural networks, and cellular automata. ... The manner of exposition is more in the tradition of theoretical physics than of mathematics: elaborate formal proofs are often replaced in the text by arguments based on new methods of analysis of the system under study. The material is presented at a level suitable to a wide circle of scientists.

**13. Mathematical modeling, by A A Samarskii and A P Mikhailov.**(Russian edition 1997).

**13.1. Review of Russian edition by: A T Barabanov.**

*Mathematical Reviews*, MR1845902

**(2002f:00011).**

This monograph is devoted to the ideas, methods and examples of mathematical modelling. By considering a wide range of problems in mechanics, physics, biology, and sociology, the authors clearly and convincingly present mathematical modelling as the intellectual core of modern information technologies in general and scientific knowledge in particular. They comprehensively demonstrate the possibilities and bases of the use of mathematical models of the fundamental laws of nature, variational principles, hierarchical structures, and the method of analogies. Modelling is presented as a universal methodology that ably embraces not only traditional fields but also human activities that are difficult to formalize such as economic, financial and social systems, the state-society relationship, international competition, etc. The authors consider fundamental approaches to the investigation of mathematical models. In this relatively small volume, they naturally avoid cumbersome or complicated constructions, but they do pay considerable attention to the description of ideas and the selection of examples that lucidly illustrate approaches to modelling. The book organically combines popular exposition and rigorous scientific reasoning. It should be of interest and use to undergraduate and graduate students, instructors, and specialists interested in studying and using methods of mathematical modelling and computational experiments.

**14. Additive schemes for problems in mathematical physics, by A A Samarskii and P N Vabishchevich.**(Russian edition 1999).

**14.1. Review of Russian edition by: Lutz Tobiska.**

*Mathematical Reviews*, MR1788271

**(2002f:65003).**

The book deals with the construction and investigation of additive difference schemes for approximating solutions of nonstationary problems in mathematical physics.

**15. Difference Schemes with Operator Factors, by A A Samarskii, P P Matus and P N Vabishchevich.**(English edition 2002).

**15.1. Review by: Jens Lorenz.**

*SIAM Review*

**46**(4) (2004), 752-753.

There are two main branches of stability theory for discretizations of partial differential equations: One is based on estimates of the Fourier symbol of frozen coefficient problems and is often called normal mode analysis or GKS stability theory. ... The other, which is the subject of the book under review, is often called the energy method. Normal mode analysis applies to more general problems, but is often very difficult to use. The energy method applies to more specific equations, which are, however, most important in applications. ... in summary, the book gives a rather comprehensive treatment of the energy method to investigate stability of difference methods applied to diffusion and wave equations. It covers an important topic of numerical analysis .... My main criticism regards the use of the English language, which is frequently inadequate. Quite a few sentences only make sense after one substitutes words like restriction for contraction, jump condition for conjugation condition, etc. The book deserved better editing.

**15.2. Review by: John C Strikwerda.**

*Mathematical Reviews*, MR1950844

**(2003k:65095).**

This text proves stability estimates for a range of finite difference schemes using operator methods. The schemes considered are two-level and three-level schemes for equations of first order in the time derivative and three-level schemes for equations of second order in the time derivative. The equations considered are primarily the standard hyperbolic and parabolic equations. There is a section on Korteweg-de Vries type equations. There is also a section on adaptive grid methods.

**16. Numerical methods for solving inverse problems of mathematical physics, by A A Samarskii and P N Vabishchevich.**(English edition 2007).

**16.1. Review by: Dinh Nho Hào.**

*Mathematical Reviews*, MR2381619

**(2008m:65002).**

The authors introduce several numerical methods for solving some standard inverse problems in partial differential equations, such as the identification of the right-hand sides in elliptic or parabolic problems, retrospective inverse problems, the identification of boundary conditions in parabolic equations, and coefficient inverse problems for parabolic equations. The book consists of eight chapters, at the end of each of which is a section for programs and a section for exercises.