THE METHOD OF STRAIGHT LINES

(A survey of works)*)

O. A. Liskovets

§ 1. Preliminary remarks

It is known what a large place in the modern mathematical literature is occupied by investigations of approximate methods for solving partial differential equations. With the aid of such equations a considerable part of the problems of natural science can be described, and the question of whether we are able to bring the solution to numerical results is very important for practice. This is why, as the range of applications of mathematics expands, computational mathematics as a whole and methods for solving the equations of mathematical physics in particular are being developed more and more intensively. This article is devoted to a group of such methods.

As is known, the method of straight lines consists in the following (the example given below is due mainly to V. I. Smirnov [10]). Suppose that in the rectangle \(R\{0 \leq x \leq 1,\ 0 \leq t \leq T\}\) the problem is posed

\[ u_t = u_{xx} + f(x,t), \quad (x,t)\in R, \]

\[ u(0,t)=u(1,t)=0,\quad 0\leq t\leq T, \tag{1} \]

\[ u(x,0)=\varphi(x),\quad 0\leq x\leq 1. \]

We shall seek its approximate solution on the straight lines

\[ t=t_n=nh,\quad n=0,1,\ldots,N\quad (h=T/N) \]

(which is what gives the method its name). To this end, at \(t=t_{n+1}\) we replace in the original equation the derivative with respect to \(t\) by the backward difference quotient:

\[ u_t(x,t_{n+1}) \approx \frac{u(x,t_{n+1})-u(x,t_n)}{h} \overset{\mathrm{def}}{=}\Delta u(x,t_n)/h. \tag{2} \]

Denoting the approximate values for \(u(x,t_n)\) by \(u_n(x)\), we obtain for these functions the system of ordinary differential equations

\[ u''_{n+1}(x)= [u_{n+1}(x)-u_n(x)]/h - f(x,t_{n+1}),\quad 0\leq n\leq N-1, \tag{3} \]

which, in accordance with the problem, must naturally be supplemented by the conditions

\[ u_{n+1}(0)=u_{n+1}(1)=0,\quad 0\leq n\leq N-1, \tag{4} \]

\[ u_0(x)=\varphi(x). \tag{5} \]

This system can be solved successively, beginning with \(n=0\). As for any approximate method, the question then arises of the closeness of this solution to the exact one and of its convergence to the exact solution as \(h\to 0\). Uniform convergence of the approximations (3)—(5) to the solution of problem (1) is easy to prove if \(u_t\) is assumed continuous in the rectangle \(R\).

Indeed, then for the errors \(\gamma_n(x)=u(x,t_n)-u_n(x)\) the equalities

\[ \gamma''_{n+1}(x)=[\gamma_{n+1}(x)-\gamma_n(x)]/h+\eta_{n+1}(x), \]

\[ \gamma_{n+1}(0)=\gamma_{n+1}(1)=0,\quad 0\leq n\leq N-1;\quad \gamma_0(x)\equiv 0, \]

hold.

*) The article is an expanded version of the introductory part of the author’s dissertation [118], including the results obtained in the dissertation.

where the error of replacing (2)

\[ \eta_{n+1}(x)=o(1)\qquad (h\to 0) \]

uniformly in \(n\) and \(x\). Now, using the a priori estimate for the solution known for the boundary-value problem

\[ y''(x)=y(x)/h+\alpha(x),\qquad y(0)=y(1)=0 \]

\[ \max_{0\le x\le 1}|y(x)|<h\max_{0\le x\le 1}|\alpha(x)|, \tag{6} \]

we obtain for the quantities \(\gamma_n=\max_{0\le x\le 1}|\gamma_n(x)|\) the relation
\(\gamma_{n+1}<\gamma_n+h\cdot o(1)\), and consequently,

\[ |u(x,t_n)-u_n(x)|\le \gamma_n<nh\cdot o(1)=t_n\cdot o(1)\le T\cdot o(1),\qquad 1\le n\le N. \]

If, moreover, there exists a continuous second derivative \(u_{tt}\), and \(M\) is the maximum of its modulus in \(R\), then
\[ |\eta_n(x)|\le \frac{1}{2}hM, \]
and therefore
\[ |\gamma_n(x)|<\frac{1}{2}t_nMh\qquad (1\le n\le N). \]

From the example considered it is clearly seen that the method of straight lines is a variety of difference approximation methods, so that its idea is close to that of the method of nets. But, unlike the latter, here only a part of the independent variables is subjected to “quantization,” i.e., replacement of continuous domains of their variation by discrete sets of values, while the derivatives in the corresponding directions are replaced by expressions approximating them. The other part of the arguments and the derivatives corresponding to them are retained, and the original differential problem is thus reduced to another, also differential, problem, but of lower dimension (in the method of nets a system of algebraic equations is obtained).

This idea was first used by E. Rothe [1] in 1930 for an equation of parabolic type. But it is quite obvious that the applicability of the method of straight lines (as well as of the method of nets, of which it is a limiting case) is much broader, and that it is applied to equations not only of parabolic type but also of other, including nonclassical, types. Before proceeding to describe the results obtained in this direction, let us make one important observation of a methodological nature.

The name “method of straight lines” is sometimes applied also to other methods, for example to the well-known variational method of L. V. Kantorovich*), which also leads to a reduction in the dimension of the problem. However, by this name we shall understand only the totality of differential-difference methods in which an independently chosen approximate solution is sought for a differential-difference analogue of the problem. At the same time, the dimension and the form of those manifolds on which the approximate solution is sought are immaterial for us, so that the subject of this article will include, in particular, methods sometimes called in the literature methods of planes or hyperplanes. The latter term is all the more meaningful since the properties of differential-difference approximating schemes depend not so much on the number of variables that remain continuous as on the number of “quantized” variables.

§ 2. EQUATIONS OF ELLIPTIC TYPE

M. G. Slobodyanskii [5] began to apply the method of straight lines as a method for the numerical solution of elliptic problems of mathematical physics in 1939. His article considers the Laplace equation. For our purposes it is more convenient to rephrase the results for the case of the Poisson equation

\[ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(x,y). \tag{7} \]

In the rectangle \(\Pi\{x_0\le x\le x_0+a,\; y_0\le y\le y_0+b\}\), on the straight lines

\[ y=y_n=y_0+nh,\qquad 0\le n\le N\quad (h=b/N), \tag{8} \]

the equation is replaced by the three-layer scheme

\[ \frac{\partial^2 u(x,y_n)}{\partial x^2}+\Delta^2 u(x,y_{n-1})/h^2=f(x,y_n)+r_n(x), \]

*) Izv. Acad. Sci. USSR, Ser. VII, No. 5, 1933.

\[ r_n(x)=\frac{h^2}{24}\left[\frac{\partial^4 u(x,\alpha_{n+1,n})}{\partial y^4}+ \frac{\partial^4 u(x,\alpha_{n-1,n})}{\partial y^4}\right], \]

\[ \pm y_n<\pm\alpha_{n\pm1,n}<\pm y_{n\pm1},\qquad 1\leq n\leq N-1. \]

The remainder term (local error) \(r_n(x)\) is discarded, and one obtains a system of ordinary differential equations

\[ u_n''(x)+\Delta^2 u_{n-1}(x)/h^2=f(x,y_n),\qquad 1\leq n\leq N-1, \tag{9} \]

which approximates equation (7) on the straight lines (8) and is solved, for example, under boundary conditions of the first kind.

By a linear combination of the other terms of equation (7) on the same three straight lines, a scheme with a remainder of higher order is constructed:

\[ \frac{1}{12}\, \frac{\partial^2\left[u(x,y_{n+1})+10u(x,y_n)+u(x,y_{n-1})\right]}{\partial x^2} +\Delta^2 u(x,y_{n-1})/h^2 \]
\[ = \frac{f(x,y_{n+1})+10f(x,y_n)+f(x,y_{n-1})}{12} +r_n(x),\qquad r_n(x)=O(h^4), \]

\[ 1\leq n\leq N-1. \]

Corresponding to it is the approximating scheme

\[ \frac{u_{n+1}''(x)+10u_n''(x)+u_{n-1}''(x)}{12} +\Delta^2 u_{n-1}(x)/h^2 \]
\[ = \frac{f(x,y_{n+1})+10f(x,y_n)+f(x,y_{n-1})}{12}, \qquad 1\leq n\leq N-1. \tag{10} \]

In an analogous way, five-layer symmetric schemes with remainders of the second and fourth orders have been constructed for the biharmonic equation.

It is noted that the general solutions of the homogeneous systems corresponding to the schemes obtained do not depend either on the boundary conditions of the problem or on the shape and dimensions of the domain, and therefore can be determined in advance for any step \(h\), so that in fact the application of the method reduces to finding the particular solution of the scheme and the constants entering the general solution of the homogeneous system. The form of the latter for systems (9) and (10) is indicated in the next paper by the same author [6], where, in addition, schemes of the method of straight lines and general solutions of homogeneous systems are constructed for various problems of elasticity theory of the second and fourth orders both in the plane and in space.

These investigations were continued by V. N. Faddeeva [8]. She obtained the general solution of system (10), bringing its orthogonal transformation to diagonal form (this part of the work is reproduced in S. G. Mikhlin [9]). In addition, she gives instructions for solving problems in domains with curvilinear boundaries.

It is also shown there how, by applying sliding interpolation, one can, with the aid of the method proposed by L. V. Kantorovich [3], obtain, for example, scheme (10).

For a large number of elliptic equations of the second order (with a broad choice of conditions on the contour), a series of schemes and the form of their solutions were indicated by K. S. Yunusov [15]. He also considered problems with nonzero data for biharmonic and polyharmonic equations (both homogeneous and nonhomogeneous).

Very diverse linear equations of the second ([105], [107]), fourth ([95], [96], [106], [107], [121]) and sixth ([107]) orders in the plane were studied by E. O. Omarov. Following [8], he expresses the general solutions of the proposed schemes in terms of the roots of certain characteristic equations. For the constructed schemes their unique solvability in a rectangular domain ([107], [121]) is proved, i.e. the possibility of finding (in a unique way) the values of the arbitrary constants entering the general solution.

The method of straight lines is applied not only in rectangular Cartesian coordinates. Thus, N. N. Ustinova [23] in a disk by Dirichlet’s problem for Laplace’s equation in polar coordinates constructs on the straight lines \(\varphi_n=2\pi n/N\), \(0\leq n\leq N-1\), an approximating scheme of type (10) with a remainder term \(O(h^4)\) and its general solution. The same, and a simpler scheme of type (9) for Poisson’s equation in polar coordinates, together with the general solutions of the corresponding homogeneous systems, are given by V. I. Lebedev [22]. An analogous scheme for the Helmholtz equation with boundary conditions of the first and second kinds was applied by S. V. Nemchinov [93].

L. P. Vinokurov [27], [73], besides rectangular Cartesian and polar coordinates, uses oblique Cartesian, cylindrical, and spherical coordinates, moreover, as

and in an earlier paper [7], sliding interpolation serves him for obtaining schemes. K. E. Chernin [66] in the problem for an infinite body first transformed the solution to cylindrical coordinates, and then, by means of the method of straight lines, found the form of the approximate solution.

One important circumstance should be noted. When the method of straight lines is used for solving elliptic equations, one is usually limited to constructing an approximating scheme and searching for an effective method of solving it, for example the general solution of the scheme or a corresponding homogeneous system of equations is given (see above, and also the papers of A. Langenbach [37], Ya. I. Alikhashkin [42], etc.). Questions of convergence and error estimates, which are of primary importance for any approximate method, are raised comparatively rarely.

The first attempt to prove convergence of the method for Laplace’s equation was undertaken in the paper [6]. The author proved for scheme (9) the maximum principle, consisting in the fact that, everywhere in \(\Pi\), \(u_N(x)\leq 0\), if this holds at the boundary points, while the left-hand sides of (9) are nonnegative in \(\Pi\) and \(u_0(x)=u_N(x)\equiv 0\). But in the subsequent derivations connected with the error estimate he made an error.*) The proof in [15] of an analogous maximum principle for scheme (10), and consequently also the error estimates based on it, is incorrect.

In [22] Poisson’s equation with zero values on a piecewise smooth boundary and with a right-hand side from the class \(W_2^1\) is considered. With the aid of embedding theorems it is asserted that, as \(h\to 0\), the set of approximations given by scheme (9) converges, in any closed subdomain, to a certain function that is unique and is the generalized solution of the problem. The same is true for scheme (10).

An interesting problem is the subject of a series of papers by E. Kh. Kostyukovich. He drew attention to the fact that in many papers ([6], [8], [15], [22], etc.) the authors, without sufficient grounds, regard their results as valid not only for rectangular, but also for trapezoidal domains. Indeed, if one projects the interior sections of the domain by the \(n\)-th straight line onto the \(x\)-axis and denotes the resulting set by \(g_n\), then all \(g_n\) coincide with one another for any number of straight lines only in the case of a suitably oriented rectangle; this means that, in the general case, the equation of the scheme connecting the values on the straight lines \(k=n-p, n-p+1,\ldots,n+q\), without additional definition, is specified only on the intersection

\[ G_n=\bigcap_{-p\leq i\leq q} g_{n+i}. \]

An analogous circumstance occurs also in polar coordinates, which remained unnoticed in [23].

If, however, one regards the equations of the scheme as holding also outside \(G_n\) (this is tacitly done in the papers just enumerated) and, having written the general solution of the scheme, tries from the boundary conditions to determine the constants entering it \(C_i\), then it turns out ([48], [54], [55], [78]) that this is not always possible. It has been proved, for example, that there exist very simple domains in which scheme (9) for the Dirichlet problem leads to a system of algebraic equations for the constants \(C_i\) with zero determinant; moreover, for arbitrarily large \(N\) there exist domains possessing this property.

To overcome these difficulties, E. Kh. Kostyukovich proposes continuously extending the boundary function of the Dirichlet problem \(\varphi(x,y)\) inside the domain and solving the \(n\)-th equation of scheme (9) on \(G_n\), assuming that \(u_n(x)=\varphi(x,y_n)\) for \(x\in g_n-G_n\). In this case it is recommended to solve the approximating scheme iteratively:

\[ u_{n,j}''(x)+\frac{1}{h^2}\,[u_{n-1,j}(x)-2u_{n,j}(x)+u_{n+1,j-1}(x)]=f(x,y_n), \]

\[ 1\leq n\leq N-1,\quad j\geq 1, \]

\[ u_{0,j}(x)\equiv \varphi(x,y_0),\quad u_{N,j}(x)\equiv \varphi(x,y_N),\quad j\geq 0, \]

taking as initial approximations functions that are linear on \(g_n\), \(1\leq n\leq N-1\). If the original solution has a uniformly continuous derivative \(\partial^3 u/\partial y^3\) in the domain, then

\[ \lim_{h\to 0}\ \lim_{j\to\infty}\ \max_{0\leq n\leq N}\ \sup_{x\in g_n} |u(x,y_n)-u_{n,j}(x)|=0 \tag{11} \]

in any trapezoidal domain with continuous lateral sides ([48], [54], [55]). This result remains valid also for an equation more general than (7).

With only continuity of \(d^3u/dy^3\), the construction is complicated by constructing in the domain a contour close to the given one. The scheme is solved inside the domain bounded by the new contour, and under infinite approach of the contours (11) is also fulfilled ([48], [54], [55]). An analogous result is valid for a general linear elliptic equation of the second order in a multidimensional domain different from a right cylinder ([115]).

*) For the class of problems under consideration, the integral error turned out to be of order \(O(h^3)\), whereas the local error is only of order \(O(h^2)\).

§ 3. EQUATIONS OF PARABOLIC TYPE

As has already been noted, E. Rothe [1] first applied the method proposed by him precisely to the parabolic equation

\[ u_{xx}=R(x,t)u_t+S(x,t,u),\qquad R>0,\quad 0\leq x\leq 1,\quad 0<t\leq T. \]

He approximates the equation by the scheme

\[ u''_{n+1}=R(x,t_{n+1})\Delta u_n/h+S(x,t_{n+1},u_n),\qquad t_n=nh,\quad n\geq 0, \]

linear with respect to the desired function \(u_{n+1}(x)\). On the basis of an a priori estimate of type (6), with the aid of which, for \(h_k=T2^{-k}\to0\), it is established that for \(t\leq t_0<T\) there exists a unique classical solution of the problem with zero boundary conditions of the 1st kind. For values of \(t\) of the form \(t^*=mT2^{-k}\) (\(m\) natural), an estimate of order \(O(h)\) was obtained for the error of the method. In a similar way, in [2] the solvability of the equation

\[ \sum_{i,j=1}^{3}(a_{ij}(x)u_{x_i})_{x_i}=R(x,t)u_t+S(x,t),\qquad R>0,\quad x=(x_1,x_2,x_3), \]

under zero boundary conditions of the 1st kind was shown, and the same error estimate for \(t=t^*\).

After these works Rothe’s method was used repeatedly to prove the solvability of linear and nonlinear parabolic equations of the 2nd order with one or several spatial variables. In this, the proof scheme is basically the same as that of E. Rothe: by means of a priori estimates one establishes compactness of the family of approximations as \(h\to0\) in a definite class of functions and, consequently, the existence of a limiting function, which turns out to be the desired solution.

Thus, for the equation

\[ u_t=\sum_{i,j=1}^{n}a_{ij}(x,t,u)u_{x_i x_j} -\sum_{i=1}^{n}a_i(x,t,u)u_{x_i} -a(x,t,u),\qquad 0\leq t\leq T, \]

O. A. Ladyzhenskaya [34], by means of the scheme

\[ \Delta u_{k-1}/h= \sum_{i,j=1}^{n}a_{ij}(x,t_k,u_{k-1})u_{k x_i x_j} -\sum_{i=1}^{n}a_i(x,t_k,u_{k-1})u_{k x_i} - \]

\[ -u_k\int_{0}^{1}a_u(x,t_k,\tau u_{k-1})\,d\tau -a(x,t_k,0),\qquad t_k=kh,\quad k\geq0, \]

proved the existence in \(L_2\) of a solution of the first boundary-value problem with zero data, and, under somewhat stronger requirements, of a smooth (classical) solution (see also [36], [57]). These same equations and the scheme under other conditions were considered by T. D. Venttsel [45], [46]; here, too, the solvability of the first boundary-value problem (“as a whole”) was shown. For homogeneous boundary conditions of the 3rd kind with derivative along the conormal, when \(a_{ij}=a_{ij}(x,t)\), analogous investigations were carried out by Chzhou Yui-lin [49], [50]. The case when the equation is linear was treated by A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik [90].

In [49], [50], with the aid of the scheme

\[ u''_k=A(x,t_k,u_{k-1})\Delta u_{k-1}/h+F(x,t_k,u_k,u'_k) \]

the solvability of the mixed problem for the equation

\[ u_{xx}=A(x,t,u)u_t+F(x,t,u,u_x),\qquad A>0, \]

with initial condition \(u(x,0)=u_0(x)\) and nonlinear boundary conditions

\[ u_x(0,t)=\varphi_1(t,u),\qquad u_x(l,t)=\varphi_2(t,u) \]

or

\[ u_t(0,t)=\psi_1(t,u,u_x),\qquad u_t(l,t)=\psi_2(t,u,u_x). \]

The subject of the paper by O. A. Oleinik, A. S. Kalashnikov, and Chzhou Yui-lin [68] is the equation \(u_t=\varphi_{xx}(x,t,u)\) (see also O. A. Oleinik [80]), while K. Kordunyan’s [77] method—

Rothe proved the existence of a bounded solution of classical mixed problems on the half-line for the equation \(u_{xx}=u_t+f(x,t,u)\).

In the last of the named works, a solution is understood in the classical sense. A priori estimates of the solutions of the scheme are obtained for the most part on the basis of the maximum principle.

Besides these, essentially theoretical, applications, Rothe’s scheme was also proposed as a practical method for solving parabolic equations of the 2nd order, when the solvability of the initial problem in a certain class is known either a priori or from other considerations. Such an application of the scheme may be found, for example, in A. A. Samarskii [65], where uniform convergence of the scheme to the solution of boundary-value problems is shown for the equation \(u_t=(k(x,t)u_x)_x+f(x,t)\) with discontinuous piecewise-smooth \(k(x,t)\), and in K. Inata and N. Kalistru [89], who approximate not only the derivative with respect to \(t\) but also the free term, and for the proof of convergence use Lax’s theorem on its relation to the stability of the scheme. Analogous locally more accurate schemes were considered (but without proof of convergence) by D. R. Hartree and J. R. Womersley [4] and in the work [8].

E. H. Kostyukovich studied the convergence of the Rothe scheme in regions of the form \(\{\varphi_1(t)\le x\le \varphi_2(t),\ 0\le t\le T\}\). It turned out ([55], [79]) that in a nonexpanding region, i.e., in the case where the functions \(\varphi_1(t)\) and \(-\varphi_2(t)\) are nondecreasing in \(t\), it converges for equation (1), but that nothing of this kind can be said for regions of general form. Moreover, in such regions the scheme is not completely determined. By a modification analogous to that described in § 2, it is possible to obtain its convergence for a general parabolic equation of the 2nd order in any region with continuous \(\varphi_1(t)\), \(\varphi_2(t)\) \((\varphi_1(t)<\varphi_2(t))\).

However, in applied questions the greatest development has been achieved by other, so-called longitudinal schemes of the method of straight lines, in which the lines are arranged along the \(t\)-axis and derivatives are approximated not with respect to time but with respect to spatial variables (Rothe’s scheme and those similar to it are called, in contradistinction to this, transverse). The advantage of longitudinal schemes is that they lead to a Cauchy problem for a system of ordinary differential equations, and not to boundary-value problems, as do transverse schemes.

A large number of works on the application of longitudinal schemes to the Cauchy problem for equations of the form

\[ \partial^k u/\partial t^k=\partial^m u/\partial x^m,\qquad k\ge 1,\quad m\ge 2, \tag{12} \]

was devoted by L. I. Kamynin. He studied classes of unique solvability of schemes

\[ d^k u_n/dt^k=\sum_{\alpha=-s}^{r} C_\alpha u_{n+\alpha},\qquad r,\ s\ge 0, \tag{13} \]

specified by growth conditions at infinity ([11], [13], [16]) (for \(k=m\) equation (12) is of hyperbolic type, but, as is shown in [16], this does not change the class of uniqueness of scheme (13)). Schemes of the form (13) with variable \(C_\alpha\) ([18], [32]), with infinite \(r\) and \(s\) ([32]), with nonlinear right-hand side ([31], [32]), and a number of other generalizations were also considered. (Similar investigations were also carried out by G. I. Bass [17] and B. L. Gurevich [28], [29], [47].) For some schemes an explicit solution was constructed in the form of an infinite series ([11], [13], [32]). In [12] convergence as \(h\to 0\) was proved for the scheme

\[ u'_n(t)=\Delta^2 u_{n-1}(t)/h^2,\qquad u_n(0)=\varphi(nh),\quad |n|<\infty, \]

to the solution of the Cauchy problem

\[ u_t=u_{xx},\qquad u(x,0)=\varphi(x),\quad |x|<\infty, \]

in the case where

\[ \varphi(x)=O\bigl(\exp C|x|\ln(1+|x|)\bigr),\qquad |x|\to\infty,\quad C\ge 0, \]

and for a somewhat modified scheme, when

\[ \varphi(x)=O\bigl(\exp C|x|^{2-\delta}\bigr),\qquad |x|\to\infty,\quad C\ge 0,\quad 0<\delta\le 2. \]

Questions concerning the application of longitudinal schemes of the method of straight lines to mixed problems were also developed. In [22], for example, convergence of the approximations almost everywhere in a rectangle to the generalized solution of boundary-value problems with zero conditions of all three types for the simplest nonhomogeneous heat equation is asserted. In doing so, a differential-difference analogue of the energy integral and the embedding theorem are used.

A very important step for applications was taken by B. M. Budak [25], [26]. He not only convinced himself of the convergence of the scheme

\[ \rho_k u'_k(t)=\Delta(\sigma_{k-1}\Delta u_{k-1}(t))/h^2-q_k u_k(t)+f_k(t), \tag{14} \]

\[ u_k(0)=\varphi(kh)=\varphi\!\left(\frac{kl}{n+1}\right),\qquad u_0(t)=\psi_1(t),\qquad u_{n+1}(t)=\psi_2(t), \]

\[ 1\leq k\leq n, \]

to the solution of the problem

\[ \rho(x)u_t=(\sigma(x)u_x)_x-q(x)u+f(x,t),\qquad \rho,\sigma>0,\quad q\geq0,\quad 0\leq x\leq l, \tag{15} \]

\[ u(x,0)=\varphi(x),\qquad u(0,t)=\psi_1(t),\qquad u(l,t)=\psi_2(t),\qquad 0\leq t\leq T, \]

but also, with the aid of the energy integral, gave a uniform error estimate of order \(O(h)\).

The same ideas were repeatedly used later in the study of line schemes for a large number of parabolic problems, including those with discontinuous elements (B. M. Budak [72], [86], [87], V. A. Morozov [119]) and quasilinear ones ([72], [87], [119]). In most cases the convergence rate of the approximations as \(h\to0\) was established ([72], [86], [87], [119]), and sometimes also an estimate of the error yielded by the method (V. A. Morozov [91]).

Other methods of proving convergence and estimating the error are also used. J. Douglas [30], for example, carries out a limiting passage in the error estimate of the grid method; N. E. Friedman [40] expands the exact and approximate solutions in a Fourier series; G. M. Komladze [64] replaces the system of ordinary differential equations for the error by a matrix integral equation, which he solves by successive approximations; R. F. Albrecht [67], for the equation

\[ \sum_{k=1}^{m} a_k(t)\frac{\partial^k u}{\partial t^k} + \sum_{i=0}^{n} b_i(x,t)\frac{\partial^i u}{\partial x^i} = f(x,t),\qquad a_m\equiv1, \]

with general boundary conditions and for comparatively general line schemes, expresses the errors through a fundamental system of solutions, etc.

Transverse schemes of general form

\[ \sum_{i=0}^{m} A_i u_{n+i}(x) = h^k\sum_{i=0}^{m} B_i\bigl(l(u_{n+i})+f(x,t_{n+i})\bigr), \qquad 0\leq n\leq [T/h]-m, \tag{16} \]

with real coefficients \(A_i,B_i\) for the equation

\[ \partial^k u/\partial t^k=l(u)+f(x,t),\qquad 0\leq t\leq T,\qquad k\geq1, \tag{16'} \]

were considered by V. I. Krylov and O. A. Liskovets [116], [125], [118]. The coefficients of the scheme (16) must be chosen so that the scheme would provide an approximation of degree \(r\geq1\) and be stable with stability exponent \(q<k+r\). Then, under a priori assumptions on the solution, using spectral analysis in a Hilbert space, one obtains for the mean-square error \(\|\varepsilon_n\|\), \(n\geq m\), an estimate of order

\[ O\!\left( h^{1-q}\sum_{i=0}^{m-1}\|\varepsilon_i\|+h^{k+r-q} \right) \]

(\(\varepsilon_i(x)\), \(0\leq i\leq m-1\), are the initial errors). Here neither the order of the derivative \(k\), nor the dimension of the space \(x\), nor the order of the operation \(l\) plays a role; it is essential only that the differential operator \(l\), together with the homogeneous boundary conditions corresponding to the problem, define a linear operator of one or more spatial variables, self-adjoint and nonnegative in \(L_2\). If this operator is bounded above by a positive constant, the requirements imposed on the scheme (16) are somewhat increased. The result remains valid also for the Cauchy problem.

A number of new convergent schemes of the form (16), not previously used, were obtained for \(k=1,2\) (see [126], [118]). In addition, it became clear that Rothe’s scheme is suitable for equations of any order.

§ 4. EQUATIONS OF HYPERBOLIC TYPE

As a special case, obtained under sliding interpolation, the schemes of the method of lines for hyperbolic equations are contained among the schemes of the method [3], suitable for equations of any type. In an independent form, however, the method of lines for hyperbolic equations began to be applied and developed considerably later.

Longitudinal schemes of the method were first investigated by L. I. Kamynin. The nature of his results is analogous to those described in § 3, i.e., for an infinite system of ordinary differential equations approximating the Cauchy problem, conditions of unique solvability are derived ([16], [32]) and, with the aid of the Fourier transform, an explicit solution is constructed in the form of a series ([16], [32], [19]). Convergence to the exact solution is not considered here.

By the method indicated in § 3, in [25], [26] a uniform error estimate of first order is obtained for the scheme of type (14) for problems of type (15) (in (14) and (15), instead of first derivatives with respect to \(t\), there are second derivatives and one more initial condition is added). For more general hyperbolic equations the same method is used in [86], where the order of the error is established (first—for discontinuous elements, second—for continuous ones), and in [91], where error estimates of order \(O(h)\) and \(O(h^2)\) are obtained for various schemes.

As for parabolic equations, other ideas and methods were also developed. In [22] the convergence of longitudinal schemes to a generalized solution of the simplest equation of string vibrations is asserted on the basis of embedding theorems. In [64] the differential system is replaced by a matrix integral equation, which is solved by successive approximations. M. Lees [69] estimates the error of a scheme in the first boundary-value problem for the equation

\[ u_{tt}-a(x,t)u_{xx}=F(x,t,u,u_x,u_t) \]

by passage to the limit in an estimate for the mesh method obtained by him with the aid of energy inequalities, and proves uniform convergence of order \(O(h^2)\). The work [67] is also partly concerned with hyperbolic equations.

Transverse schemes of the method of lines were also studied in mixed problems and in the Cauchy problem for hyperbolic equations. Such are, for example, [89], where a boundary-value problem for a second-order equation is considered, and [116], [125], [118], [126], in which the application of schemes of type (16) to hyperbolic equations of type \((16')\) is studied. The methods and results of these works are analogous to those set out in § 3. In general, as is seen from the preceding, almost all the methods and ideas used for parabolic equations (apart from the maximum principle) are also suitable for the study of schemes of the method of lines approximating corresponding problems for hyperbolic equations.

A special group is formed by problems for equations in the (first) canonical form

\[ u_{xy}=f(x,y,u,u_x,u_y). \tag{17} \]

In the theory of parabolic equations there are usually no direct analogues of these problems.

The simplest of such problems—the Goursat–Darboux problem with data on characteristics—for equation (17) without first derivatives was considered in [25], [26]. The scheme

\[ u'_{n+1}(y)-u'_n(y)=h f(x_n,y,u_n),\qquad x_n=x_0+nh, \tag{18} \]

is constructed, and by successive majorization of the solutions of the equations for the error an estimate of the latter of order \(O(h)\) is derived. The same order is possessed by the error estimate for a scheme of type (18) for the linear equation (17) without the derivative \(u_x\), obtained in [64] by successive approximations.

On the basis of the idea of the method of integral relations, V. I. Krylov and V. V. Bobkov [103] constructed a more accurate scheme for equation (17) with linearly entering \(u_x, u_y\).

It is asserted that the result is also valid for an equation of the form

\[ u_{xy}=a(x,y)u_x+f(x,y,u,u_y). \tag{17'} \]

Subsequently V. V. Bobkov [99], [114] constructed \((k+1)\)-layer schemes \((k\geq 1)\) for an equation without first derivatives, whose error has order \(O(h^{k+1})\) (depending on the initial errors). The error is estimated with the aid of an a priori estimate for the solution of a system of ordinary differential equations.

The same method was simultaneously applied by V. I. Krylov and O. A. Liskovets [104], [118] in the study of general schemes

\[ \sum_{i=0}^{m} A_i\left[u'_{n+i}(y)-a(x_{n+i},y)u_{n+i}(y)\right]= \]

\[ = h \sum_{i=0}^{m} B_i \left[f(x_{n+i}, y, u_{n+i}(y), u'_{n+i}(y)) - a_x(x_{n+i}, y) u_{n+i}(y)\right], \]

\[ x_n = x_0 + nh,\qquad m \geqslant 1, \]

for equation \((17')\). In view of the arbitrariness of the real coefficients \(A_i, B_i\), a class of stable schemes is singled out, consisting of those schemes with such \(A_i\) that all the roots of the polynomial

\[ \sum_{i=0}^{m} A_i z^i \]

belong to the unit disk, with only simple roots allowed on its boundary. The error estimates derived for stable schemes and for its first derivative have order \(O(h^r + \varepsilon + \varepsilon')\), where \(r\) is the degree of the residual of the scheme, and \(\varepsilon\) and \(\varepsilon'\) are majorants of the errors and of their first derivatives on the initial lines \(x_i\), \(0 \leqslant i \leqslant m-1\).

A more general problem in the domain \(\{0 \leqslant x \leqslant X,\; g(x) \leqslant y \leqslant Y\}\) with data on the straight line \(x=0\) and on the curve \(y=g(x)\) \((g(0)=0,\; g'(x) \geqslant 0)\) for equation (17) and a scheme obtained by replacing the derivatives with respect to \(x\) by the divided left difference was considered by B. M. Budak and A. D. Gorbunov [51]. The solution of the scheme makes it possible, in addition to the desired function itself, to approximate both of its first derivatives. The total error of these approximations was estimated and has first order.

The same problem, as a special case of Picard’s problem, was studied in papers by V. V. Bobkov [113] and O. A. Liskovets [117] (see also [114] and [118]). The methods and results here are the same as in the publications mentioned above [99], [103], and [104]. These same authors investigated the application of two-layer straight-line schemes to the solution of the Cauchy problem and some of its generalizations for hyperbolic equations of canonical form ([112], [114] and [117], [118]).

§ 5. EQUATIONS OF MIXED TYPE, SYSTEMS OF EQUATIONS. INTEGRO-DIFFERENTIAL AND OPERATOR EQUATIONS

The method of straight lines is also applied to equations of mixed type. Thus, I. A. Kodachigov [76] constructed the general solution of the method’s scheme for the Lavrent’ev–Bitsadze equation. Similar investigations were also carried out for schemes obtained by means of the method of integral relations.

A number of works are devoted to systems of differential equations. The parabolic system

\[ \partial u_k/\partial t - (\alpha_k + i \beta_k) \sum_{j=1}^{n} \partial^2 u_k/\partial x_j^2 = f(x,t),\qquad \alpha_k > 0,\qquad 1 \leqslant k \leqslant m, \]

with boundary conditions of the third kind was considered by V. I. Lebedev [21]. As in the previously mentioned work [22], here for the longitudinal scheme a differential-difference energy integral is introduced and, on the basis of embedding theorems, convergence of the scheme to a generalized solution of the problem from the class \(W_2^1\) is proved. T. D. Venttsel’ [44], [46] established, by the Rothe method, solvability of the first boundary-value problem for the system

\[ \bar{u}_t = \bar{u}_{xx} + B(x,t,\bar{u})\bar{u}_x + C(x,t,\bar{u})\bar{u} + \bar{f}(x,t) \]

(\(B, C\) are square matrices). In the third boundary-value problem for the simplest parabolic system E. A. Grigorieva [52] proved uniform convergence of the Rothe scheme to the exact solution, starting from a correct (in Sobolev’s sense) approximation of the difference operator by an integral operator with a kernel in the form of Green’s function.

Hyperbolic systems of equations were studied in [26] by the same methods as were individual equations. In the Goursat–Darboux problem for the system \(\bar{u}_{xy} = A(x,y)\bar{u} + \bar{f}(x,y)\) (\(A\) is a square matrix) and in the first boundary-value problem for the system

\[ R(x)\bar{u}_{tt} - (S(x)\bar{u}_x)_x + Q(x)\bar{u} = \bar{f}(x,t) \]

with positive definite matrices \(R, S\) and a nonnegative definite matrix \(Q\), uniform estimates of order \(O(h)\) were derived for the approximations given by schemes of types (18) and (14), respectively. For the generalized system of telegraph equations

\[ L\bar{i}_y + R\bar{i} = -\bar{v}_x,\qquad C\bar{v}_y + G\bar{v} = -\bar{i}_x \]

(the matrices \(L, C\) are symmetric positive definite, \(R, G\) are nonnegative definite), with prescribed \(\bar{v}(0,y)\), \(\bar{v}(x,0)\), \(\bar{i}(x,0)\), and \(\bar{i}(l,y)\), a uniformly

a convergent scheme of the method of straight lines, for which the error estimate has order \(O(\sqrt h)\). The same results are reproduced in [43].

Systems of equations of the form

\[ \frac{\partial^2 u_i}{\partial x \partial y} = \sum_{k=1}^{s} a_{ik}(x,y)\,\frac{\partial u_k}{\partial x} + F_i\left(x,y,u_1,\ldots,u_s,\frac{\partial u_i}{\partial y}\right), \qquad 1\leqslant i\leqslant s, \]

and a number of problems for them, including the Goursat–Darboux, Picard, and Cauchy problems, are considered in the works already mentioned [99], [103], [112]—[114] and [104], [117], [118]. The essence of the methods applied here and of the results obtained does not differ from those described in § 4.

There are also other works on the use of the method of straight lines for solving hyperbolic systems of equations. Let us name two more works: [55], where, for a system of first-order linear equations, convergence of the scheme to a generalized or smooth solution is shown, and [98] by N. I. Chelnokov, where a system of 2 nonlinear equations is solved.

The method of straight lines turns out to be applicable not only to purely differential problems. Thus, K. Yu. Yosupov [24] proposes using it to solve integro-differential equations of second order of the hyperbolic type with constant coefficients, and derives the form of the solutions of the transverse scheme on successive straight lines, and also gives an expression for the error from which it is not difficult to obtain its uniform first-order estimate.

Still more general—operator equations—were considered by O. A. Ladyzhenskaya. For example, for the operator Cauchy problem

\[ du/dt + (S_1(t)+S_2(t))u = f(t),\qquad 0\leqslant t\leqslant l;\quad u(0)=\varphi, \tag{19} \]

where \(u(t)\) is a function of \(t\) with values in a Hilbert space \(H\), \(S_1\) is a linear unbounded self-adjoint positive definite operator in \(H\) with parameter \(t\), and the linear operator \(S_2\) is subordinate to it in a certain sense, in [33] Rothe’s scheme is constructed

\[ \Delta u_k/h + (S_1(t_{k+1})+S_2(t_{k+1}))u_{k+1}=f(t_{k+1}),\qquad u_0=\varphi\quad (t_k=kh,\ k\geqslant 0), \]

and, with the aid of an inequality of energy type, the solvability of problem (19) and the convergence of the approximations to its generalized solution are proved. (Analogous results for equation (19) with nonlinear free term \(f(u,t)\) were subsequently obtained by Sh. I. Ibragimov [10]. The scheme, in his notation, has the form \(f(u_k,t_k)\).)

Here, too, an equation of Schrödinger type is considered

\[ du/dt + iS(t)=f(t),\qquad 0\leqslant t\leqslant l, \]

where \(S(t)\) is a self-adjoint operator with domain of definition independent of \(t\). By the substitution

\[ \Delta u_k/h+\frac{1}{2}iS(t_{k+1})(u_k+u_{k+1})=f(t_{k+1}),\qquad u_0=u(0)\quad (t_k=kh,\ k\geqslant 0), \]

the existence of a generalized solution of the Cauchy problem is proved (see also [20] and [35]).

By the same method, in [33] solvability in the generalized sense is shown for the Cauchy problem for a “hyperbolic” operator equation analogous to (19) (with \(u_t\) replaced by \(u_{tt}\)), and the possibility of generalizations is asserted. Most fully, the latter are carried out in [56], where, in particular, the solvability of the Cauchy problem for the linear equation

\[ \sum_{k=0}^{m} S_k(t)\frac{d^k u}{dt^k}=f(t),\qquad m\geqslant 1, \]

is investigated.

In this case, each time the approximating scheme is constructed depending on the properties and subordination of the operators \(S_k\) in such a way that some inequality of energy type holds for it.

S. G. Krein and O. I. Prozovskaya [102] use the explicit difference scheme

\[ \Delta u_{k-1}/h=Au_{k-1}\qquad (1\leqslant k\leqslant N,\quad Nh=T),\qquad u_0=v_0, \tag{20} \]

for solving the Hadamard-ill-posed problem of the inverse heat-conduction type

\[ v'(t)=Av(t)\qquad (0\leqslant t\leqslant T),\quad v(0)=v_0, \]

with a self-adjoint operator \(A>0\) in a Hilbert space. In a certain class of correctness, convergence of the scheme to the desired solution is proved, and a uniform...

for \(t \in [0,T_1]\subset[0,T]\) an error estimate of order \(O(h)\). For a more general operator \(A\) from a Banach space, convergence has been proved both for scheme (20) and for the analogous longitudinal scheme of the method of lines

\[ v_n'(t)=A_n v_n(t), \qquad v_n(0)=v_n^0, \]

provided \(A_n\) is properly consistent with \(A\) and \(v_n^0\) with \(v_0\).

Finally, for problems

\[ u^{(k)}(t)=L(t)u(t)+f(t) \quad (0\leqslant t\leqslant T), \qquad u^{(i)}(0)=\varphi_i \quad (0\leqslant i\leqslant k-1), \]

the results of papers [116], [118], [125], [126] stated earlier remain valid for general schemes of the form

\[ \sum_{i=0}^{m} A_i u_{n+i} = h^k \sum_{i=0}^{m} B_i \bigl(L_{n+i}u_{n+i}+f_{n+i}\bigr), \]

provided only that the operator \(L(t)\) has the required properties, since its differentiability is nowhere required in the arguments.

Before concluding, we note that brief surveys of works on the method of lines are contained in the papers of M. I. Vishik, A. D. Myshkis and O. A. Oleinik ([61], § 4, 5), and of M. K. Gavurin and L. V. Kantorovich ([62], § 12), and a special paragraph is devoted to this method in the textbook of I. S. Berezin and N. P. Zhidkov [60].

In view of the existence of recent surveys by O. M. Belotserkovskii and P. I. Chushkin [85], V. V. Bobkov and V. I. Krylov [123], this article and its bibliography mention almost no works in which the construction of an approximating differential-difference scheme is based on the idea of the method of integral relations. Some works were not reflected in the article also because their main emphasis is not on investigations in which the convergence of the method is established.

In conclusion the article gives a bibliography on the method of lines. The works in it are arranged in chronological order, and within a single year—in alphabetical order.* The reports are assigned according to the time of their presentation, not according to the time of publication. Some works on the subject considered, apparently, have remained unknown to me. I hope that their number is not very large.

I express my deep gratitude to V. V. Bobkov and O. A. Ladyzhenskaya, who read the article in manuscript, and also to V. T. Ivanov and E. Kh. Kostyukovich. I express special gratitude to V. I. Krylov for his constant attention and useful advice, on whose initiative this article was written.

References

1. Rothe E. Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math. Ann., 102, 650—670, 1930.

2. Rothe E. Über die Wärmeleitungsgleichung mit nichtkonstanten Koeffizienten im räumlichen Falle. Math. Ann., 104, H. 3, 340—362, 1931.

3. Kantorovich L. V. On one method of approximate solution of differential equations in partial derivatives. Dokl. Akad. Nauk SSSR, 2, No. 9, 532—536, 1934.

4. Hartree D. R., Womersley J. R. A Method for the Numerical or Mechanical Solution of Certain Types of Partial Differential Equations. Proc. of Royal Soc. (London), Ser. A, 161, No. 906, 353—366, 1937.

5. Slobodyanskii M. G. A method of approximate integration of equations with partial derivatives and its application to problems of the theory of elasticity. PMM, 3, issue 1, 75—81, 1939.

6. Slobodyanskii M. G. Spatial problems of the theory of elasticity for prismatic bodies. Uchen. zap. MGU, issue 39. Mechanics, 103—144, 1940.

7. Vinokurov L. P. An approximate method for solving plane problems of the theory of elasticity. Tr. Kharkov. inzh.-str. in-ta, issue 2, 47—123, 1949.

8. Faddeeva V. N. The method of lines in application to certain boundary-value problems. Tr. Mat. in-ta im. V. A. Steklova, 28, 73—103, 1949.

9. Mikhlin S. G. Direct methods in mathematical physics. GITTL, ch. 7, 1950.

10. Smirnov V. I. A course of higher mathematics, 4. GITTL, 737—739, 1951.

* Works beginning with [129] have been added to the list additionally.

1. Kamynin L. I. Various uniqueness theorems for the heat-conduction equation and systems of finite-difference differential equations. DAN SSSR, 82, No. 1, 13—16, 1952; On the applicability of the method of finite differences to the solution of the heat-conduction equation. I. Uniqueness of the solution of a system of finite-difference equations. Izv. AN SSSR, ser. matem., 17, No. 2, 163—180, 1953.

2. Kamynin L. I. On the convergence of the finite-difference process for the heat-conduction equation. DAN SSSR, 85, No. 4, 701—704, 1952; On the applicability of the method of finite differences to the solution of the heat-conduction equation. II. Convergence of the finite-difference process for the heat-conduction equation. Izv. AN SSSR, ser. matem., 17, No. 3, 249—268, 1953.

3. Kamynin L. I. Construction of an explicit solution of an infinite system of ordinary differential equations with constant coefficients. DAN SSSR, 93, No. 3, 397—400, 1953.

4. Tikhonov A. N., Samarskii A. A. Equations of mathematical physics. GITTL, 1953, pp. 494—502.

5. Yunusov K. S. Application of the method of straight lines to the solution of boundary-value problems for partial differential equations. Abstract of Cand. Sci. dissertation, Kazan Univ., Kazan, 1953; Uch. zap. Kazan. ped. in-ta, phys.-math. fac., issue 10, 57—83, 1955.

6. Kamynin L. I. On one defect of the method of straight lines. DAN SSSR, 95, No. 1, 13—16, 1954.

7. Bass G. I. Formulas for solving the Cauchy problem for certain differential-difference equations. DAN SSSR, 100, No. 4, 613—616, 1955.

8. Kamynin L. I. On the uniqueness of the solution of the Cauchy problem in the class of rapidly increasing functions for an infinite system of ordinary differential equations. DAN SSSR, 103, No. 4, 545—548, 1955.

9. Kamynin L. I. On the behavior of the solution of a finite-difference analogue of the wave equation. PMM, 19, No. 5, 589—598, 1955.

10. Ladyzhenskaya O. A. On the solution of nonstationary operator equations of various types. DAN SSSR, 102, No. 2, 207—210, 1955.

11. Lebedev V. I. On one system of parabolic equations. DAN SSSR, 103, No. 5, 763—766, 1955.

12. Lebedev V. I. Equations and convergence of the differential-difference method (the method of straight lines). Vestn. Mosk. un-ta, ser. phys.-math. and natural sciences, No. 10, 47—57, 1955.

13. Ustinova N. N. On one modification of the method of straight lines. Uch. zap. Kazan. un-ta, 115, book 14 (Mathematics), 159—167, 1955.

14. Yusupov K. Yu. An approximate method for solving one class of integro-differential equations. Uch. zap. Kazan. ped. in-ta, phys.-math. fac., issue 10, 85—108, 1955.

15. Budak B. M. On the method of straight lines for certain boundary-value problems. DAN SSSR, 109, No. 1, 9—12, 1956.

16. Budak B. M. On the method of straight lines for certain boundary-value problems. Vestn. Mosk. un-ta, ser. matem., No. 1, 3—11, 1956.

17. Vinokurov L. P. Direct methods for solving spatial and contact problems for arrays of foundations, ch. 1. Publ. Kharkov Univ., Kharkov, 1956.

18. Gurevich B. L. New types of spaces of basic and generalized functions and Cauchy problems for systems of differential-difference equations. DAN SSSR, 108, No. 6, 1001—1003, 1956.

19. Gurevich B. L. New types of spaces of basic and generalized functions and classes of uniqueness of the generalized Cauchy problem. Tr. 3rd All-Union Math. Congress, 1, Publ. AN SSSR, Moscow, 1956, p. 114.

20. Douglas Jim, Jr. On the Errors in Analogue Solutions of Heat Conduction Problems. Quart. Appl. Math., 14, No. 3, 333—335, 1956.

21. Kamynin L. I. On the Cauchy problem for an infinite system of ordinary differential equations. DAN SSSR, 109, No. 3, 446—449, 1956.

22. Kamynin L. I. Solution of the Cauchy problem for an infinite system of ordinary differential equations. Vestn. Mosk. un-ta, ser. phys.-math. and natural sciences, No. 6, 3—10, 1956.

23. Ladyzhenskaya O. A. On the solution of nonstationary operator equations. Matem. sb., 39 (81): 4, 1956, pp. 491—524.

24. Ladyzhenskaya O. A. The first boundary-value problem for quasilinear parabolic equations. DAN SSSR, 107, No. 5, 636—639, 1956.

25. Ladyzhenskaya O. A. The method of finite differences in the theory of partial differential equations. Tr. 3rd All-Union Math. Congress, 2. Publ. AN SSSR, Moscow, 1956, pp. 13—16; UMN, 12, issue 5(77), 123—148, 1957.

26. Ladyzhenskaya O. A. The first boundary-value problem for quasilinear parabolic equations and the Cauchy problem for quasilinear hyperbolic equations as a whole. Tr. 3rd All-Union Math. Congress, 4. Publ. AN SSSR, Moscow, 1959, pp. 29—32.

1. Langenbach A. Approximate solution of the biharmonic equation in the case of a trapezoidal domain. Vestn. Leningr. Univ., Ser. Math., Mech., Astr., No. 13, issue 3, 119—123, 1956.

2. Misztal F. Metoda przybliżonego rozwiązywania równań różniczkowych cząstkowych. Zastosowania mat., 2, No. 4, 416—425, 1956.

3. Myshkis A. D., Abolina V. E., Zhdanovich V. F., Kostyukovich E. Kh., Letin A. Ya., Kharitonenko P. I., Shlopak A. S. A mixed problem for linear hyperbolic systems in the plane. Proceedings of the 3rd All-Union Mathematical Congress, 1. Publishing House of the Academy of Sciences of the USSR, Moscow, 1956, pp. 61—63.

4. Friedmann N. E. The Truncation Error in a Semi-discrete Analog of the Heat Equation. J. Math. and Phys., 35, No. 3, 299—308, 1956.

5. Chistyakova E. M. Application of the finite-difference method to the solution of mixed problems for certain hyperbolic and parabolic equations and systems. Candidate dissertation, Pedagogical Institute named after M. Pokrovskii. Leningrad, 1956.

6. Alikhashkin Ya. I. Solution of the problem on an imperfect well by the method of straight lines. Computational Mathematics, collection 1, 1957, pp. 136—152.

7. Budak B. M. On the method of straight lines for certain boundary-value problems for systems of partial differential equations. Dokl. Akad. Nauk SSSR, 112, No. 2, 187—190, 1957.

8. Ventsel T. D. On certain quasilinear parabolic systems. Dokl. Akad. Nauk SSSR, 117, No. 1, 21—24, 1957.

9. Ventsel T. D. The first boundary-value problem and the Cauchy problem for a quasilinear parabolic equation with many spatial variables. Matem. sb., 41(83): 4, 1957, pp. 499—520.

10. Ventsel T. D. On certain quasilinear equations and systems. Author’s abstract of Candidate dissertation, Moscow State University, Moscow, 1957.

11. Gurevich B. L. New spaces of basic and generalized functions and the Cauchy problem for certain operator equations. Author’s abstract of Candidate dissertation, Kharkov State University, Kharkov, 1957.

12. Kostyukovich E. Kh. On the convergence of the method of straight lines under various schemes of its application to the solution of the first boundary-value problem for elliptic-type differential equations. Scientific Notes of Grodno Pedagogical Institute, Ser. Math., issue 2, 60—78, 1957.

13. Chzhou Yu-lin. Boundary-value problems for nonlinear parabolic equations. Dokl. Akad. Nauk SSSR, 117, No. 2, 195—198, 1957; Matem. sb., 47(89): 4, 1959, pp. 431—484.

14. Chzhou Yu-lin. Boundary-value problems for nonlinear parabolic and elliptic equations. Author’s abstract of Candidate dissertation, Moscow State University, Moscow, 1957.

15. Budak B. M., Gorbunov A. D. The method of straight lines for solving one nonlinear boundary-value problem in a domain with curvilinear boundary. Dokl. Akad. Nauk SSSR, 118, No. 5, 858—861, 1958; Vestn. Mosk. Univ., Ser. Math., No. 3, 3—12, 1958.

16. Grigorieva E. A. The method of straight lines in mixed problems for parabolic systems. Dokl. Akad. Nauk SSSR, 119, No. 4, 648—651, 1958.

17. Kaminin L. I. Mbi klasën e unicitetit të zgjidhjes se problemit Koshi ekuacionin parabolik analog me diferenca të fundme. Bul. Univ. shtetër. Tiranës. Ser. shkenc. natyr., 12, No. 2, 22—28, 1958.

18. Kostyukovich E. Kh. On the convergence of the method of straight lines under various schemes of its application to the solution of certain boundary-value problems. Dokl. Akad. Nauk SSSR, 118, No. 3, 433—435, 1958.

19. Kostyukovich E. Kh. On the convergence of the method of straight lines under various schemes of its application to the solution of boundary-value problems of mathematical physics in the plane. Candidate dissertation, Belarusian State University, Minsk, 1958.

20. Ladyzhenskaya O. A. On nonstationary operator equations and their applications to linear problems of mathematical physics. Matem. sb., 45(87): 2, 1958, pp. 123—158.

21. Ladyzhenskaya O. A. Solution in the large of the first boundary-value problem for quasilinear parabolic equations. Tr. MMO, 7, 1958, pp. 149—176.

22. Oleinik O. A., Kalashnikov A. S., Chzhou Yu-lin. The Cauchy problem and boundary-value problems for equations of the nonstationary filtration type. Izv. Akad. Nauk SSSR, Ser. Math., 22, No. 5, 667—704, 1958.

23. Rakhmatullina L. F. On the question of applying the method of straight lines. Scientific Notes of Udmurt Pedagogical Institute, issue 12, 65—68, 1958.

24. Berezin I. S., Zhidkov N. P. Methods of Computation, 2. Fizmatgiz, Moscow, 1959, Ch. 10, § 8.

25. Vishik M. I., Myshkis A. D., Oleinik O. A. Partial differential equations. Mathematics in the USSR over 40 years, 1, Fizmatgiz, Moscow, 1959, pp. 563—636.

26. Gavurin M. K., Kantorovich L. V. Approximate and numerical methods. Ibid., pp. 809—855.

27. Kärnhoḷm Gunnar. A Flow Problem Solved by Strip Method. Chalmers. Tekn. Högskol. Handl., No. 208, 1959.

1. Komladze G. M. Solution of certain problems of mathematical physics by the method of straight lines. Tr. Tbilissk. Ped. Inst., 14, 1959, pp. 75–117.

2. Samarskii A. A. On the convergence of the Rothe method for the heat-conduction equation with a discontinuous coefficient of thermal conductivity. Nauchn. dokl. vyssh. shk., Fiz.-matem. nauki, No. 1, 48–53, 1959.

3. Chernin K. E. Solution of one axisymmetric problem by the method of straight lines. Tr. Matem. Inst. im. V. A. Steklova, 53, 1959, pp. 302–306.

4. Albrecht Rudolf F. Approximation to the Solution of Partial Differential Equations by the Solutions of Ordinary Differential Equations. Numer. Math., 2, No. 4, 245–262, 1960.

5. Venttsel’ T. D. On a problem with a free boundary for the heat-conduction equation. DAN SSSR, 131, No. 5, 1000–1003, 1960.

6. Lees Milton. Energy Inequalities for the Solution of Differential Equations. Trans. Amer. Math. Soc., 94, No. 1, 58–73, 1960.

7. Maruashvili T. I. The method of straight lines for an equation of parabolic type. Tr. VTs AN GruzSSR, 1, 215–230, 1960.

8. Rozen N. V. The method of straight lines for the problem of the flow of air over a mountain ridge. DAN BSSR, 4, No. 10, 409–412, 1960.

9. Budak B. M. On the method of straight lines for certain quasilinear boundary-value problems of parabolic type. ZhVM i MF, 1, No. 6, 1105–1112, 1961.

10. Vinokurov L. P. A method of approximate calculation of the stability of plates loaded by forces in the middle plane. Tr. Kharkovsk. Aviats. Inst., issue 18, 3–18, 1961.

11. Kamynin L. I. On the stability of difference parabolic equations. DAN SSSR, 136, No. 6, 1287–1290, 1961.

12. Karp V. N. Application of the method of straight lines to the solution of boundary-value problems of mathematical physics. Nauchn. zap. Odessk. Politekhn. Inst., issue 34, 34–38, 1961.

13. Kodachigov I. A. Solution of the Darboux–Goursat equation by the method of straight lines. Uchen. zap. kafedry matem. Taganrogsk. Ped. Inst., 51–58, 1961.

14. Corduneanu C. Approximation des solutions d’une équation parabolique dans un domaine non borné. Mathematica (RPR), 3, No. 2, 217–224, 1961.

15. Kostokovich E. Kh. Investigation of nonrectangular contours in connection with the application of the method of straight lines to the solution of the first boundary-value problem. Uchen. zap. Grodnensk. Ped. Inst., issue 5, 31–41, 1961.

16. Kostokovich E. Kh. Solution by the method of straight lines of a mixed problem for parabolic equations. Ibid., pp. 43–54.

17. Oleinik O. A. On the convergence of certain difference schemes. DAN SSSR, 137, No. 3, 523–526, 1961.

18. Petrov Yu. P. Fundamentals of calculating the bending of plates by a discrete method. Tr. Kharkovsk. Aviats. Inst., issue 18, 67–84, 1961.

19. Petrov Yu. P. Calculation of the bending of elastic rectangular plates by a discrete method. Ibid., pp. 85–101.

20. Petrov Yu. P. Calculation of the bending of plates with a linear variation of thickness by a discrete method. Ibid., pp. 103–115.

21. Shirinov A. N. Solution of equations of hydromechanics by the method of finite differences. Nauchn. zap. Odessk. Politekhn. Inst., 34, 78–85, 1961.

22. Belotserkovskii O. M., Chushkin P. I. A numerical method of integral relations. ZhVM i MF, 2, No. 5, 731–759, 1962.

23. Budak B. M. On homogeneous differential-difference schemes of the 2nd order of accuracy for parabolic and hyperbolic equations with discontinuous coefficients. DAN SSSR, 142, No. 5, 986–989, 1962; Vestn. Mosk. Univ., Ser. I, No. 2, 7–13, 1962.

24. Budak B. M. On the method of straight lines for solving certain quasilinear boundary-value problems of parabolic type with discontinuous data. Vestn. Mosk. Univ., Ser. I, No. 3, 3–8, 1962.

25. Buledza A. V. On the application of the method of straight lines to the solution of certain boundary-value problems for the Poisson equation. Dokl. i soobshch. Uzhgorodsk. Univ., Ser. Fiz.-matem. i istor. nauk, No. 5, 92–96, 1962.

26. Ignat C., Calistru N. La méthode des droites pour la résolution approximative de certains problèmes à valeurs initiales. An stiint. Univ. Jasi, Sek. 1, 8, No. 1, 85–96, 1962.

27. Il’in A. M., Kalashnikov A. S., Oleinik O. A. Linear equations of the 2nd order of parabolic type. UMN, 17, issue 3, 3–146, 1962.

28. Morozov V. A. Convergence of the method of straight lines for certain boundary-value problems. Vestn. Mosk. Univ., Ser. I, No. 5, 25–33, 1962.

29. Murray W. D., Landis Fred. The Effect of Spacewise Lumping on the Solution Accuracy of the One-dimensional Diffusion Equations. Trans. ASME, E 29, No. 4, 629–636, 1962.

30. Nemchinov S. V. On the solution by the non-iterative grid method of boundary-value problems

for partial differential equations with periodic boundary conditions. DAN SSSR, 146, No. 6, 1263—1266, 1962.

1. N. I. Chenin, S. V. On the solution by the grid method of problems for second-order partial differential equations of elliptic type. ZhVM i MF, 2, No. 3, 418—436, 1962.

2. E. O. Omarov. Solution of the equation for the bending of a rectangular plate with 4 clamped sides. Reports of the 2nd Siberian Conf. on Math. and Mech., TTU. Tomsk, 1962, pp. 42—44.

3. E. O. Omarov. Application of the method of straight lines to the solution of boundary-value problems for a single partial differential equation of elliptic type in the case of a trapezoidal domain. Ibid., pp. 44—46; Approximate solution by the method of straight lines of a single fourth-order partial differential equation of elliptic type in the case of a trapezoidal domain. Proc. scientific postgraduate conf. for 1962 (abstracts). Kazan, 1962, pp. 153—158.

4. F. Thompson. Analysis and prediction of weather by numerical methods. IL, 1962, ch. 9.

5. N. I. Chelnokov. Application of the method of straight lines for integrating equations of unsteady, slowly varying motion in open watercourses with the aid of computing devices. Tr. MЭI, issue 41, 1962, pp. 173—185.

6. V. V. Bobkov. On solving Goursat’s problem by the method of integral relations. Vestsi AN BSSR, ser. phys.-techn., No. 4, 14—24, 1963.

7. B. M. Budak, F. P. Vasil’ev. Convergence and error estimate of the method of straight lines for solving certain filtration problems. Collected works of the Computing Center of Moscow University, issue 2, 1963, pp. 211—238.

8. Yu. M. Ibrahimov, Sh. I. Approximate solution of the Cauchy problem for an evolution equation with an unbounded operator. DAN AzSSR, 19, No. 11, 9—14, 1963.

9. S. G. Krein, O. I. Prozovskaya. On approximate methods for solving ill-posed problems. ZhVM i MF, 3, No. 1, 120—130, 1963.

10. V. I. Krylov, V. V. Bobkov. On the method of integral relations for the Goursat problem. DAN BSSR, 7, No. 7, 433—438, 1963.

11. V. I. Krylov, O. A. Liskovets. Error estimate of the method of straight lines for the Goursat problem. DAN BSSR, 7, No. 8, 505—508, 1963.

12. E. O. Omarov. Approximate solution by the method of straight lines of the Dirichlet problem for a partial differential equation of elliptic type. Izv. AN UzSSR, ser. phys.-math. sciences, No. 1, 21—25, 1963; Problems of differential equations and mechanics of mining rocks. Tr. Sect. Math. and Mech. AN KazSSR, 2, 45—48, 1963.

13. E. O. Omarov. Approximate solution of the partial differential equation of the form \(d^4u/dx^4 + a d^4u/dx^2dy^2 + b d^4u/dy^4 = f(x, y)\) in the case of a trapezoidal domain by the method of straight lines. Izv. AN UzSSR, ser. phys.-math. sciences, No. 4, 33—38, 1963.

14. E. O. Omarov. Application of the method of straight lines to the solution of boundary-value problems for certain partial differential equations. Author’s abstract of Cand. diss., KGU. Kazan, 1963.

15. H. H. Rosenbrock. Some General Implicit Processes for the Numerical Solution of Differential Equations. Comput. J., 5, No. 4, 329—330, 1963.

16. E. I. Sarmin, L. A. Chudov. On the stability of numerical integration of systems of ordinary differential equations arising in the application of the method of straight lines. ZhVM i MF, 3, No. 6, 1122—1125, 1963.

17. Kare Hellan. Application of a Numerical Procedure to the Analysis of thin Rectangular Plates of Variable Thicknesses. Acta polytechn. scand. Sivil Engng and Build. Constr. Ser., 1963, No. 16.

18. A. B. Bartman, E. I. Berezovsky, N. G. Kondrashev, V. B. Ryvkin. On the solution of linear heat-transfer problems with variable coefficients by approximating them by piecewise-constant ones. 2nd All-Union Conf. on Heat and Mass Transfer, Institute of Heat and Mass Transfer. Minsk, 1964, report 7—3.

19. V. V. Bobkov. On the question of convergence of the method of integral relations. DAN BSSR, 8, No. 1, 5—9, 1964.

20. V. V. Bobkov. Error estimates of the method of integral relations in solving certain problems for hyperbolic equations. Vestsi AN BSSR, ser. phys.-techn., No. 1, 18—22, 1964.

21. V. V. Bobkov. The method of integral relations for equations and systems of hyperbolic type. Cand. diss., BSU. Minsk, 1964.

22. E. Kh. Kostyukovich. Application of the method of planes to the solution of the first boundary-value problem for an elliptic equation with many variables. 1st Belarusian Math. Conf. (abstracts). Minsk, 1964, 58—59; Justification of the method of planes in solving the first boundary-value problem for an elliptic equation with many spatial variables. Izv. vuzov, Mat., No. 4, 64—74, 1965.

23. V. I. Krylov, O. A. Liskovets. The method of “straight lines” for nonstationary mixed problems and an estimate of the mean-square error. DAN BSSR, 8, No. 6, 353—356, 1964.

1. Liskovets O. A. Solution of problems for hyperbolic equations by the method of straight lines. Error estimates. DAN BSSR, 8, No. 10, 623—626, 1964.

2. Liskovets O. A. Error estimates and investigation of the convergence of certain variants of the method of straight lines (hyperplanes) for nonstationary problems. Candidate dissertation, BSU, Minsk, 1964.

3. Morozov V. A. On difference-differential schemes of second-order accuracy for quasi-linear problems of parabolic type with discontinuous elements. Vestn. Mosk. un-ta, ser. I, No. 2, 12—22, 1964.

4. Nemchinov S. V., Libov S. L. A straight-line method of increased accuracy for solving boundary-value problems for the Helmholtz equation on a grid of points in a rectangle. ZhVM i MF, 4, No. 4, 771—773, 1964.

5. Omarov E. O. Approximate solution by the method of straight lines of a single partial differential equation of elliptic type. I. Ibid., No. 3, 585—592.

6. Pavlyuk I. A. Asymptotic images of solutions (by the method of straight lines) of a mixed problem for a hyperbolic equation. DAN URSR, No. 6, 726—729, 1964.

7. Bobkov V. V., Krylov V. I. The method of integral relations for equations and systems of hyperbolic type. (Survey of studies of convergence and error estimates.) Differential Equations, 1, No. 2, 230—243, 1965.

8. Iskra K. K. On one scheme of the method of straight lines. Abstracts of reports of the scientific session dedicated to the 25th anniversary of the Grodno Pedagogical Institute, 107—110, 1965.

9. Liskovets O. A. The method of hyperplanes in nonstationary problems with a self-adjoint operator. Differential Equations, 1, No. 2, 255—259, 1965.

10. Liskovets O. A. The method of straight lines for one-dimensional nonstationary mixed problems and an estimate of the mean-square error. ZhVM i MF, 5, No. 2, 360—363, 1965.

11. Khramov N. E. Calculation of the flow around a sphere by a nonuniform gas flow. PMM, 29, issue 1, 175—177, 1965.

12. Chushkin P. I., Li-kan. Determination of the parameters of supersonic two-dimensional gas flows. ZhVM i MF, 5, No. 1, 57—66, 1965.

13. Paschkis V. and Baker H. D. A Method for Determining Unsteady State Heat Transfer by Means of an Electrical Analogy. Trans. ASME, 64, 105—112, 1942.

14. Crank J., Nicolson P. A. Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of Heat-Conduction Type. Proc. Cambr. Phil. Soc., 43, 1, 50—67, 1947.

15. Albasiny E. L. The Solution of Non-Linear Heat-Conduction Problems on the Pilot Ace, Proc. Instn Electr. Engrs, B 103, Suppl. No. 1, 158—162, 1956.

16. Vinokurov L. P. Solution of a plane problem of the theory of elasticity by a discrete method, with determination of the displacements in the form of a sum of principal and additional displacements. Tr. Khar’kovsk. inzh.-stroit. in-ta, issue 8, 1958, pp. 3—19.

17. Grigoryants N. I. Calculation of rod systems under the action of a load varying arbitrarily in time. Ibid., pp. 61—83.

18. Hartree D. R. A Method for the Numerical Integration of the Linear Diffusion Equation. Proc. Cambridge Philos. Soc., 54, No. 2, 207—213, 1958.

19. Matyáš J. Successive Approximations Process in Analog Solution of Partial Differential Equations by Difference Method. ZAMM, 40, No. 10—11, 488—493, 1960.

20. Lovass-Nagy V. Über die Anwendung der Matrizenrechnung zur Berechnung zweidimensionaler Temperaturfelder. ZAMM, 42, No. 3, 110—119, 1962.

21. Landau H. G. The Accuracy of the Analog Solution of Heat Conduction Problems. J. Soc. Industr. and Appl. Math., 11, No. 3, 564—578, 1963.

22. Dem’yashkina E. Ya. Application of the method of straight lines to the approximate solution of one equation. Reports of the 3rd Siberian Conference on Mathematics and Mechanics, TSU, Tomsk, 1964, p. 101.

23. Kucherenko E. I. The Galerkin–straight-line method for solving boundary-value problems. Ibid., p. 127.

24. Wadsworth M., Wragg A. The Numerical Solution of the Heat Conduction Equation in One Dimension. Proc. Cambridge Philos. Soc., 60, No. 4, 897—907, 1964.

25. Cherpin N. E. On one method for solving an equation of parabolic type. Tr. Arkt. i Antarkt. NII, 271, 1964, pp. 45—49.

26. Ivanov V. T. Solution and investigation of the problem of an imperfect borehole. Uchen. zap. Bashkirsk. un-ta, ser. matem. nauk, issue 20, No. 1, 1965, pp. 17—28.

27. Ivanov V. T. Solution of certain special equations by the method of straight lines. Ibid., pp. 29—32.

28. Ivanov V. T. Error estimate for the solution, by the method of straight lines, of certain boundary-value problems for partial differential equations. Ibid., No. 2, pp. 9—14.

29. Ivanov V. T. On the method of straight lines for elliptic-type equations with discontinuous coefficients. Ibid., pp. 140—147.

30. Ivanov V. T., Trofimov A. N., Migunov M. I. Calculation of current distribution on an electrode by the method of straight lines and least squares. Ibid., pp. 60—65.

1. Ivankov V. T., Yakimova M. P. Solution of the problem of an imperfect well by a differential-difference Fourier method. Izv. vuzov, ser. “Neft’ i gaz,” No. 7, 1965.

2. Katskova O. N., Chushkin P. I. Three-dimensional supersonic equilibrium flow of a gas around bodies at an angle of attack. ZhVM i MF, 5, No. 3, 503–518, 1965.

3. Leonchuk M. P., Trofimov A. S., Kurbatov I. M. On the numerical solution of one problem of optimal control of a nuclear reactor. Ibid., pp. 558–561.

4. Trofimov A. N., Ivanov V. T. Calculation of the distribution of current on a cathode by the method of straight lines. Elektrokhimiya, 1, No. 2, 224–226, 1965.

5. Wadsworth M., Wragg A. On Matrix Methods for the Solution of Partial Differential Equations. Proc. Cambridge Philos. Soc., 61, No. 1, 129–132, 1965.

6. Nemchinov S. V., Libov A. S. The use of differential-difference identities for solving boundary-value problems for the Helmholtz equation with increased accuracy by the non-iterative method of grids. Nauchn. tr. Tashkentsk. un-ta, issue 259, 189–203, 1964.

7. Cobble M. H. Quasi-Analytic Solution of the Diffusion Equation. J. Franklin Inst., 279, No. 2, 110–123, 1965.

8. Sarmin E. I. On the application of the method of straight lines to the solution of boundary-value problems for certain non-self-adjoint two-dimensional elliptic equations of the 2nd order. ZhVM i MF, 5, No. 5, 945–949, 1965.

Received by the editors
May 27, 1965

Institute of Mathematics
Academy of Sciences of the BSSR