Solving Systems of Differential Equations of Block ... - Springer Link

c Pleiades Publishing, Ltd., 2015. ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2015, Vol. 9, No. 1, pp. 1–10. c K.R. Aida-zade, Ye.R. Ashrafova, 2014, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2014, Vol. XVII, No. 4, pp. 3–13. Original Russian Text

Solving Systems of Differential Equations of Block Structure with Nonseparated Boundary Conditions K. R. Aida-zade1, 2* and Ye. R. Ashrafova2** 1

Azerbaijan State Oil Academy, pr. Azadlyg 20, AZ 1010 Baku, Azerbaijan

2

Institute of Cybernetics of the Azerbaijan National Academy of Sciences, pr. B. Vakhabzade 9, AZ 1141 Baku, Azerbaijan Received April 18, 2014

Abstract—We study the solution of a system of higher-dimensional ordinary differential equations of block structure. Some separate subsystems are connected with each other by the nonseparated boundary conditions caused by an arbitrary relation between the boundary values of the solutions to the subsystems. For numerical solution, we propose some scheme of the method of transferring the boundary conditions that takes into account the specific characteristics of the systems under consideration. The results of numerical experiments are given. DOI: 10.1134/S1990478915010019 Keywords: system of differential equations, block structure, nonseparated boundary conditions, method of transfer of conditions, Cauchy problem, Runge–Kutta method

INTRODUCTION We propose an approach to the numerical solution of a system consisting of many independent subsystems of nonautonomous linear ordinary differential equations. The subsystems of differential equations are connected with each other by some nonseparated boundary conditions, and these connections are characterized by weakly and arbitrarily filled matrices. Similar problems describe the mathematical models of many complex objects and technological processes whose mathematical modeling involves the discretization (decomposition) methods with respect to time or space variables [1–9]. In general, the mathematical statement of the problem is represented as a two-point problem with respect to a higher-dimensional system of ordinary differential equations of block structure, and the well-known numerical methods can be used for its solution, in particular, the sweep method [10–17]. In this article, taking into account the block structure of the system of differential equations and the weak but arbitrary filling of the boundary condition matrix, we propose a version of the method of transfer of the boundary conditions. The advantage of this approach in comparison with the direct use of transfer methods in general form is obvious since here the transfer is carried out only with respect to those variables whose coefficients in the boundary conditions are nonzero; moreover, the transfer is carried out with the use of the only subsystem of differential equations involving the variable that is being transferred. We show the results of the numerical experiments obtained by solving a test problem based on the problem of calculating an unsteady fluid motion in a fragment of a pipeline network of complex structure. * **

E-mail: [email protected] E-mail: [email protected]

1

2

AIDA-ZADE, ASHRAFOVA

1. STATEMENT OF THE PROBLEM Consider the system consisting of L independent subsystems of nonautonomous linear differential equations: dy i (x) x ∈ [0, li ], = Ai (x)y i (x) + B i (x), dx y i (·) ∈ Rni , i = 1, . . . , L.

(1)

Here Ai (x) and B i (x) are some given continuous ni -dimensional square matrix and vector functions respectively; moreover, Ai (x) = const, x ∈ (0, li ); the unknown ni -dimensional vector-functions y i (x) are continuously differentiable for x ∈ [0, li ]; li > 0 are given, and i = 1, . . . , L. Solutions y i (x), i = 1, . . . , L, to subsystems in (1) are connected by the initial and boundary conditions which we write in general form: (2)

Gy(0) + Qy(l) = R, L

where G = (gij ) and Q = (qij ) are given square matrices of size n × n, n = i=1 ni , and the rank of the extended matrix rang(G, Q) = n; and R = (r 1 , . . . , r n ) is a given n-dimensional vector. In (2) and bellow, we use the following notation: y(x) = y11 (x), . . . , yn1 1 (x), y12 (x), . . . , yn2 2 (x), . . . , y1L (x), . . . , ynLL (x) , y(0) = y11 (0), . . . , yn1 1 (0), y12 (0), . . . , yn2 2 (0), . . . , y1L (0), . . . , ynLL (0) , y(l) = y11 (l1 ), . . . , yn1 1 (l1 ), y12 (l2 ), . . . , yn2 2 (l2 ), . . . , y1L (lL ), . . . , ynLL (lL ) . Here stands for transposition. The matrix G has the following structure: ⎤ ⎡ 11 11 11 12 12 12 1L 1L 1L ⎢ g1 g2 . . . gn1 g1 g2 . . . gn2 . . . g1 g2 . . . gnL ⎥ ⎥ ⎢ 21 21 ⎢ g1 g2 . . . gn211 g122 g222 . . . gn222 . . . g12L g22L . . . gn1L ⎥ L⎥ ⎢ G=⎢ ⎥. ⎢... ... ... ... ... ... ... ... ... ... ... ... ... ⎥ ⎦ ⎣ n1 n1 n1 n2 n2 n2 nL nL 1L g1 g2 . . . gn1 g1 g2 . . . gn2 . . . g1 g2 . . . gnL We also assume that most of the entries in the matrix are zero, and its nonzero entries correspond to the presence of a connection between the initial and final states of the separate sections of the complex object under study. The matrix Q has the same structure. In what follows, we need the componentwise representation of each of the constraints (2): g1i1 y11 (0) + g2i1 y21 (0) + · · · + gni11 yn1 1 (0) + g1i2 y12 (0) + · · · + gni22 yn2 2 (0) + g1iL y1L (0) + · · · + gniLL ynLL (0) + q1i1 y11 (l1 ) + q2i1 y21 (l1 ) + · · · + qni11 yn1 1 (l1 ) + q1i2 y12 (l2 ) + · · · + qni22 yn2 2 (l2 ) + q1iL y1L (lL ) + · · · + qniLL ynLL (lL ) = ri , which we write in vector form: L L ij j G y (0) + Qij y j (lj ) = r i , j=1

Gij =

j=1

g1ij , . . . , gnijj ,

i = 1, . . . , n,

i = 1, . . . , n, (3)

Qij = q1ij , . . . , qnijj .

Problem (1), (2) is a two-point boundary value problem. Rather numerous articles are devoted to the study of the existence and uniqueness of solutions to boundary value problems in the general case [18, 19]. Note however that the available conditions for the existence and uniqueness of solutions to autonomous systems are not constructive and use a fundamental matrix of solutions whose construction for nonautonomous systems is a difficult problem. Therefore, in practice, the solvability of the problem is JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

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checked during the application of the numerical method for solving the problem. Henceforth, we assume that the problem in question is well posed; i.e., the problem has a solution and that is unique. Problem (1), (2) has the following peculiarities: • the subsystems of differential equations of (1) are mutually independent; • solutions to the subsystems y i (x), i = 1, . . . , L, are connected by nonseparated boundary conditions characterized by weakly but arbitrarily filled connection matrices G and Q; • a large number of subsystems and so a large order of the system (1) in whole. This problem can describe a mathematical model of a multilink dynamic object with lumped parameters of a complex configuration whose every section is described by a subsystem of ordinary differential equations independent of the other sections. The ends of the sections themselves are linked in an arbitrary manner; the connections between them are described by (2) or (3), namely, by the nonzero entries of G and Q. For real objects, the number of connections (i.e., the nonzero coefficients in (2), (3)) is much less than the total dimension n of the problem. Such kind of models can be obtained in applications of the decomposition methods in the mathematical modeling of complex and higher-dimensional technical objects and technological processes [1–6, 20–22]. For the objects with distributed parameters in time and space, the decomposition (or discretization) can be carried out with respect to the space variables or time [1–9]. To the mathematical statement (1), (2), there is also reduced the problem of computing the dynamics of artificial neural networks of complicated structure and huge dimension. In this case, each neuron is described by ordinary differential equations, and communication between neurons is carried out at the expense of the interdependence of their initial and final states. We can assume that the mathematical models of artificial neural networks result from the decomposition of a complex dynamic object with respect to the space variables. To the problem under consideration, we can also reduce the problem of calculating the transient processes of motions of a fluid in some pipeline networks of a complex structure. The mathematical models of processes of this kind are described by the systems of partial differential equations consisting of subsystems of hyperbolic equations describing the motion process in each separate section. At the junction points of the sections, there are fulfilled the conditions of continuity of the flow and the material balance, which are defined by conditions of the form (2). Application of the method of lines for the time or space variables (an analog to the use of decomposition) reduces the problem of the calculation of the motion modes of raw materials in a transport network to the problem of the form (1), (2). 2. NUMERICAL SOLUTION OF THE PROBLEM For solving (1), (2) directly, we can use various available schemes of the sweep method [10–17]. Such an approach, involving the above peculiarities of the problem, is inefficient, especially if it is applied many times, for example, to the problem of an optimal control of an object described by the mathematical model (1), (2). Below we expose an approach that is an analog of the method of transfer of conditions which involves the peculiarities of system (1) and give the corresponding formulas and algorithms not requiring solving all subsystems of (1) simultaneously. As well as all methods of transfer of conditions, the proposed approach consists in the replacement in (2) of the values y(0) (or y(l)) at the expense of transferring to the right (left) by some equivalent combinations of the values y(l) (or y(0) in transferring to the left). In result, instead of (2) we obtain n conditions of the form Qy(l) =R (4) in transferring conditions (2) to the right or of the form Gy(0) =R

(5)

in transferring (2) to the left. Owing to the weak occupancy of the matrices G and Q and to avoid dealing with matrix operations, we propose to transfer each condition in (3) separately. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

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In some practical problems, many conditions in (2), instead of the general form, can be separated or, moreover, can coincide with the Cauchy conditions on the left- or right-hand sides. Therefore, in choosing the direction of the transfer of the conditions to the left or to the right, we determine which of the sides contains more local conditions and then transfer the remaining conditions to this side. Consider an arbitrary ith condition in (2), i = 1, . . . , n, having representation (3). Transfer the ith condition in (3) to the right-hand side; i.e., obtain the equivalent condition q˜1i1 y11 (l1 ) + q˜2i1 y21 (l1 ) + · · · + qñi11 yn1 1 (l1 ) + q˜1i2 y12 (l2 ) + · · · + qñi22 yn2 2 (l2 ) + q˜1iL y1L (lL ) + · · · + qñiLL ynLL (lL ) = r˜i , which can be represented as L

ij y j (lj ) = r˜i , Q

(6)

j=1

ij and r˜i are some yet unknown coefficients; here the nj -dimensional vector y j (lj ) defines the where Q unknown values of the solution to the jth subsystem in (1) at the right end of [0, lj ]. Obtain the condition of the form (6) step-by-step. Suppose that the coefficients Gi1 , Gi2 , . . . , GiL are not all zero. Otherwise, there is no need of transferring the ith condition to the right because this condition involves only the values y(l). Suppose that the first nonzero coefficient is Giν = 0nν , 1 ≤ ν ≤ L, and Gij = 0nj for j < ν (0nν denotes the nν -dimensional vector whose all components are zero). Definition 1. We say that an nν -dimensional vector-function α(x) and a function β(x) such that α(0) = Giν ,

β(0) = r i ,

(7) solution y ν (x)

to the νth subsystem transfer the ith condition in (3) to the right if, given an arbitrary of (1) dy ν (x) = Aν (x)y ν (x) + B ν (x), x ∈ [0, lν ], (8) dx we have ⎡ ⎤ L L Gij y j (0) + Qij y j (lj )⎦ = β(x) (9) α (x)y ν (x) + ⎣ j=ν+1

j=1

for all x ∈ [0, lν ] Clearly, with (7) taken into account, for x = 0 condition (9) coincides with the ith condition in (3). Call the functions α(x) and β(x) sweep functions. Inserting the values of α(x) and β(x) for x = lν in (8), we obtain the following equality that is equivalent to the ith condition: L j=ν+1

ij j

G

y (0) +

L

ij y j (lj ) = r˜i , Q

j=1

ij = Qij , j = 1, . . . , L, and j = ν. iν = Qiν + α(lν ), r˜i = β(lν ), Q where Q The sweep functions α(x) and β(x) transferring the conditions are nonunique. We can use the functions that are proposed in Theorem 1. Suppose that an nν -dimensional vector function α(x) and a function β(x) for x ∈ (0, lν ] are solutions to the following Cauchy problems: dα(x) = −Aν (x)α(x), α(0) = Giν , dx (10) dβ(x) ν i = B (x)α(x), β(0) = r . dx Then these functions satisfy (9) for x ∈ [0, lν ]. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

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Proof. Let α(x) and β(x) be yet arbitrary differentiable functions satisfying (9) and (7). Differentiate (9): dα (x) ν dβ(x) dy ν (x) y (x) + α (x) = . dx dx dx Reckoning with (8) and combining summands, we obtain

dβ(x) dα (x) ν ν ν + α (x)A (x) y (x) + − + α (x)B (x) = 0. dx dx

(11)

Involving the arbitrariness of α(x) and β(x) together with the necessity of (11) for all solutions y ν (x) to (8), require the fulfillment of equality to zero of the expression in parentheses. This implies that α(x) and β(x) are solutions to the Cauchy problem (10). The above procedure repeats for the next variable y j (0), j > ν, for which the coefficient Gij = 0nj until the ith condition does not involve the components of the vector y(0) = (y 1 (0), . . . , y L (0)). After that we must pass to the (i + 1)th condition in (2) until these procedures are implemented for all constraints and a condition of the form of (4) is obtained. Solving the system of nth-order algebraic equations (4) gives the vector y(l) = (y 1 (l1 ), . . . , y L (lL )) . For determining the desired vector-functions y ν (x) for x ∈ [0, l], the components y ν (l), ν = 1, . . . , L, of the found vector y(l) are used as the initial values for the corresponding Cauchy problems with respect to each separate subsystem in (1) solved in the reverse order: from x = lν to x = 0, ν = 1, . . . , L. The transfer of conditions to the left for obtaining conditions of the form (5) equivalent to (2) is carried out similarly. The proof of Theorem 1 is complete. Suppose that, in the ith condition of (2), some of the vector coefficients Qi1 , Qi2 , . . . , QiL are nonzero; for example, the first of them is Qiν = 0nν , while Qij = 0nj for j = 1, ν − 1. Definition 2. We say that an nν -dimensional vector-function α(x) and a function β(x) such that α(lν ) = Qiν ,

β(lν ) = r i ,

(12)

transfer the ith condition in (3) to the left if, for an arbitrary solution y ν (x) to the νth subsystem in (7), we have ⎡ ⎤ L L Gij y j (0) + Qij y j (lj )⎦ = β(x). (13) α (x)y ν (x) + ⎣ j=1

j=ν+1

for all x ∈ [0, lν ]. Clearly, with account taken of (1), for x = lν , condition (13) coincides with the ith condition. If these sweep functions are known then (13) for x = 0 takes the form L j=1

ij y j (0) + G

L

Qij y j (lj ) = β(0),

(14)

j=ν+1

ij = Gij , j = 1, . . . , L, and j = ν. Condition (14) is equivalent in the ith iν = Giν + α(0), G where G condition in (2) and involves one variable less defined at the right end than before the transfer. Similarly to Theorem 1, we prove Theorem 2. Suppose that an nν -dimensional vector-function α(x) and a function β(x) are solutions to the following Cauchy problems for x ∈ [0, lν ]: dα(x) = −Aν (x)α(x), dx dβ(x) = B ν (x)α(x), dx Then these functions satisfy (13). JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

α(lν ) = Qiν , (15) β(lν ) = r i .

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Clearly, the Cauchy problems (15) are solved in the reverse order: from x = lν to x = 0. The procedures of the transfer of the values of y(l) occurring in (2) are carried out consecutively for all conditions and all components of the vector y(l) = (y 1 (l1 ), . . . , y L (lL )) . While determining the transfer functions by solving the Cauchy problems (10) or (15), we can face the problem of the presence of rapidly growing solutions, and hence with a large error in the so-obtained solutions. Namely, if all eigenvalues of the matrices Aν (x) are positive and there are rather large numbers among them then systems (10) have rapidly growing solutions. In this case, it is better to transfer the conditions to the left with using system (15). If all eigenvalues of the matrices Aν (x) are negative and among them there are numbers with large moduli then systems (10) are stable and conditions must be transferred to the right. If the matrices Aν (x) simultaneously have positive and negative eigenvalues of large modulus then both systems (10) and (15) are unstable, and their use will lead to a huge error. In this case, we will use sweep the functions v(x), w(x), and s(x) satisfying the following condition, equivalent to the ith condition in (2): ⎡ ⎤ L L Gij y j (0) + Qij y j (lj )⎦ = w(x), x ∈ [0, lν ], (16) v(x)y ν (x) + s(x) ⎣ j=ν+1

j=1

proposed in Theorem 3. Suppose that Giν Rnν = 0, an nν -dimensional vector-function v(x) and functions w(x) and s(x), x ∈ [0, lν ], are solutions to the following systems of differential equations: dv(x) = s0 (x)v(x) − Aν (x)v(x), dx dw(x) = s0 (x)v(x) − B ν (x)w(x), dx ds(x) = s0 (x)s(x), dx s0 (x) = (v(x)Aν (x)v (x)) − v(x)B(x)w(x)/(v(x)2 + w2 (x)).

(17) (18) (19) (20)

If they satisfy the initial conditions v(0) = Giν ,

w(0) = r i ,

s(0) = 1

(21)

then they enjoy condition (16): these functions transfer the ith equation to the right. If they satisfy the conditions v(lν ) = Qiν ,

w(lν ) = r i ,

s(lν ) = 1

(22)

then they transfer the ith condition to the left. In addition, the numerical transfer scheme is stable because of the fulfillment of the condition x ∈ [0, lν ].

v(x)2Rnν + w2 (x) = const,

(23)

Proof. Suppose that α(x) and β(x) are defined from (10) and satisfy condition (9) of transfer to the right. Multiplying both sides of (9) by an arbitrary scalar function s(x) satisfying only the condition s(0) = 1, we obtain ⎡ ⎤ L L Gij y j (0) + Qij y j (lj )⎦ = s(x)β(x) s(x)α(x)y ν (x) + s(x) ⎣ j=ν+1

j=1

that coincides with (16) if we put v(x) = s(x)α(x),

w(x) = s(x)β(x);

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these functions obviously enjoy (21). Require the fulfillment of (23) for s(x). Differentiate (24) (sometimes we omit the arguments of the functions for brevity): ds(x) dα(x) dw(x) ds(x) dβ(x) dv(x) = α(x) + s(x) , = β(x) + s(x) . dx dx dx dx dx dx From this, reckoning with (9) and (24), it follows that dv(x) 1 ds(x) = v(x) − Aν (x)v(x), dx s dx 1 ds(x) dw(x) = w(x) + B ν (x)v(x). dx s dx

(25) (26)

Differentiate (23): dw(x) dv(x) dv (x) v(x) + v +2 w(x) = 0. dx dx dx Inserting (24) and (25) in this equality, we have 1 ds 2 ds 2 1 ds v v − v Av + v v − v A v + w + 2B ν vw = 0, s dx s dx s dx ν v Av − wB v 1 ds = = s0 . s dx v2 + w2 Inserting (26) in (24), (25) and involving the initial conditions (16), we conclude that functions v(x), w(x), and s(x) satisfying system (17)–(20) transfer the νth condition to the right by (16). Analogously, we prove the second part of the theorem and state that if the functions v(x), w(x), and s(x) satisfy the initial conditions (22) and system (17)–(20) then they transfer the ith condition to the left by (16). The proof of Theorem 3 is complete. The order of transfer is arbitrary; i.e., we can transfer some variable from one side to the other and thereafter transfer another variable in all conditions. Similarly, we can transfer the values of all variables to one side for one condition and thereafter carry out this procedure for the remaining conditions consecutively. Remark. As follows from the above, the transfer of each of the conditions in (2) is carried out independently of each other; consequently, this procedure is easily implemented by a parallel computation method. After solving the system of algebraic equations (4) or (5), the process of solution of the Cauchy problems is well parallelizable with respect to the subsystems of (1). 3. RESULTS OF NUMERICAL EXPERIMENTS Consider the results of numerical solution of the following boundary value problem with respect to a system of five subsystems of differential equations: dy11 dx dy12 dx dy13 dx dy14 dx dy15 dx

= y21 , = y22 , = y23 , = y24 , = y25 ,

dy21 dx dy22 dx dy23 dx dy24 dx dy25 dx

= −y11 + 2y21 + x2 − 4x,

x ∈ [0, 1],

= −y12 − 3y22 + 3x − 5.5ex/2 − 3 sin x + 9,

x ∈ [0, 1],

= y13 + 0.5y23 − 2ex/2 − x + 0.5,

x ∈ [0, 1],

= −0.25y14 + y24 + 0.125x2 − x + 0.5,

x ∈ [0, 1],

= −0.25y15 + y25 + 0.083x3 − x2 + 2x − 0.75,

x ∈ [0, 1],


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(27)

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The scheme of the motion of a fluid in a pipeline network consisting of five sections.

under the following ten conditions: y11 (0) + y12 (0) + y13 (0) = 0, y21 (0) − y23 (0) = 0, y14 (0)

y22 (0) − y23 (0) = 0,

= 4,

y11 (1) = 4.437, y14 (1) + y13 (1) y23 (1) − y25 (1) = 0,

(28)

+

y15 (0) = −1, y12 (1) = 0.243, y15 (1) = 9.726, y14 (1) − y15 (1) =

(29) (30) (31) (32) (33)

0.

In this problem, L = 5, nν = 2, lν = 1, ν = 1, 5; its exact solution is given by the functions y11 (x) = x2 + 2ex − 2,

y12 (x) = 3x − 2ex/2 + cos x,

y13 (x) = x + 2ex/2 − 1,

y14 (x) = 0.5x2 + 2ex/2 + 2,

y15 (x) = 0.3x3 + 2ex/2 − 3,

y21 (x) = 2x + 2ex ,

y22 (x) = 3 − ex/2 − sin x,

y23 (x) = 1 + ex/2 ,

y24 (x) = x + ex/2 ,

y25 (x) = 0.9x2 + ex/2 .

(34)

The problem simulates the calculation of the regimes of an unsteady motion of a fluid in a pipeline network consisting of five sections (see the figure). The above equations can be obtained hypothetically by using the method of lines for approximation of the time derivative in the hyperbolic equations describing the motion along each section. Conditions (28) and (32) mimic the laws of the material balance of the raw materials at the vertices, conditions (29) and (33) simulate the continuity of the flow (equal pressures at the sections adjacent to the common vertices), and conditions (30)–(31) determine the given flow rates of the raw materials by sources (inflows and outflows in the network). In the figure, the flow directions in the network are shown formally; i.e., the calculations of actual motion regimes can be such that the consumption on some section can be negative, which indicates that the formal direction for this section and the actual motion direction are opposite to each other. As is seen from the ten conditions (28)–(33), the left- and right-hand sides both contain five conditions. The analysis of the eigenvalues of each of the matrices Aν shows that the subsystems do not have rapidly growing solutions. Therefore, the transfer of conditions (28)–(30) was made to the right by the functions that are solutions to the Cauchy problem (10). For the numerical solution of the Cauchy problem, we used the Runge–Kutta method of the fourth order with meshsize h = 0.01. In view of the already existing conditions (31)–(33) on the right-hand side and by the transfer of conditions (28)–(30) to the right, we obtained an algebraic system of the form (4) in which the matrix Q JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

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equals ⎛ ⎜1 − 0.4323 ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 =⎜ 0 Q ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎝ 0

0

1 −6.3618

1 −0.7869

0

0

0 20.0855

0 −0.6065

0

0

0

0

0 −0.6065

0

0

0

0

0

0

1 −0.6321

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

1

0

0

0

0

1

0

0

0

0

1

0

0

⎞ 0 ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ 1 − 0.6321⎟ ⎟, ⎟ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ 0 1 ⎟ ⎟ ⎟ −1 0 ⎟ ⎠ −1 0 0

and the right-hand side is given by the vector = − 0.5659, 8.6332, −0.6001, 4.1231, −1.0435, 4.4365, 0.2428, 9.7256, 0, 0 . R Using the Gauss method, we found the vector y(l) = (4.4366, 7.4366, 0.2429, 0.5098, 3.2974, 2.6487, 5.7974, 2.6487, 0.6308, 2.6487) such that the maximal deviation of its components from the exact values obtained from (34) is less than 10−8 . Next, for obtaining y(x), we must solve the Cauchy problems for the separate independent subsystems of (27) with the initial conditions at the right end.

CONCLUSION As a rule, higher-dimensional systems of differential equations which arise in many applications have block structure, and the conditions relating the initial and final values of the unknown functions are nonseparated with weakly filled Jacobi matrix. In general, they are a special case of two-point boundary value problems, and for their numerical solution it is possible to use different schemes of the sweep method. For a more efficient numerical solution of the problem, this paper proposes an approach based on the idea of the transfer of conditions but substantially using the peculiarities of the problem. The illustrative test problem in the article is obtained after applying the method of lines for calculating the regimes of fluid motion by the example of a complex pipeline transport network, wherein, at every linear section, the motion is described by a system of two hyperbolic partial differential equations. The proposed approach can be used in calculation for the computer models of complex and large systems with lumped and distributed parameters obtained by using the decomposition methods of mathematical modeling.

ACKNOWLEDGMENTS The authors were supported by the Science Development Foundation of the President of the Republic of Azerbaijan (project no. EIF/GAM–2–2013–2(8)–25/06/1). JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS

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Vol. 9 No. 1 2015