# Books by Gurii Ivanovich Marchuk

Gurii Ivanovich Marchuk has written, or edited, a very large number of books. They are all originally written in Russian but some have been translated into English. Below we give extracts from reviews of some of the books which have English translations.

**1. Numerical methods for nuclear reactor calculations (1959), by Gurii Ivanovich Marchuk.**

**1.1. Review by: R R Coveyou.**

*Mathematical Reviews*MR0104354

**(21 #3109)**.

This important book fills a real gap in the literature pertaining to the theory of neutron chain reactors and nuclear reactors. Among others, Glasstone and Edlund [

*The elements of nuclear reactor theory*, 1952] furnish a comprehensive survey of the elementary aspects of the theory; Davison and Sykes [

*Neutron transport theory*, 1957] a mathematically and physically sophisticated view of the problem of neutron transport; and Weinberg and Wigner [

*The physical theory of neutron chain reactors*, 1958] a scholarly treatment of the physics of neutron chain reactors. None of these books attempts to discuss the problems arising from the actual calculation of the performance of a chain-reactor system. The book under review does just this, and very successfully. ... As the author himself emphasises, this book is not the place to acquire an understanding of the basic principles of nuclear reactor theory, a prerequisite for its use is a study of one or all of the books mentioned above. If the book has a real weakness, it lies in the physics; not all the evaluations of the approximations involved would be agreed to by other workers in this field. But, given a system of equations from which to work, one will find here many fruitful methods for translating these equations into concrete results. The author states correctly that the book should be valuable to graduate students, engineers, and specialists in the field of nuclear reactor calculation ...

**1.2. Review by: Elizabeth H Cuthill.**

*Mathematics of Computation*

**15**(73) (1961), 100-103.

A more accurate title for this book would have been Numerical Methods for Nuclear Reactor Physics Calculations. As stated by the author, "the book is an attempt at a more or less systematic exposition of numerical methods for the calculation of thermal, intermediate, and fast neutron reactors. Particular attention is devoted to the problems of critical mass, the space-energy neutron flux distribution, and the neutron importance." This is the first book to become available in English that is entirely devoted to this area of calculation, and as such is a helpful addition to our literature. The book is clearly not addressed to mathematicians. A familiarity with nuclear reactor theory is assumed; terms such as "cross section" are used without definition.

**2. Numerical Methods in Weather Prediction (1974), by Gurii Ivanovich Marchuk.**

**2.1. From the Publisher.**

Numerical Methods in Weather Prediction focuses on the numerical methods for solving problems of weather prediction and explains the aspect of the general circulation of the atmosphere. This book explores the development in the science of meteorology, which provides investigators with improved means of studying physical processes by mathematical stimulation. Organised into eight chapters, this book starts with an overview of the significant physical factors that are instrumental in enriching the theoretical models of weather prediction. This text then examines the system of hydrodynamic equations and the equation of heat transfer related to large-scale atmospheric processes. Other chapters consider the quasi-geostrophic approximation model, which is the basis for concepts of the dynamics of atmospheric motions and instrumental in establishing the basic features and laws of evolution of meteorological variables as applied to large-scale processes. The final chapter deals with the adjustment of the humidity field. This book is a valuable resource for meteorologists.

**2.1. Review by: P D Thompson.**

*American Scientist*

**63**(4) (1975), 463-464.

This book deals primarily with numerical methods and the analysis of their stability and accuracy, with specific reference to several aspects of the problem of weather prediction. In view of the generality of the latter problem, the "splitting-up" methods are likely to have wide applicability to more specialised hydrodynamical and thermodynamical questions. These methods, which cope separately and sequentially with the processes of local conservation, dynamic adjustment, and diffusion at different fractional time-stages during each unitary cycle, also have the advantage that complicated difference-operators may be reduced to products of a few elementary operators. Although this is primarily a book for specialists in computing and numerical analysis, it is valuable reading for any student of non-classical hydrodynamics. The exposition is detailed but clear, and the translation is excellent.

**3. Methods of numerical mathematics (1975), by Gurii Ivanovich Marchuk.**

**3.1. From the Publisher.**

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 mathematical physics which, in the course of their solution, can be reduced to simpler and theoretically better developed problems allowing effective algorithmic realisation 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 primarily 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.

**3.2. Review by: Murli M Gupta.**

*Mathematical Reviews*MR0381226

**(52 #2123)**.

The author has written an excellent book on the contemporary methods of computational mathematics. This book includes a discussion and a description of various new methods of numerical analysis such as variational methods, the method of splines and the fast Fourier transform. Detailed analyses of convergence, stability, accuracy, etc., are given of most of the methods that can be used for solving large problems of mathematical physics by modern computers. ... This book is essentially a reference book. It is well written ...

**3.3. Review by: Bertil Gustafsson.**

*Mathematics of Computation*

**31**(137) (1977), 321-322.

This book is a translation of the original Russian edition which was published in 1973. The title is quite general, but the main content is numerical methods for initial or boundary value problems in Mathematical Physics. ... As the author says in the preface, he has concentrated on basic ideas, and that is a good approach. The methods are presented in such a way that they can be easily generalised to more complicated problems. However, since the author emphasises implicit methods, it would have been natural to include a discussion about methods for solving nonlinear systems of algebraic equations. Other topics which are treated very briefly or not at all, are explicit difference schemes, higher order schemes and the choice of boundary conditions for approximations where this is not trivial. The book is written in a very clear and nice way. It is well suited to give a good understanding of the methods treated, and it should be a good help to physicists and engineers who already have some knowledge of practical problems.

**4. Modelling and Optimization of Complex Systems (1978), by Gurii Ivanovich Marchuk.**

**Review by: David Covall.**

*Biometrics*

**37**(4) (1981), 868.

True to the aims of this series, the text reports, informally, developments in the fields of optimal control and mathematical programming. The areas to which these developments have been applied include biology, environmental and socio-economic systems. The majority of the applications are in biology, particularly immunology. The first half of the book deals almost exclusively with the modelling of immune systems and with their identification. Although there is considerable overlap in the formulation of immune-system models, the individual analyses exhibit interesting, often subtle differences. The consensus, however, indicates that more precise and experimental analysis of the immune system is required for further model development. The remainder of the book consists of a mixture of theoretical and applied works involving primarily on-going projects in the USSR. ... Overall, the content is not geared for those seeking an introductory text in modelling and optimisation. The level of presentation is more appropriate for those already highly aware of current theory and technology in Russia.

**5. Mathematical Methods in Clinical Practice (1980), by G I Marchuk and N I Nisevich.**

**5.1. Review by: E L Forker.**

*American Scientist*

**70**(1) (1982), 86.

Despite a title that promises more, this book, translated from Russian, is narrowly restricted to a consideration of mathematical models that purport to describe the clinical course and the prognosis of viral hepatitis in children. American clinicians will be disappointed if they expect to find here fresh insights into the epidemiology or management of viral hepatitis, or if they hope to learn a practically useful mathematical approach to this group of diseases. The book does provide an interesting, if tangential, glimpse of medical practice in the USSR.

**6. The Monte Carlo methods in atmospheric optics (1980), by G I Marchuk, G A Mikhailov, M A Nazaraliev, R A Darbinjan, B A Kargin and B S Elepov.**

**6.1. From the Publisher.**

This monograph is devoted to urgent questions of the theory and applications of the Monte Carlo method for solving problems of atmospheric optics and hydro-optics. The importance of these problems has grown because of the increasing need to interpret optical observations, and to estimate radiative balance precisely for weather forecasting. Inhomogeneity and sphericity of the atmosphere, absorption in atmospheric layers, multiple scattering and polarisation of light, all create difficulties in solving these problems by traditional methods of computational mathematics. Particular difficulty arises when one must solve non-stationary problems of the theory of transfer of narrow beams that are connected with the estimation of spatial location and time characteristics of the radiation field. The most universal method for solving those problems is the Monte Carlo method, which is a numerical simulation of the radiative-transfer process. This process can be regarded as a Markov chain of photon collisions in a medium, which result in scattering or absorption. The Monte Carlo technique consists in computational simulation of that chain and in constructing statistical estimates of the desired functionals. The authors of this book have contributed to the development of mathematical methods of simulation and to the interpretation of optical observations. A series of general method using Monte Carlo techniques has been developed. The present book includes theories and algorithms of simulation. Numerical results corroborate the possibilities and give an impressive prospect of the applications of Monte Carlo methods.

**6.2. From the Preface.**

The book deals with general applications of the Monte Carlo method to radiative transfer problems. A series of effective algorithms is given for estimating the linear functionals that depend on the solution of the transfer equation. In order to reduce statistical errors, modifications based on asymptotic solutions of the Milne problem are elaborated. General algorithms are proposed for solving systems of integral equations of the second kind and also algorithms for estimating the polarisation characteristics of the light. Use of symmetry and other peculiarities of problems enable the authors to construct effective local estimates for calculating the multiple-scattering radiation field at desired points of the phase space. The corresponding algorithms of the dependent-sampling method are proposed. The general formulation of inverse problems is given and numerical algorithms are proposed for solving these problems by the use of linearisation, for which the required derivatives are calculated by use of the Monte Carlo method. Algorithms are also given for estimating the correlation function of the strong random fluctuation of light in a turbulent medium. How the radiation field characteristics depend on the various parameters of the optical model, as well as on observation and illumination conditions, is investigated. The book is directed to specialists in applied mathematics and physics, and to students and postgraduates studying Monte Carlo methods

**7. Difference methods and their extrapolations (1983), by G I Marchuk and V V Shaidurov.**

**7.1. From the Publisher.**

The stimulus for the present work is the growing need for more accurate numerical methods. The rapid advances in computer technology have not provided the resources for computations which make use of methods with low accuracy. The computational speed of computers is continually increasing, while memory still remains a problem when one handles large arrays. More accurate numerical methods allow us to reduce the overall computation time by of magnitude. several orders The problem of finding the most efficient methods for the numerical solution of equations, under the assumption of fixed array size, is therefore of paramount importance. Advances in the applied sciences, such as aerodynamics, hydrodynamics, particle transport, and scattering, have increased the demands placed on numerical mathematics. New mathematical models, describing various physical phenomena in greater detail than ever before, create new demands on applied mathematics, and have acted as a major impetus to the development of computer science. For example, when investigating the stability of a fluid flowing around an object one needs to solve the low viscosity form of certain hydrodynamic equations describing the fluid flow. The usual numerical methods for doing so require the introduction of a "computational viscosity," which usually exceeds the physical value; the results obtained thus present a distorted picture of the phenomena under study. A similar situation arises in the study of behaviour of the oceans, assuming weak turbulence. Many additional examples of this type can be given.

**7.2. Review by: R S Anderssen.**

*Mathematical Reviews*MR0705477

**(85g:65003)**.

A good illustration that the course of human history influences the directions of mathematics is the rapid growth in the study of extrapolation techniques since the advent of the computer. The basic idea of extrapolation is simple, but its implementation requires explicit knowledge of the leading terms in the discretisation error associated with the finite difference approximations being used. This book is concerned with the derivation and use of such knowledge about the discretisation error for a number of different approximation procedures for various problems. Except for quite minor points, it is an excellent and lucid translation of the original Russian text ... Though lucidly written (and translated), the book is not an introduction for a beginner without some experience with the concepts and techniques of numerical analysis. Though suitable as an advanced textbook on extrapolation, it should be viewed more as a source book, since it is really a useful compendium on extrapolation and its various ramifications.

**8. Mathematical models in immunology (1983), by Gurii Ivanovich Marchuk.**

**8.1. From the Publisher.**

Unified approach to mathematical modelling of host's immune response to viral and bacterial challenge is presented. Models are formulated by the systems of delay-differential equations within the framework of the Burnet's principle of clonal selection, the major histocompatibility complex restricted recognition of antigens by T-cells ; consider T- and B-cells of one specificity and fixed affinity antibodies to pathogen's antigen, and are derived using the birth-death cell population balances. The models are used for quantitative analysis of the viral hepatitis B infection, influenza A virus infection, acute pneumonia and viral-bacterial complications in lung. An approach to estimating parameters of a particular patient is suggested. Adjoint equations are used for sensitivity analysis of the models.

**8.2. Review by: J M Skowronski.**

*Mathematical Reviews*MR0832185

**(87c:92013)**.

The book is devoted to mathematical-numerical models in theoretical and experimental immunology. It encompasses classical as well as new results on the subject, the latter obtained by the author at a laboratory of the Academy of Sciences in the USSR. The introductory chapter outlines the problems concerned, both mathematical and medical. Chapter 2 introduces the growth model of viruses, plasma cells and antibodies in blood, combined with the rate of infection in a separate equation. The model is then discussed analytically. In Chapter 3 some additional refining factors are introduced. Chapter 4 models the blood-making process and Chapter 5 suggests approximate and numerical methods for analysing the therapeutic procedures.

**8.3. Review by: B Schneider.**

*Biometrics*

**42**(4) (1986), 1003.

This is a translation from the Russian. In the foreword by Academician R V Petrov, member of the USSR Academy of Medical Sciences, the book is described as a "watershed book on Immunology," where "the author pries into the future and unveils this future to the reader." I doubt whether this is correct for the future of either immunology or mathematics; however, for future cooperation between mathematicians and immunologists this book may provide a very helpful step forward. Mathematicians are introduced very intelligibly to the basic concepts of immunology and hemopoiesis, and immunologists are informed about the use of ordinary differential equations to describe biological processes. The attention given to elementary analytical methods and to their careful and extensive development is one of the advantages of the book. ... Overall, the book can be recommended to biometricians mainly because of its very clear exposition of the basic concepts in immunology and hemopoiesis, and for the construction in terms of differential equations and discussion of mathematical models for various aspects of these processes.

**9. Ocean tides. Mathematical models and numerical experiments (1984), by G I Marchuk and B A Kagan.**

**9.1. From the Publisher.**

*Ocean Tides: Mathematical Models and Numerical Experiments*discusses the mathematical concepts involved in understanding the behaviour of oceanic tides. The book utilises mathematical models and equations to interpret physical peculiarities of tidal generation. The text first presents the essential information on the theory of tide, and then proceeds to tackling the studies on the equations of tidal dynamics. Next, the book covers the numerical methods for the solution of the equations of tidal dynamics. Chapter 4 deals with the tides in the World Ocean, while Chapter 5 talks about the bottom boundary layer in tidal flows. The last chapter tackles the vertical structure of internal tidal waves. The book will be of great interest to individuals whose profession involves the direct interaction with tides, such as mariners, marine biologists, and oceanographers.

**9.2. Table of Contents.**

Introduction

1. Indispensable Information on the Theory of Tides

1.1 Forces Inducing Ocean Tides

1.2 Tidal Potential

1.3 Equations of Tidal Dynamics

1.4 Additional Potentials of Deformation

1.5 Boundary Conditions

1.6 References

2. Studies on the Equations of Tidal Dynamics

2.1 Formulation of the Problem

2.2 Basic Ideas and Definitions

2.3 Uniqueness Theorem

2.4 A Priori Estimates

2.5 Existence Theorem

2.6 On the Existence of a Periodic Solution of the Equations of Tidal Dynamics

2.7 Conjugate Equations of Tidal Dynamics

2.8 The Perturbation Theory

2.9 The Spectral Problem

2.10 References

3. Numerical Methods for the Solution of the Equations of Tidal Dynamics

3.1 Method of Boundary Values

3.2 Hn-Method

3.3 Modified Variant of the Hn-Method

3.4 The Method of Fractional Steps

3.5 A Modified Variant of the Method of Fractional Steps

3.6 References

4. Tides in the World Ocean

4.1 Empirical Cotidal Charts

4.2 Basic Features of the Spatial Distribution of Tides in the World Ocean

4.3 An Example of Numerical Modelling of Tides in the World Ocean

4.4 Some Other Calculations of Tides in the World Ocean

4.5 Numerical Experiments on Tidal Dynamics in the World Ocean

4.6 Estimation of the Rate of Tidal Energy Dissipation in the Open Ocean

4.7 References

5. The Bottom Boundary Layer in Tidal Flows

5.1 Some Definitions

5.2 Experimental Data

5.3 Theoretical Models of the Bottom Boundary Layer in Tidal Flows

5.4 On the Law of Drag in Tidal Flow

5.5 References

6. Vertical Structure of Internal Tidal Waves

6.1 Generation of Internal Tidal Waves

6.2 Qualitative Analysis of the Equations for Internal Waves

6.3 Vertical Structure of the Internal Tidal Waves in a Realistically Stratified Ocean

6.4 References

**9.3. Review by: Robert Pearson.**

*Mathematical Reviews*MR0743753

**(86f:86002)**.

Chapter 4 of this book is entitled "Tides in the world ocean". This chapter is the central theme of the book. The first three chapters lead up to this discussion. Chapter 1 gives the necessary background for the topic, considering, from a very symbolic point of view, the forces, potential, equations and boundary conditions. Chapter 2 studies dynamics from a theoretical point of view, and Chapter 3 gives the numerics. The central chapter gives many examples of theoretically derived tidal lines, and some empirical charts. The authors consider that the variation in empirical charts leaves everything unreliable. They discuss and compare numerically derived results. They observe that none accurately reflects observations. Chapter 5 deals with the bottom boundary layer, because it is a possible reason for discrepancies, while Chapter 6 deals with the vertical structure of internal tides. The reviewer believes that the first three chapters may be a bit daunting for graduate students without a very, very strong mathematical background. However Chapters 4 and 5 form a basis for a general discussion on deep ocean tides, probably even for advanced undergraduates.

**9.4. Review by: Myrl Hendershott.**

*Science, New Series*

**226**(4677) (1984), 961-962.

This book appeared in Russian in 1977. Its aim appears to have been to draw together, for Russian researchers, both historical and modern aspects of the dynamical theory of tides. In this aim the authors have been successful. Students of the (predominantly English-language) literature on tides will thus find here much that is familiar, as is important in a self-contained discussion, although occasionally in unfamiliar notation and from an unconventional point of view. ... On the whole , this book presents an accurate and self-contained account of tidal studies up to the date of original publication; it is the first account in English that adequately represents Russian work on ocean tides. A sequel by the same author's was published by Gidrometeorizdat, Leningrad, in 1983. The present English translation is thus somewhat behind the most recent developments, both in the U.S.S.R. and abroad, although acknowledgement of this fact is made in an appendix and an updated bibliography.

**10. Mathematical models in environmental problems (1986), by Gurii Ivanovich Marchuk.**

**10.1. Author's Abstract.**

Environment, its condition and pollution control are becoming pivotal problems of science, since life on this planet is globally and regionally a universal concern. The extent of industrial pollution of the atmosphere and the sea is so great that is harming living conditions in large industrialised areas and beginning to produce global effects changing the radiation balance of the Earth by larger concentrations of carbon dioxide and aerosols and thinning the ozone layer in the atmosphere. Polluting emissions and acid rains affect greater still environmental processes in industrialised areas.

**10.2. Review by: J M Skowronski.**

*Mathematical Reviews*MR0842433

**(87e:92043)**.

The parabolic differential equation model of diffusion is applied to the study of industrial output into an environment. When the output consists of a lumped substance, it falls to the ground in accordance with the Stokes law. Velocity and direction of wind are considered in the forcing terms. The aim is to define regions of possible influence of unwanted aerosols. Chapter 1 establishes basic equations of air-transport and diffusion, Chapter 2 discusses the problem of coupled equations, and Chapter 3 gives numerical methods for the solution. Optimisation problems are discussed in Chapter 4 (multicriteria) and the economics of planning a defence of the environment are considered in Chapter 5. Chapters 6-10 are devoted to specific problems and case studies.

**11. Mathematical modelling of ocean circulation (1988), by G I Marchuk and A S Sarkisyan.**

**11.1. From the Publisher.**

Ocean dynamics, seasonal currents and further characteristics of oceans may be analysed by applying numerical algorithms. In this volume suitable algorithms are presented and the results of applications are thoroughly discussed. On this basis both simplified and sophisticated models of ocean circulation are constructed taking temperature, salinity, the complex form of bottom relief and the contours of coast lines into account. The monograph is addressed to oceanographers working in the field of mathematical modelling and ocean hydrological characteristics, as well as to students in physics, mechanics and oceanography departments.

**11.2. From the Introduction.**

The problems of ocean dynamics present more and more complex tasks for investigators, based on the continuously sophistication of theoretical models, which are applied with the help of universal and efficient algorithms of numerical mathematics. The present level of our knowledge in the field of mathematical physics and numerical mathematics allows one to give rather complete theoretical analysis of basic statements of problems as well as numerical algorithms. Our task is to perform such analysis and also to analyse the results of calculations in order to improve our knowledge of the mechanism of large-scale hydrological processes occurring in the World Ocean. The new level of numerical mathematics has essentially influenced , the formation of new solution methods of ocean dynamics problems, among which an important one is the splitting method, which has been already widely practised in various fields of science and engineering. A number of monographs by N N Yanenko, A A Samarsky, G I Marchuk and others are devoted to the description of this methods. But the methods of the splitting theory require extensive creative work for their application to concrete problems, which are peculiar, as a rule, in problem formulation. The success of the application of these methods is related to the deep understanding of the essence of the described processes.

**11.2. Review by: J Sündermann.**

*GeoJournal*

**21**(1/2), IGU Regional Conference: Asian Pacific Countries (1990), 110-111.

Both authors are well-known internationally as pioneers in the field of mathematical ocean circulation modelling, especially with respect to the development and implementation of appropriate numerical algorithms. Marchuk, currently president of Academy of Sciences, has investigated the basic mathematics of the most widely used models and was a co-originator of the splitting-up-technique, which is utilised extensively in the Soviet Union. Sarkisyan was involved with the development of diagnostic and quasi-diagnostic models and with equatorial dynamics. The two authors represent the well-established "Russian School" in numerical hydrodynamics. The volume is an interesting mixture of textbook and recipe book for modellers with examples of practical applications. ... [The] organisation of the book is indeed useful for the advanced modeller who wants a quick comprehensive view of model classes, of practical solution techniques, including the relevant systems of algebraic equations, and of typical results. Above all, the reader will learn about the philosophy and the approach of the Russian School which is less well known in the West. Thus, he will benefit from this book if he simultaneously pays attention to the relevant western publications. The very general title of the book, of course, suggests a comprehensive survey of ocean modelling. This, however, is not the case, because many decisive contributions to the topic from the rest of the world have been left out.

**12. Adjoint equations and analysis of complex systems (1995), by Gurii Ivanovich Marchuk.**

**12.1. From the Publisher.**

New statements of problems arose recently demanding thorough analysis. Notice, first of all, the statements of problems using adjoint equations which gradually became part of our life. Adjoint equations are capable to bring fresh ideas to various problems of new technology based on linear and nonlinear processes. They became part of golden fund of science through quantum mechanics, theory of nuclear reactors, optimal control, and finally helped in solving many problems on the basis of perturbation method and sensitivity theory. To emphasise the important role of adjoint problems in science one should mention four-dimensional analysis problem and solution of inverse problems. This range of problems includes first of all problems of global climate changes on our planet, state of environment and protection of environment against pollution, preservation of the biosphere in conditions of vigorous growth of population, intensive development of industry, and many others. All this required complex study of large systems: interaction between the atmosphere and oceans and continents in the theory of climate, cenoses in the biosphere affected by pollution of natural and anthropogenic origin. Problems of local and global perturbations and models sensitivity to input data join into common complex system.

**12.2. Review by: Ian Gladwell.**

*Mathematical Reviews*MR1336381

**(96e:00011)**.

This is an unusual book in many ways. As the title implies, the concentration is on adjoint equations (mainly for time-dependent problems). The aim is identification and measurement in complex systems, specifically in environmental mathematics involving time-dependent partial differential equation models. The book is divided into three parts: a theoretical part of four chapters with the title "Adjoint equations and perturbation theory", a modelling part with the title "Problems of environment and optimisation methods on the basis of adjoint equations", and a final part consisting of two appendices on the numerical techniques used and a bibliography of 280 items with the most recent references in 1994.

**13. Adjoint equations and perturbation algorithms in nonlinear problems (1996), by Gurii I Marchuk, Valeri I Agoshkov and Victor P Shutyaev.**

**13.1. From the Publisher.**

Sparked by demands inherent to the mathematical study of pollution, intensive industry, global warming, and the biosphere, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems is the first book ever to systematically present the theory of adjoint equations for nonlinear problems, as well as their application to perturbation algorithms. This new approach facilitates analysis of observational data, the application of adjoint equations to retrospective study of processes governed by imitation models, and the study of computer models themselves. Specifically, the book discusses:

(i) Principles for constructing adjoint operators in nonlinear problems

(ii) Properties of adjoint operators and solvability conditions for adjoint equations

(iii) Perturbation algorithms using the adjoint equations theory for nonlinear problems in transport theory, quasilinear motion, substance transfer, and nonlinear data assimilation

(iv) Known results on adjoint equations and perturbation algorithms in nonlinear problems

This ground-breaking text contains some results that have no analogs in the scientific literature, opening unbounded possibilities in construction and application of adjoint equations to nonlinear problems of mathematical physics.

**13.2. Review by: I K Marek.**

*Mathematical Reviews*MR1413046

**(98d:47161)**.

The concept of "adjoint operator'' is widely used in the theory of differential equations for a formally adjoint operator. In the context of functional analysis, the definition of the adjoint operator is more sophisticated and essentially depends on boundary conditions. But in order to introduce an adjoint operator and the corresponding adjoint equation one needs first to formulate the concept of "dual space''. It turns out that, in doing so, there exist several possibilities. ... Adjoint equations are increasingly widespread throughout mathematics and its applications to problems of diffusion, environment protection models, the theory of climate and its changes, mathematical problems of processing the information provided by satellites, mathematical models in immunology, and others. A rational approach to the solution of inverse problems and to mathematical experiment planning is evolving parallel to the development of adjoint equation techniques. In Chapter 9 some applications of adjoint equations to some problems of practical interest are examined. This nicely written book can be recommended to a broad community of mathematically oriented specialists in various fields of applied sciences.

**13.3. Review by: Thomas P Svobodny.**

*SIAM Review*

**38**(2) (1996), 353-355.

This book possesses two obvious turnoffs: the price and the title. I cannot explain the price, but I will explain the title. The complex systems of the title refer to models of complicated physical phenomena that involve systems of partial differential equations whose solutions are functions of time and are defined over spatial regions with complicated geometry. There may also be nonlinear coupling among the variables that leads to especially complicated mathematical behaviour. Environmental modelling is becoming extremely important as more scientists become convinced that our time on earth is running out if better control is not exercised over some of our activities. Of importance, for example, is to model the effect of human-emitted aerosols on the radiative balance of the earth (the greenhouse effect). Global environmental modelling leads to very complex systems and some of these models make up the emphasis of this book. The author is mainly concerned with averaged models for transport and diffusion of airborne substances. The particular problems are the global transport of pollutants, the impact of industrial emission, and related meteorological questions such as the global transfer of energy from tropical ocean to temperate atmosphere. as well as problems of weather forecasting.

**14. Mathematical modelling of immune response in infectious diseases (1997), by Gurii Ivanovich Marchuk.**

**14.1. From the Publisher.**

Beginning his work on the monograph to be published in English, this author tried to present more or less general notions of the possibilities of mathematics in the new and rapidly developing science of infectious immunology, describing the processes of an organism's defence against antigen invasions. The results presented in this monograph are based on the construction and application of closed models of immune response to infections which makes it possible to approach problems of optimising the treatment of chronic and hypertoxic forms of diseases. The author, being a mathematician, had creative long-lasting contacts with immunologists, geneticist, biologists, and clinicians. As far back as 1976 it resulted in the organisation of a special seminar in the Computing Centre of Siberian Branch of the USSR Academy of Sciences on mathematical models in immunology. The seminar attracted the attention of a wide circle of leading specialists in various fields of science. All these made it possible to approach, from a more or less united stand point, the construction of models of immune response, the mathematical description of the models, and interpretation of results.

**14.2. Review by: Lutz Edler.**

*Mathematical Reviews*MR1448406

**(98k:92009)**.

The monograph presents a total of 323 pages of logical concepts and mathematical models for a unified system of immunological processes and hypotheses. The first part of the book starts with models of the immune response at a more logical level without much mathematical formalism, dealing in 6 chapters with (1) General knowledge, hypotheses, and problems, (2) Survey of mathematical models in immunology, (3) Simple mathematical model of infectious disease, (4) Mathematical modelling of antiviral and antibacterial immune responses, (5) Identification of parameters of models and (6) Numerical realisation algorithms for mathematical models. The second part considers the quantitative description of the mechanisms of viral and bacterial infections in a human organism, and the parameter identification methods for mathematical models.

**14.3. Review by: Lee A Segel.**

*SIAM Review*

**40**(4) (1998), 1014-1015.

Audaciously, a Russian school of theoreticians led by the author have attempted medically significant models of the immune system's fight against disease. The fashion in the West, by contrast, has been developing fragmental submodels to enhance understanding of the enormously complex immune system. The models in this book take the form of nonlinear delay differential equations. In the model for viral hepatitis B, for example, there are 10 equations and 45 parameters. The book recounts methods for efficient solution of the initial value problem when the equations are stiff. Parameter identification and sensitivity analysis are discussed extensively. Marchuk's models fulfil his sensible goal of reproducing a quantitative "generalised picture" of the course of disease. They pioneer in their incorporation of the effects of disease damage. They indicate a direction for studying the influence of drugs. Yet the modelling is indeed at "an initial stage," as Marchuk himself asserts several times. ... All but a handful of physicians will find the mathematics opaque and significant medical conclusions few. Applied mathematicians wishing an introduction to immunology should probably look elsewhere among the 358 publications cited, but they will be helped by a 25 page literature survey. Working theoretical immunologists will probably be alternately and inconsistently uncomfortable with oversimplifications and with the proliferation of equations and parameters. Yet these theorists should have the book available, for they will be inspired to and assisted in further work by the author's first steps along a road that they too will wish eventually to take. One day, large scale mathematical models of the immune system will surely play a role in fighting disease.

Last Updated April 2020