1.1. Review by: Alston Scott Householder.
Mathematical Reviews MR0121956 (22 #12683).

According to the introduction, this text is prepared for use at the University of Kiev for the course on machine computing, and is, moreover, the first one. Its purpose is to acquaint the student with "approximate methods of mathematics" and with the use of simple computing machines. While references are made to electronic computers, the first chapter describes the abacus, the "arithmometer", and the slide rule, and concludes with a section on the use of tables.

Chapter 5 deals with ordinary differential equations, and Chapter 7 is entitled "Methods of mathematical physics connected with the solution of linear algebraic equations", which is to say, boundary-value problems and integral equations. Elsewhere, and perhaps even here, a good course in calculus should provide an adequate mathematical background for the book. The organisation is uniform throughout: the theory of a method is presented in simple terms, then simple illustrative examples are given. Each chapter contains at least one set of problems as "laboratory work".

Chapter 1 begins with a discussion of computational error; Chapter 2 deals with numerical and graphical methods for the solution of algebraic and transcendental equations; Chapter 3 with interpolation; Chapter 4 with numerical differentiation and integration; Chapter 6 with "linear algebra".

Chapter 2, for example, begins with the method of Lobachevsky, properly so-called, since they do not describe Gräffe's convenient algorithm; then goes on to the method of false position, Horner's method, Newton's method, and graphical methods. In Chapter 5, for the characteristic-value problem, only the escalator method and the method of Krylov are given. Each chapter is concluded with a short list of references, mostly textbooks, a fair number non-Russian but in Russian translation.

In some instances, one can criticise the selection of methods. This is especially so for the characteristic-value problem, and the refusal to use Gräffe's algorithm seems inexcusable. Usually, however, the selection is reasonable, and even inevitable, and the presentation is lucid. Considered as a practicium and not as a treatise, it should do quite well.

2.1. Review by: H J Norton.
Mathematical Reviews MR0168135 (29 #5399).

In two dimensions some finite-difference approximations to linear partial differential equations may be written in the form (1). By using the transformation of the [matrix B] the author reduces equations (1) to the equations (2) The solution of (2) may then be written in explicit form.

In the first chapter the solution of equations (2) is discussed and also the reduction of B to diagonal form. The second chapter deals with the derivation and transformation of approximations (1) for Laplace's, Poisson's, and the biharmonic equations, and also equations of parabolic and hyperbolic type with constant coefficients. The extension of the method to curvilinear boundaries and three-dimensional problems is described. Brief consideration is given of equations with variable coefficients.

This method would appear to be useful in avoiding round-off errors when a small mesh, and consequently a large number of nodes, is required.

3.1. Review by: Stefan Drobot.
Mathematical Reviews MR0182181 (31 #6404).

This is a standard textbook of an "applied" type. It consists of six chapters with the following headings. (1) Problems in physics and mechanics leading to the basic equations of mathematical physics. (2) General problems in the theory of partial differential equations. (3) Elliptic equations. General properties of harmonic functions. Theory of potential. Solution of boundary value problems. (4) Parabolic equations. Basic boundary-value problems. General properties of the solutions of the equation of heat conductivity. (5) Hyperbolic equations. Basic boundary-value problems. Stationary oscillations. Propagation and dispersion of waves. (6) Method of separation of variables applied to hyperbolic, parabolic and elliptic equations. Eigenvalue and eigenfunction problems. Elementary theory of special functions. (7) Methods of integral transforms.

4.1. Review by: Petru P Caraman.
Mathematical Reviews MR200464 (34 #357).

Lipman Bers and I N Vekua in their monographs have considered the solutions of certain linear systems of two first order partial differential equations as generalized analytical functions, and have obtained the extension of the most important results of the theory of analytical functions in the case of these classes more general functions. Of the different systems considered by Bers and Vekua, the author deals with Beltrami systems ...

The author's monograph is based on his previous papers on generalisation in the case of $(p, q)$-analytic functions of the Cauchy formula and integral, analytic continuation, some topological and differential properties and the theory of singularities and residues belonging to the theory of analytical functions. Then, his results on the integral representation and the inversion of the $p$-analytical functions where $p = x^{k}$ allows him to solve the problem on the contour for these functions. The last chapter is devoted to the applications of the $(p, q)$-analytic functions in the theory of homogeneous and non-homogeneous media, in the theory of elasticity and in the theory of torsion of bodies of revolution. This book is intended for specialists in the field of the theory of functions of a complex variable, partial differential equations and mechanics. It contains 67 figures and a bibliography of 165 titles.

4.2. Review of 1973 2nd edition by: Petru P Caraman.
zbMATH 0257.30040

This is the second edition of the author's book "Generalization of the theory of analytic functions of a complex variable. $p$-analytic and $(p, q)$-analytic functions and some of their applications" (1965). In this edition we find new results of the author and of his students published after 1965; some of the author's results are published now for the first time. Thus, in the first chapter there are, for instance, results on affine transformations of equivalence classes of $p$-analytic and $(p, q)$-analytic functions, in the second chapter, results on the integral representation of $p$-analytic functions and its inversion formula, in the third chapter, results on application of $p$-analytic functions for the solution of concrete axial-symmetric problems of the theory of elasticity. In order not to increase too much the size of the book, for some questions, connected to the considered problems, the author preferred only to mention them and to refer to the corresponding papers and books.

5.1. Note.

This English text is an authorised translation of [2] above incorporating revisions and new material supplied by the author. It was translated from the Russian by G J Tee and the translation was edited by K L Stewart. Polozii is "Doctor of Physico-Mathematical Science, Professor and Director of the Faculty of Computational Mathematics at the University of Kiev."

There is also a German translation which appeared in 1966 under the title Numerische Lösung von Randwertproblemen der mathematischen Physik und Funktionen diskreten Arguments.

5.2. Description by the Publisher.

Pure and Applied Mathematics, Volume 79: The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics presents the numerical solution of two-dimensional and three-dimensional boundary-value problems of mathematical physics. This book focuses on the second-order and fourth-order linear differential equations. Organised into two chapters, this volume begins with an overview of ordinary finite-difference equations and the general solutions of certain specific finite-difference equations. This text then examines the various methods of successive approximation that are used exclusively for solving finite-difference equations. This book discusses as well the established formula of summary representation for certain finite-difference operators that are associated with partial differential equations of mathematical physics. The final chapter deals with the formula of summary representation to enable the researcher to write the solution of the corresponding systems of linear algebraic equations in a simple form. This book is a valuable resource for mathematicians and physicists.

5.3. Author's Preface to the English Edition.

This monograph describes the so-called method of summary representation and P-transformation for the numerical solution of finite-difference equations approximating to boundary-value problems of mathematical physics, in two, three or more dimensions. This method may be regarded as an attempt to lay the foundations for one of the possible approaches to the development of "numerical-analytical" methods in mathematics.

The extent to which this attempt has been successful and useful can be judged by the reader.

The results of a number of computations, made according to the method of summary representation, have been added as a Supplement to the English edition of this monograph. These results have been published since the appearance of the Russian edition in 1962. (G N Polozhii, Chislennoye Resheniye Dvumernykh I Trekhmernykh Krayevykh Zadach Matematicheskoi Fiziki, I Funktsii Diskretnogo Argumenta, Kiev University Press, 1962).

The author wishes to express his gratitude to the British scientists Mr G J Tee and Professor L Fox (Director of the Oxford University Computing Laboratory), and to Mr G Alexander (ex Pergamon Press), for showing much interest in this monograph, and for their initiative in having this English edition published.

5.4. Preface.

When boundary-value problems of mathematical physics involving partial differential equations are solved approximately by reducing them to corresponding finite-difference boundary-value problems, considerable difficulties arise in the solution of the corresponding systems of linear algebraic equations unless the number of these equations is comparatively small.

This book presents the author's investigations into the numerical solution of two-dimensional and three-dimensional boundary-value problems of mathematical physics, primarily those connected with second-order and fourth-order linear differential equations.

To begin with, during our development of the general concept of "analysis of the finitely small", we introduce special functions of discrete argument and establish special formulae for addition of their arguments. We also construct an original apparatus, namely the so-called P-transformation.

Next we present a method which enables boundary-value problems for partial differential equations, corresponding to two-dimensional or three-dimensional boundary-value problems of mathematical physics, to be solved either explicitly or in the form of comparatively simple formulae containing only a small number of parameters, the numerical values of which are determined from a correspondingly small number of linear algebraic equations. As a result of this, only a comparatively small amount of computation is required for finding the solutions to many quite different types of problem in mathematical physics. This raises the possibility of avoiding large computational errors, and such a method also has definite advantages in comparison with tabular forms of solution. It greatly extends the practical possibility of finding completely satisfactory numerical solutions to problems of mathematical physics, particularly for comparatively fine-meshed nets.

This book should prove to be useful for the numerical solution of every type of boundary-value problem of mathematical physics and its applications to engineering practice, particularly in those cases where much importance is attached to the accuracy of the solution.

The results of this investigation were obtained by the author during the 1960/61 academic year when he was a "spetskurs" student at the Kiev (Order of Lenin) State University (named after T G Shevchenko), specialising in the faculty of Computational Mathematics.

The author would be pleased to receive comments and suggestions concerning this book from its readers.

5.5. Introduction.

Of all the approximate methods for the solution of boundary-value problems of mathematical physics, the most prominent is the method consisting of reduction of these problems to corresponding finite-difference boundary-value problems, or (what is the same thing), to systems of algebraic equations. This method is known in the literature under the names of the finite-difference method, or the method of nets. By means of the finite-difference method we may successfully apply one of the most fundamental concepts of approximate mathematical methods; namely, the approximation of various functional spaces by more restricted spaces with a finite (or denumerably infinite) number of dimensions.

The following fundamental questions arise when the finite-difference method is used for the solution of boundary-value problems of mathematical physics:

1. The problem of the existence and uniqueness of the solution of the corresponding finite-difference boundary-value problem.
   
   
2. The problem of the convergence of the solution of the finite-difference problem to the exact solution of the corresponding boundary-value problem of mathematical physics.
   
   
3. The problem of the rate of convergence, or, for that matter, the problem of estimating the error of the method; i.e. the problem of estimating the difference between the exact solution of the boundary-value problem of mathematical physics and the exact solution of the corresponding finite-difference problem.
   
   
4. The problem of finding the solution of the finite-difference boundary-value problem, or, equally, of the corresponding system of algebraic equations.

From the extensive literature devoted to the finite-difference method and its applications to the solution of actual problems of mathematical physics, and also to the proof of theorems concerning the existence of solutions of boundary-value problems for partial differential equations, we confine ourselves to indicating the works and the references in the survey article. This latter article contains a fairly complete bibliography of works on the finite-difference method (principally by Soviet scientists) over the last forty years. A survey of this work and of the results of applying the finite-difference method for proving theorems concerning the existence of solutions of boundary-value problems for partial differential equations is given in the work.

Two-dimensional and three-dimensional boundary-value problems of mathematical physics are connected with definite partial differential equations. Among these differential equations we may note first of all Laplace's equation, Poisson's equation, the heat-conduction equation, the wave equation, the biharmonic equation, the equation of a vibrating beam, the equation of a vibrating plate and general partial differential equations with constant coefficients, and also equations whose coefficients depend upon the independent variables in some one or other special manner. Certain problems from various branches of mathematical physics are connected with non-linear second-order partial differential equations and other problems are connected with certain sixth-order or eighth-order partial differential equations with constant coefficients.

It may be considered that the fundamental questions (a), (b) and (c) of the previous paragraph, concerning the finite-difference method, have been adequately investigated for the partial differential equations most frequently arising in mathematical physics. In many cases the results of the investigations of these problems made by various authors leave nothing to be desired. However, this cannot be said of the question (d), concerning the numerical solution of finite-difference boundary-value problems, or of the corresponding systems of algebraic equations. This is due to the fact that if we wish to make the error of the finite-difference method as small as possible, then we must use a very small step for the net. Because of this, the corresponding system of algebraic equations becomes extremely unwieldy, consisting of a large number of equations with a correspondingly large number of unknowns. Significant difficulties are found to arise in the solving of such systems with a large number of unknowns. This is particularly so in the case of finite-difference boundary-value problems corresponding to elliptic differential equations, when the solution of the system of algebraic equations cannot be found by an explicit scheme, i.e. it does not reduce to a direct computation by steps. The best-known methods for solving such systems of algebraic equations are the method of elimination, the method of successive approximations and the method of relaxation. However, each of these methods can be regarded as being wholly satisfactory only when the finite-difference problem consists of a fairly small number of algebraic equations. Otherwise significant difficulties arise ...

In our view, the line of advance from such difficulties must proceed simultaneously along two directions, which in a certain sense are opposed to one another:

1. Along the line of increased operating speed of computers, with the aim of increasing the number of arithmetic operations which can practicably be performed;
   
   
2. Along the line of creating new, more modern, methods of computational mathematics, which are applicable to specific fairly wide classes of mathematical problems; and which, as well as having small inherent error, do not lead to the danger of the accumulation of large computational errors, such as would arise after performing a large number of arithmetic operations.

However far we may have progressed in the first direction, we cannot hope to get satisfactory solutions for many classes of mathematical problems unless progress is made in the second direction. [We would not have spoken of this had we not encountered the contrary opinion; namely, that increased operating speed of computers solves all problems immediately.]

Therefore, for all linear two-dimensional and three-dimensional boundary-value problems of mathematical physics, the problem of finding new, more modern, methods for solving the corresponding finite-difference boundary-value problems (or, equally, the corresponding systems of linear algebraic equations) should be regarded as being exceptionally important when the system contains a large number of equations.

The works [referenced] are definitely interesting from this viewpoint. Another very interesting method is that of "marching", or matrix factorisation, which has been developed in recent years, principally at the Steklov Mathematical Institute of the USSR Academy of Sciences. This method may advantageously be applied for the solution of one-dimensional and two-dimensional finite-difference boundary-value problems, connected with linear second-order differential equations. A characteristic feature of this method is that the entire process of solving is reduced to a certain stable step-wise computation. In the two-dimensional case, in contrast to the one-dimensional case, each step involves the inversion of a matrix.

As direct evidence of not only the importance of the aforesaid problems, but also of their difficulty, we may cite the appearance in the literature of special tables, devised for the solution of two-dimensional boundary-value problems for Laplace's equation. [These tables give the numerical solutions to boundary-value problems for Laplace's equation in the case of certain specific types of two-dimensional regions, for which Green's Function is known in explicit form from the equations of mathematical physics.] It could be considered that further evidence of this is provided by the introduction of networks of electro-integrators into computational technique, enabling one to solve finite-difference boundary-value problems connected with linear second-order elliptic differential equations, with a number of unknowns which may be as large as 1000-1200 in the most modern machines. The same could be said of fluid integrators designed for the approximate solution of problems connected with the differential equation of heat-conduction. All of this evidence becomes particularly convincing, if we reflect that the manufacture and operation of such devices require much expenditure of labour and materials.

In this book we present our investigations into the aforesaid problem of the solution of finite-difference boundary-value problems, or of systems of linear algebraic equations corresponding to two- and three-dimensional boundary-value problems of mathematical physics, which are connected with linear partial differential equations.

We present a new method for the solution of the aforesaid finite-difference boundary-value problems: hereafter we shall call this the method of summary representation and $F$-transformation or, simply, the method of summary representation.

The essence of this method consists of finding the general solutions of finite-difference boundary-value problems, corresponding to boundary-value problems of mathematical physics for plane or solid regions of general shape, either in explicit form or in the form of comparatively simple so-called formulae of summary representation, containing a small number of parameters, which are determined by a correspondingly small number of linear algebraic equations.

Particularly effective results are obtained when the method is applied to any boundary-value problem for the fundamental equations of mathematical physics: Laplace's equation, Poisson's equation, the heat-conduction equation, the wave equation, the biharmonic equation, the equation of vibrations of a beam and certain other partial differential equations, frequently occurring in mathematical physics and technology.

A characteristic feature of our method is that the great majority of the unknowns appearing in the finite-difference boundary-value problem do not enter directly into the calculation during the finding of the solution. This results in a comparatively small amount of computation being needed for finding the solution, and also it leads to the possibility of avoiding unnecessarily large computational error. This indicates the importance of the method, inasmuch as the number of unknowns appearing in finite-difference boundary-value problems tends to be rather large: not only for the purpose of finding the approximate solution at a large number of points, but also in order to minimise the error of the method. Finally, we remark that the presentation of our solution in the form of formulae of summary representation has definite advantages, in our view, over the tabular form of presentation of the solution: at any rate, it does not require the construction of large numerical tables, from which only a few numbers will be needed in the end; and moreover it is a promising approach for qualitative and analytic investigations of the summary characteristics of boundary-value problems. [By summary characteristics of boundary-value problems we mean such quantities connected with the solution of boundary-value problems as have direct interest for applications. The need to determine these numerically ordinarily leads to the solution of boundary-value problems.]

The construction of a formula of summary representation may be explained as a transition from a given "local finite-difference operator" (expressing the value of the solution at one point in terms of its values at neighbouring points) to a "global operator", which gives an expression for the solution at a set of nodes over some one or other net region, in terms of the values of the solution at nodes which are close to its boundary. From this point of view (i.e. the transition from the "local operator" to the "global operator"), the method of summary representation proves to be, in our opinion, quite natural and valid.

On the one hand the method of summary representation is a numerical method, but on the other hand it may be regarded as an original discrete analogue to the classical methods of integral representation in mathematical physics, particularly the theory of potential.

In view of this latter aspect, the method of summary representation possesses a number of analytical properties which are very useful and convenient when specific classes of problems are being solved.

In particular, in many cases in which the classical theory of potential leads to Fredholm integral equations of the first or second type, the method of summary representation leads to systems of linear algebraic equations in comparatively few unknowns. The idea of fitting together individual solutions can be accomplished in an exceptionally simple manner in the method of summary representation, and does not require any additional investigation to be performed in each individual case, such as is needed by certain other methods.

Although the classical methods of finite integral transformations have many advantageous features, they cannot be applied directly to any given rectangle if the boundary conditions given on the sides of the rectangle are not all of the same type. But, in many such cases, the method of summary representation requires only the solving of an appropriate auxiliary system of linear algebraic equations, in a comparatively small number of unknowns.

In contrast to many of the previously known exact (or approximate) methods of mathematical physics, the method of summary representation is not crucially dependent upon the form of the boundary conditions of a problem. It generally happens that if a solution to a boundary value problem for a given region has been found by the method of summary representation, then by proceeding from this solution we may fairly readily find the solution to any "perturbed boundary-value problem", obtained from the original problem by means of a small alteration to the region, or by a change in the type of boundary condition on some part of its boundary. As in the previous paragraph, the solution of the perturbed problem requires only the solving of an appropriate auxiliary system of linear algebraic equations, in a comparatively small number of unknowns.

Let $D$ be a region contained within a certain rectangle $G$, and let us be required, for example, to find the solution of the Dirichlet problem for Laplace's equation over the region $D$. We shall attempt to find this solution by extending the required function so that it is defined over the rectangle $G$. Such an endeavour is natural, since the Dirichlet problem for Laplace's equation (or for Poisson's equation) can be solved in a simple manner for a rectangular region. However, nothing will come of this attempt, since in order to extend the required function from $D$ to $G$ we would need to know not only the values of the required function everywhere on the boundary of $D$, but also the values of the normal derivative of the required function. But if this latter quantity is known then there is no need to extend the required function, since the solution at any point in the region $D$ may be expressed as the sum of the logarithmic potential due to a simple boundary layer, plus the inverse potential due to a dipole layer. But this idea is not wholly useless, if the given problem is replaced by a finite-difference formulation, which then is solved by the method of summary representation for the rectangle $G$. Indeed, if in this case we regard the finite-difference Laplace equation as being satisfied at all nodes outside $D$ as well as inside it, whilst the finite-difference Poisson equation (with unknown right-hand sides) is regarded as being satisfied at nodes on the boundary of $D$, then it is easy to construct a system of linear algebraic equations in these unknown right-hand sides of the Poisson equations. After solving for these and substituting their values in the formulae of summary representation, we get an explicit solution to the finite-difference Poisson equation over the rectangle $G$, under the boundary conditions chosen by us on the boundary of $G$. This solution gives, within the region $D$, the required solution for the Dirichlet problem for the finite-difference Laplace's equation in the region $D$. Thus, from the point of view of the method of summary representation, the idea of extending the required solution beyond the limits of the given region proves to be very useful - it results in a major reduction in the order of the system of linear algebraic equations corresponding to the given boundary-value problem.

If we look upon the finite-difference formulation of a problem in mathematical physics as a system of linear algebraic equations, then we may conveniently distinguish the method of summary representation from the general algebraic methods for solving systems of linear algebraic equations. This is due to the fact that the method of summary representation gives considerable advantages in economy of computational effort and the number of arithmetical operations, if instead of seeking the unknown numerical values of the function at all nodes of the net lying within the region under consideration, we seek only values at certain nodes of the net, i.e. we perform a selective calculation.

This same circumstance conveniently distinguishes the method of summary representation from the other methods known for solving boundary-value problems for partial finite-difference equations, which may be characterised as methods of over-all calculation. By this, we mean that in order to find the numerical solution even at a single node of the net, the computational process needs to find the unknown values of the function (or their approximations) at all nodes of the net.

It is not without interest to direct attention to one circumstance which is connected with the concept of stability of computations. When methods of over-all calculation are used, it is difficult to reduce the step of the net by a considerable factor, not only because of the finite capacity of computers, but also in a number of cases because of the danger of large computational errors arising. Examples show that when the step of the net is reduced, the corresponding system of linear algebraic equations becomes more ill-conditioned, and the convergence rates of iterative methods for solving these equations are reduced. However, it is readily seen from examples that the situation is quite different for the method of summary representation: the computational error may actually reduce when the step of the net is reduced. This is due to the fact that, before performing a selective calculation by the method of summary representation for an individual node (or a given set of nodes) in the net region, a preliminary transformation of the formula of summary representation may be made, converting it to a form which is suitable for the selective computation. This is quite analogous to the computation of some function $f (x)$, e.g.

          $f(x) = \Large\frac {\sin x}{x}\\$

for small values of $x$, where we should first transform this formula by means of a Taylor series expansion. Otherwise, if the numerator and denominator were evaluated separately and then divided, the computational error would increase as x approached zero. But after the above transformation has been performed, we get instead a reduction of the computational error as $x$ approaches zero.

We have here indicated certain general characteristic features of the method of summary representations, which have been deduced partly in a purely theoretical manner, and partly on the basis of experience of practical calculations on both desk machines and electronic computers. However, this does not by any means indicate that the method of summary representation has great advantages over many other methods for every problem of mathematical physics for which practical solutions are required. Such advantages can be said to exist, for the finite-difference formulations of problems in mathematical physics, only in those cases in which the number of nodes in the net is fairly large and in which formulae of summary representation have been constructed for the given class of boundary-value problems in mathematical physics. But such a construction, except for the fundamental equations of mathematical physics with constant coefficients, gives rise to quite definite difficulties. In order to overcome these difficulties, some preliminary work is needed for the investigation (and, in some cases, for the tabulation) of the so-called "special functions of discrete argument" and the "special matrices of type †" which we introduce. To some extent, this is analogous to the situation which arises when certain classical methods are being applied for solving specific classes of problems in mathematical physics, wherein it is necessary to carry out some preliminary work on the investigation and tabulation of a whole series of special functions, which are not included amongst the standard mathematical functions.

The book consists of two chapters, plus a supplement.

In the first chapter, following the general concept of developing the "analysis of the finitely small", we construct a general theory of the problem of eigenvalues and eigenvectors in the class of functions of discrete argument. This bears a complete analogy to the well-known Sturm-Liouville problem in the theory of differential equations. We investigate matrices of type $\Pi$ (which are, in a certain sense, close to Hermitian matrices) and their corresponding orthogonal matrices, which are orthogonal with a certain weight $\rho$. As a result, we obtain an entire series of matrices having important properties, including among their number one of the few matrices appearing previously in the literature whose eigenvalues and fundamental matrix have been found explicitly. We expound the general principles for the construction of special functions of discrete argument, by means of which we may express the general solution of a general linear second-order finite-difference equation. General reduction formulae (i.e. formulae for the addition of arguments) are established for these special functions. It is shown that, for any matrix of type $\Pi$, the corresponding fundamental matrix may be transformed into its own inverse by transposing it and post-multiplying it by a weight-matrix $\rho$, where $\rho$ is a specific diagonal matrix.

In the second chapter we introduce formulae of summary representation for various two-dimensional boundary-value problems of mathematical physics, connected with Laplace's equation, Poisson's equation, the biharmonic equation, the generalised heat-conduction equation, the generalised wave equation and the equation of transverse vibrations of a beam. This is done with the aid of $P$-transformations of special types. Then the formulae of summary representation which we have obtained are applied to the solution of boundary-value problems, initially for regions of rectangular shape and then for regions of completely general shape. As an illustration we present a numerical example which is simple from our viewpoint: solving approximately a boundary-value problem for Laplace's equation for a region which is nearly a rectangle, over which is drawn a net containing several million nodes.

At the end of the chapter, formulae of summary representation are derived by means of $P$-transformations of general form, and the method which we have devised is extended to the solution of partial difference equations corresponding to boundary-value problems of mathematical physics which are connected with partial differential equations with variable coefficients.

5.6. Review by: Leslie Fox.
J. London Math. Soc. 42 (1) (1967), 374-375.

The method of finite differences has been used for many years for solving partial differential equations. It turns the differential problem into an algebraic problem, and poses questions about how to solve the algebraic equations and about the relation between their solution and that of the differential system. The method of summary representation is concerned only with the algebraic problem, and is effectively a new method for solving finite-difference equations.

It is essentially the discrete analogy of the use of Fourier, Laplace or other integral transforms for the continuous problem. Moreover it is closely related to work of Bickley and McNamee (Phil. Trans. A252, 69-131, 1960), and absence of a reference to this paper is rather surprising. Quoting briefly from this paper, for an elliptic second-order equation with rectangular boundary the finite-difference equations can be represented in the form $AZ + ZB = F$, where $Z$ is the rectangular matrix of values of the wanted function at mesh points, $A$ and $B$ are matrices representing operations in each coordinate direction, and F is a known matrix.
...
There are rigorous relevant discussions of one-dimensional finite-difference operators, the adjoint and self-adjoint problems, the nature of the eigenvalues, and the nature and form of the solutions of the inhomogenous equation. Special "functions of discrete argument", and "reduction formulae" to simplify the computation of the solutions, are introduced for the case of variable coefficients.

A feature of the method of summary representation is that the general solutions contain arbitrary constants, which are found by solving a small number of algebraic equations with a view to satisfying some of the boundary conditions, a phrase which would also include, for example, the treatment of points near a curvilinear boundary or the common boundary of two or more rectangles. The second chapter applies the results of the first to solve various problems, including Laplace's equation in a rectangle, the sum of two rectangles, and in a more general boundary, all with various types of boundary conditions; similar problems in three dimensions; biharmonic and more general fourth-order equations; parabolic and hyperbolic problems in two and three independent variables; and the equation of transverse vibrations of beams. A final section considers the case of variable coefficients in two-dimensional problems.

A supplement to this English edition gives further practical examples, to problems of torsion, bending of beams and plates, and "infiltration under pressure", and some small extensions of the theory including the treatment of polar coordinates.

This is an interesting book, and will undoubtedly stimulate much further work. The technique is clearly very valuable for rectangular regions. Indeed it is noticeable that most of the examples have rectangular or near-rectangular boundaries, and claims of its efficiency for more general problems will need more supporting evidence. Certainly there is some justification for the search for methods more analogous to those of classical mathematics, in which the computation of a particular solution is obtained from combinations of general solutions. This book is an important step in that direction.

The translation, admirably performed by Mr J G Tee, would appear to follow very closely the original Russian text. There are some misprints not listed in the Errata slip, and I found irritating the numbering of the equations. ,... The wealth of manipulative detail also does not make for easy reading, but the effort is certainly worth making and the price makes this a good book to buy.

5.7. Review by: Alan Solomon.
Mathematics of Computation 21 (97) (1967), 123-124.

In this book a method is given for the numerical solution of a class of boundary and initial value problems for second and fourth order linear partial differential equations. The method rests on the transformation of the finite difference approximation to the differential equation into a vector difference equation in one variable, and the explicit solution of the latter. This explicit solution contains open constants determined by the initial or boundary conditions.
...
The method is most effective when applied to the basic differential equations of mathematical physics for rectangular regions with many mesh points, for it does not require excessive computation and so prevents the accumulation of computational errors
...
The author stresses the fact that the development of new and more efficient mathematical methods is fully as important as the development of faster computing machines; his method represents a useful and interesting step in this direction. The book is divided into two chapters and one appendix. In Chapter 1, explicit solutions are obtained for various difference equations in one variable; in Chapter 2 problems associated with Laplace's equation, the wave and heat equations and others are examined, using the author's method. In the Appendix, additional examples and some extensions of the theory are given. The presentation of the material is at times hard to follow, and no clear explanation of the author's method is given at any point. There are some slight misprints, and two references are missing. Nevertheless, the book is of definite value and interest.