A Boundary Value Problem for the Linearized Haemodynamic Equations on a Graph

Differential Equations, Vol. 40, No. 1, 2004, pp. 94–104. Translated from Differentsial’nye Uravneniya, Vol. 40, No. 1, 2004, pp. 87–97. c 2004 by Ashmetkov, Mukhin, Sosnin, Favorskii. Original Russian Text Copyright

NUMERICAL METHODS. FINITE-DIFFERENCE EQUATIONS

A Boundary Value Problem for the Linearized Haemodynamic Equations on a Graph I. V. Ashmetkov, S. I. Mukhin, N. V. Sosnin, and A. P. Favorskii Moscow State University, Moscow, Russia Received September 10, 2003

INTRODUCTION The system of linearized haemodynamic (LHD) equations was considered in [1–3], where special boundary value problems were posed and solved for it. The general boundary value problem for the LHD equations in a single vessel was posed and solved in [4]. The resulting exact solutions of the LHD equations were used as efficient tests for verifying the CVSS software [5]. Conditions under which there is a resonance growth in the pressure and velocity amplitudes were obtained. The resonance phenomenon was also considered in [6, 7] in some special cases. The present paper continues the series [1–3]. Here we pose and study the general boundary value problem for the LHD equations on a graph of vessels. As a result, we obtain a simple sufficient condition for instability of the solution. 1. A BOUNDARY VALUE PROBLEM ON A GRAPH WITH A SINGLE EDGE 1. Statement of the Problem and Analytic Solution Following [4], we consider the general boundary value problem for the LHD equations [6] on a graph consisting of a single edge: pt + u ¯px + %¯ c2 ux = 0, ut + %−1 px + u ¯ux = 0, 0 < x < l, p(x, 0) = ϕ(x), u(x, 0) = ψ(x), 0 ≤ x ≤ l, α1 p(0, t) + β1 u(0, t) = µ1 (t), α2 p(l, t) + β2 u(l, t) = µ2 (t),

t > 0, (1) t ≥ 0.

Here p¯ = const, u ¯ = const is a stationary solution of the haemodynamic equations [6], p(x, t) and u(x, t) are the deviations of the pressure and velocity from the stationary values, ϕ(x) and ψ(x) are the initial perturbations of the stationary values of the p pressure and velocity, α1 , β1 , α2 , and β2 are given constants, µ1 and µ2 are given functions, c¯ = S (¯ p) /(%θ) is the propagation velocity of small perturbations inside the vessel, S(P ) is a known dependence between the cross-section area dS(P ) and the pressure inside the vessel, and θ = > 0. Throughout the following, we assume dP P =p¯ that the stationary flow in question is “subsonic”, i.e., satisfies |¯ u| < c¯. The general solution of the LHD equations is given by the functions p(x, t) and u(x, t) of the form [6]


%¯ c + p(x, t) = f x − λ+ t − f − x − λ− t , 2 (2)


1 + u(x, t) = f x − λ+ t + f − x − λ− t , 2 + − where f and f are two propagating waves of an arbitrary shape, λ+ = u ¯ + c¯ > 0, and λ− = u ¯ − c¯ < 0. The specific form of f + and f − on their respective domains of each of them is chosen so as to satisfy the initial and boundary conditions in problem (1). If 0 ≤ x ≤ l and t ≥ 0, then the domain of f + (x − λ+ t) is the half-line (−∞, l] and the domain of f − (x − λ− t) is the half-line [0, +∞). c 2004 MAIK “Nauka/Interperiodica” 0012-2661/04/4001-0094

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By substituting formulas (2) into the initial conditions in problem (1), one can find f + and f − on the interval [0, l], which is the common part of their domains: f + (z) =

1 ϕ(z) + ψ(z), %¯ c

f − (z) = −

1 ϕ(z) + ψ(z), %¯ c

0 ≤ z ≤ l.

(3)

To find f + and f − on the remaining parts of their domains, we use the boundary conditions. The graph on which problem (1) is to be solved consists of a single edge bounded by two vertices x = 0 and x = l. At the vertex x = 0, one has the incident wave f − (x − λ− t) and the outgoing wave f + (x − λ+ t). For the vertex x = l, the incident wave is f + (x − λ+ t) and the outgoing wave is f − (x − λ− t). By substituting (2) into the boundary conditions in problem (1) and by solving the resulting relation at the vertex x = 0 for f + and the relation at the vertex x = l for f − , we obtain the two expressions

f + −λ+ t = k1 f − −λ− t + ν1 µ1 (t), (4)

− − + + f l − λ t = k2 f l − λ t + ν2 µ2 (t), t ≥ 0, (5) c − β1 c + β2 α1 %¯ α2 %¯ and k2 = are the reflection constants [4, 8] of the vertices x = 0 α1 %¯ c + β1 α2 %¯ c − β2 and x = l, respectively, ν1 = 2/(β1 + α1 %¯ c1 ), and ν2 = 2/(β2 − α2 %¯ c). In (4), we perform the change of variables z = −λ+ t. Then relation (4) acquires the form  −   z  λ f + (z) = k1 f − z ≤ 0. (6) z + ν µ 1 1 − + , λ+ λ

where k1 =

Likewise, by performing the change of variables z = l − λ− t in (5), we obtain     l λ+ z − + + ∗ f (z) = k2 f λ t + − z + ν2 µ 2 , z ≥ l, − λ λ− λ−

(7)

where t∗ = l/λ+ − l/λ− . Relations (6) and (7) form a system of functional equations for two unknown functions f + and f − . This system can be formally rewritten in matrix form; to this end, we introduce the T column vectors V = (f + (z), f − (z)) ,   −    T T   λ l z  λ+ z − + + ∗ ˜ V = f λ t + −z z ,f , M = µ1 − + , µ2 − λ+ λ λ λ− λ− and the matrices Tg = diag (k1 , k2 ) and G = diag (ν1 , ν2 ). Then relations (6) and (7) acquire the form V = Tg V˜ + GM . This reflects the fact that the waves that are outgoing waves for all vertices of a graph are compositions of waves incident on the graph vertices and waves induced by the boundary conditions. The entries of the matrix Tg are the transport coefficients [8] of all vertices of the graph. Note that det Tg = k1 k2 for problem (1). + Let to  us return  Eqs. (6) and (7).These  are two linear relations containing four unknowns f (z), + − λ λ f + λ+ t∗ + − z , f − (z), and f − z . Any two of them can be eliminated. For example, by + λ  −  λ λ eliminating f − (z) and f − z , we obtain the recursion relation λ+    z 
l z + + + ∗ f (z) = k1 k2 f z + λ t + ν1 µ1 − + + ν2 k1 µ2 , (8) − λ λ− λ+ which is valid for the argument values z ≤ l − λ+ t∗ . It follows from (8) that the value of f + at an arbitrary point z can be computed via the value of the same function at the point lying by λ+ t∗ DIFFERENTIAL EQUATIONS

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to the right on the real line. Let us now apply relation (8) to the term f + (z + λ+ t∗ ) occurring on the right-hand side of itself; then we obtain an expression for the function f + (z) via its values on the interval [l − λ+ t∗ , l]. The values of f + on [l − λ+ t∗ , l] are given by formulas (3) and (6), which imply that  1   if 0 ≤ z ≤ l  ϕ(z) + ψ(z) %¯ c + +      f (z) ≡ Φ (z) =  z  2 1 λ− λ−   µ1 − + − k1 − ϕ z + ψ z if l − λ+ t∗ ≤ z ≤ 0.  α1 %¯ c + β1 λ %¯ c λ+ λ+  +

Likewise, by eliminating f (z) and f

+

relation for f − (z) : −

f (z) = k1 k2 f

−

− ∗

z+λ t




λ+ λ t + −z λ + ∗

 from (6) and (7), we obtain a recursion

   l z  z ∗ , + ν1 k2 µ1 −t − − + ν2 µ2 − λ λ− λ−

(9)

where z ≥ −λ− t∗ . It follows from (9) that the value of f − at an arbitrary point z can be computed via the value of the same function at the point lying by −λ− t∗ to the left on the real line. By applying relation (9) again to f − on the right-hand side of itself, we obtain an expression for the function f − (z) via its values on [0, −λ− t∗ ]. The values of f − on [0, −λ− t∗ ] are given by formulas (3) and (7), which imply that  1   ϕ(z) + ψ(z) if 0 ≤ z < l  %¯ c  − −       f (z) ≡ Φ (z) = 1 z λ+ l λ+   if l ≤ z < −λ− t∗ . z + ψ z + ν2 µ 2 − − + − ϕ  −k2 − − %¯ c λ λ λ λ This method for computing f + (z) and f − (z) was implemented in [4], where an analytic expression for the solution of problem (1) was obtained. The analysis of this solution shows that if |k1 k2 | = |det Tg | > 1, then the amplitude of the pressure and velocity waves in a single vessel infinitely grows in the course of time. If |det Tg | < 1, then the initial perturbations of pressure and velocity decay in the course of time, and the amplitude of oscillations caused by boundary influences bounded in magnitude do not grow. If det Tg = 1, then periodic boundary influences with period T = t∗ /n, n = 1, 2, . . . , lead to oscillations of the pressure and velocity in a single vessel with a linearly growing amplitude of oscillations. For det Tg = −1, the amplitude of the oscillations remains bounded for arbitrary boundary regimes. 2. Statement and Solution of a Finite-Difference Problem for Velocity Waves As was mentioned in the previous section, relation (8) in the form
f + (z) = k1 k2 f + z + λ+ t∗ + g(z) for z < l − λ+ t∗ , where

   z  l z g(z) = ν1 µ1 − + + ν2 k1 µ2 , − λ λ− λ+

implies that the value of f + for an arbitrary value of the argument in the interval (−∞, l − λ+ t∗ ) is determined by the value of the same function f + at the point whose absolute value is less by the constant λ+ t∗ and by the contribution of the boundary functions g(z). On the half-line z ∈ (−∞, l), we introduce the uniform grid with nodes zj = −jh + z0 , j = 0, 1, . . . , where h = λ+ t∗ and l − λ+ t∗ ≤ z0 ≤ l. We set f + (zj ) = yj and g (zj ) = gj . Note that y0 = Φ+ (z0 ) is given and det Tg = k1 k2 . DIFFERENTIAL EQUATIONS

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Then relation (8) written out at the grid points acquires the form yj = det Tg yj−1 + gj ,

y0 = Φ+ (z0 ) ;

j = 1, 2, . . . ,

i.e., we have obtained the Cauchy problem for a linear nonhomogeneous first-order finite-difference equation with constant coefficients. Its solution has the form [9] j

yj = (det Tg ) y0 +

j X

j−k

(det Tg )

gk ,

j = 1, 2, . . .

(10)

k=1

h z i Let z < l − λ+ t∗ be arbitrary. We take z0 = z + N + λ+ t∗ , where N + = − + ∗ . (Here [·] is λ t the integer part of a number.) Then zj = z for j = N + ; i.e., z coincides with the grid point zN + . Consequently, yN + = f + (z), and, by (10), we obtain the relation N+

+

f (z) = (det Tg )

Φ

+ + ∗

+

z+N λ t




N X +

+

N + −k

(det Tg )


g −kλ+ t∗ + z + N + λ+ t∗ .

k=1

By substituting the expression for g and by replacing the summation variable by the formula N + − k = n, we obtain    l z  z ∗ ∗ f (z) = (det Tg ) ν1 µ1 −nt − + + ν2 k1 µ2 − nt − + λ λ− λ n=0
+ N + (det Tg ) Φ+ z + N + λ+ t∗ , +



+ NX −1

n

which coincides with the result in [4]. One can solve Eq. (9) for the function f − (z) in a similar way. After introducing a uniform grid on the half-line z ∈ (0, +∞), we obtain the expression −

f (z) =

   l z  z ∗ ν1 k2 µ1 −(n + 1)t∗ − − + ν2 µ2 − nt − λ λ− λ− n=0
N− + (det Tg ) Φ− z + N − λ− t∗ , − NX −1



n

(det Tg )

which coincides with a similar result in [4]. An analysis of the expression (10) for the grid function yj results in some conclusions concerning the behavior of the amplitude of pulse waves in a single vessel. The grid function yj in (10) satisfies the estimate [9] j X j j−k |yj | ≤ |det Tg | |y0 | + |det Tg | |gk | , j = 1, 2, . . . k=1

If g is bounded, that is, |gj | ≤ A, then j

j

|yj | ≤ |det Tg | |y0 | + A

1 − |det Tg | , 1 − |det Tg |

j = 1, 2, . . .

Hence if |det Tg | = |k1 k2 | < 1, then the grid function yj remains bounded for arbitrarily large j. Consequently, the running wave amplitude is bounded at every time provided that |det Tg | < 1. If |det Tg | = |k1 k2 | > 1, then from (10), we find that the running wave amplitude grows infinitely. DIFFERENTIAL EQUATIONS

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2. A BOUNDARY VALUE PROBLEM ON A GRAPH CONSISTING OF TWO EDGES 1. Statement of the Boundary Value Problem Consider a graph consisting of two bounded edges meeting at a single vertex. Let the common vertex of the two edges (the internal vertex) have the number 0, and let the two boundary vertices have the subscripts 1 and 2. Let the edge bounded by vertex 1 have the number 1, and let the edge bounded by vertex 2 have the number 2. On each edge, we introduce a coordinate system of its own. Without loss of generality, we assume that the origin of each of these coordinate systems coincides with the internal vertex (vertex 0) of the graph. Suppose that the homogeneous LHD equations [2, 6] pit + u ¯i pixi + %¯ c2i uixi = 0,

1 uit + pixi + u ¯i uixi = 0, %

0 < xi < li ,

t > 0,

i = 1, 2,

0 ≤ xi ≤ li ,

i = 1, 2,

(11)

are valid on each edge of the graph and the conditions pi (xi , 0) = ϕi (xi ) ,

ui (xi , 0) = ψi (xi ) ,

(12)

are satisfied at the initial time t = 0. Here p¯i = const and u ¯i = const are the stationary background values of pressure and velocity on each edge [6], and pi (xi , t) and ui (xi , t) are the deviations of the pressure and velocity from the stationary values; the functions ϕi (x) and ψi (x)q are the initial perturbations of the stationary values of the pressure and velocity; the constant c¯i = Si (¯ pi ) /(%θ¯i ) is the propagation velocity of small perturbations along the edge with number i (i = 1, 2); the function Si (P ) is a given dependence between the cross-section area and the pressure on the edge (P ) dS i with index i; and θ¯i = > 0 (i = 1, 2) is the parameter specifying the elasticity of dP P =p¯i the vessel (the graph edge). From now on, we assume that the stationary flow is subsonic, i.e., satisfies |¯ ui | < c¯i . Suppose that the additional transmission conditions of the form s¯1 u1 (0, t) + θ¯1 u ¯1 p1 (0, t) + s¯2 u2 (0, t) + θ¯2 u ¯2 p2 (0, t) = 0, 0 0 0 0 α1 p1 (0, t) + β1 u1 (0, t) = α2 p2 (0, t) + β2 u2 (0, t), t ≥ 0,

(13)

are satisfied at the internal vertex of the graph. Relations (13) are obtained by the linearization of the transmission conditions at the internal vertex of the graph [1]. We assume that the following conditions are satisfied at the boundary vertices of the graph: α11 p1 (l1 , t) + β11 u1 (l1 , t) = µ1 (t),

α22 p2 (l2 , t) + β22 u2 (l2 , t) = µ2 (t),

t ≥ 0.

(14)

They are obtained by the linearization of the standard boundary conditions in boundary value problems of haemodynamics [4, 8]. 2. An Application of the Continuation Method to the Boundary Value Problem on a Graph Consisting of Two Edges The general solution of the LHD equations (11) on
each edge of the graph
can be+represented by − − a combination (2) of two running waves fi+ xi − λ+ x t and f − λ t , where λi = u ¯i + c¯i > 0 i i i i and λ− = u ¯ − c ¯ < 0, since |¯ u | < c ¯ , i = 1, 2. i i i i i The specific form of the functions fi+ and fi− on the domain of each of them should be chosen so as to satisfy the initial conditions (12), the transmission conditions (13), and the boundary conditions (14).
+ + If 0 ≤ xi ≤ li and t ≥ 0 in our problem, x then the domain of f i − λi t is the half-line i
(−∞, li ], and the domain of fi− xi − λ− i t is the half-line [0, +∞). DIFFERENTIAL EQUATIONS

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By a substitution into the initial conditions (12), we obtain the relations fi+ (xi ) =

1 ϕi (xi ) + ψi (xi ) , %¯ ci

fi− (xi ) = −

1 ϕi (xi ) + ψi (xi ) , %¯ ci

0 ≤ xi ≤ li ,

i = 1, 2,

which determines the functions fi+ and fi− on the intervals [0, li ]. To find the functions fi+ and fi− on the remaining parts of their domains, we use the transmission conditions (13) and the boundary conditions (14). As a result, from the boundary conditions, we obtain

i + fi− li − λ− li − λ+ t ≥ 0, i = 1, 2. (15) i t = νi µi (t) + ki→i fi i t , ci + βii 2 αii %¯ i , and k = is the reflection constant [4, 8] of the wave i→i i i i i β − α %¯ c α %¯ c j i − βi j j i
fi+ xi − λ+ i t on the vertex with index i, where i is the index of the boundary vertex and j is the index of the edge bounded by the vertex with index i [we have j = i in (15)]. The transmission conditions imply that


0 − 0 − f1+ −λ+ −λ− −λ− 1 t = k1→1 f1 1 t + k2→1 f2 2t ,


(16) 0 − 0 − f2+ −λ+ −λ− −λ− t ≥ 0. 2 t = k1→2 f1 1 t + k2→2 f2 2t , Here νi =

0 0 0 0 Here k1→1 , k2→1 , k1→2 , and k2→2 are the transport coefficients of the vertex with index 0 for the pulse velocity wave, which are computed by the formulas [8] # " 2

X s¯l (1 + m ¯ l) %α0i c¯i − βi0 0 0 0 0 0 2 ki→i = , i = 1, 2, %αi c¯i + βi − 2¯ si %αi c¯i − βi m ¯i 0 0 %α0i c¯i + βi0 %α c ¯ + β l l l l=1 # " 2 X


s ¯ (1 + m ¯ ) l l 0 kj→i , %α0i c¯i + βi0 %α0j c¯j + βj0 = −2¯ sj %α0j c¯j − βj0 m ¯j %α0l c¯l + βl0 l=1

where either i = 1, j = 2 or i = 2, j = 1, and m ¯i = u ¯i /¯ ci . We introduce the vectors
+
−
−

T V = f1+ −λ+ −λ+ l1 − λ− l2 − λ− , 1 t , f2 2 t , f1 1 t , f2 2t




T − + + V˜ = f1− −λ− −λ− l1 − λ+ l2 − λ+ , 1 t , f2 2 t , f1 1 t , f2 2t T

M = (0, 0, µ1 (t), µ2 (t)) ,
2 0 the block diagonal matrix Tg = diag (T 0 , T 1 , T 2 ), where T 0 = kj→i (i and j are the row and i,j=1 1 1 2 2 column indices, respectively), T = (k1→1 ), and T = k2→2 , and the matrix G = diag (0, 0, ν1 , ν2 ). Then system (15), (16) acquires the form V = Tg V˜ + GM . The entries of Tg are the transport coefficients for all vertices of the graph in question. Namely, T 0 is the matrix of transport coefficients of the vertex with index 0, T 1 is the matrix (consisting of the single entry) of transport coefficients of the vertex with index 1, and T 2 is the matrix of transport coefficients of the vertex with index 2. Note that the determinant of Tg is equal to det Tg = det T 0 × det T 1 × det T 2 . Let us return to the construction of solutions of Eqs. (15) and (16). These are a linear system of four equations for four unknown functions f1+ , f1− , f2+ , and f2− . By transforming this function system under the assumption that the value of the variable t in (15) and (16) is large enough to ensure that formal transformations do not lead outside the domains of f1+ , f1− , f2+ , and f2− , one can obtain, for example, recursion relations for the functions f1+ and f1− [10]

0 1 ∗ 2 0 ± ∗ f1± (z) = k1→1 z + λ± k1→1 f1± z + λ± 1 t1 + k2→2 k2→2 f1 1 t2
(17) ∗ ± ∗ ± − det Tg f1± z + λ± 1 t1 + λ1 t2 + g1 (z), DIFFERENTIAL EQUATIONS

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where



g1+ (z)

   l1 l1 z z 0 2 ∗ = − + − det T k2→2 ν1 µ1 − t2 − + λ− λ1 λ− λ1 1 1  l z 2 0 + k2→1 ν2 µ 2 − + , λ− λ1 2 0 k1→1 ν1 µ 1

+ ∗ + ∗ z ≤ l1 − λ+ 1 t1 − λ1 t2 for the function f1 ,



g1− (z)

   l1 l1 z z ∗ 0 2 ∗ ∗ = ν1 µ 1 − t1 − − − k2→2 k2→2 ν1 µ1 − t1 − t2 − − λ+ λ1 λ+ λ1 1 1   l2 z 0 1 ∗ + k2→1 k1→1 ν2 µ 2 − − t1 − − λ2 λ1

l1 l1 l2 l2 ∗ + − − and t2 = + − − . It follows λ1 λ1 λ2 λ2 from (17) that the value of f1+ at z can be expressed via the values of the same function at the three + ∗ + ∗ + ∗ ∗ other points z + λ+ 1 t1 , z + λ1 t2 , and z + λ1 t1 + λ1 t2 lying on the numerical axis to the right of z. This is supplemented by the contribution of the boundary conditions determined by the functions µ1 and µ2 . The value of f1− at z can be expressed via the values of the same function at three − ∗ − ∗ − ∗ ∗ other points z + λ− 1 t1 , z + λ1 t2 , and z + λ1 t1 + λ1 t2 lying on the real line to the left of z. This is supplemented by the contribution of the boundary conditions determined by the functions µ1 and µ2 . The recursion relations for f2+ and f2− have a similar form [10]. − ∗ ∗ ∗ and z ≥ −λ− 1 (t1 + t2 ) for the function f1 . Recall that t1 =

3. Recursion Relations for Velocity Waves on a Graph with Two Isochronous Edges Let the parameters of both edges be such that the time of propagation of the running waves along each of the two edges is the same in both directions, i.e., t∗1 = t∗2 = t∗ . Edges for which such a condition is satisfied are said to be isochronous. Then the recursion relations obtained in the preceding item acquire the form of three-point recursion relations that can be solved analytically. Namely, for the functions f1+ and f1− , we have
±

0 1 2 0 ∗ ∗ f1± (z) = k1→1 f 1 z + λ± − det Tg f1± z + 2λ± + g1± (z), k1→1 + k2→2 k2→2 (18) 1 t 1 t + − ∗ − ∗ where z ≤ l1 − 2λ+ 1 t for the function f1 and z ≥ −2λ1 t for the function f1 . For the functions f2+ and f2− , we have
±

0 1 2 0 ∗ ∗ f2± (z) = k1→1 f 2 z + λ± − det Tg f2± z + 2λ± + g2± (z), k1→1 + k2→2 k2→2 2 t 2 t

(19)

+ − ∗ − ∗ where z ≤ l2 − 2λ+ 2 t for f2 and z ≥ −2λ2 t for f2 . It follows from (18) and (19) that to find the values of f1+ , f1− , f2+ , and f2− on their entire − ∗ domains,it suffices
to know the values of f1+ on the interval l1 − 2λ+ 1 t , l1 , the values of f1 on the − ∗ + + ∗ interval 0, −2λ values of f2 on the interval l2 − 2λ2 t , l2 , and the values of f2− on  1 t −, the
the interval 0, −2λ2 t∗ . We denote f1+ , f1− , f2+ , and f2− on the corresponding initial intervals − + − ± by Φ+ 1 , Φ1 , Φ2 , and Φ2 , respectively. The specific form of the functions Φ1,2 can be found in [10].

4. Statement and Solution of the Finite-Difference Problem for the Velocity Waves Just as in the preceding item, the function f1+ (z) satisfies the three-point recursion relation (18) + + ∗ on the half-line z ≤ l1 − 2λ+ 1 t with the initial condition f1 (z) = Φ1 (z) defined on the interval  ∗ l1 − 2λ+ 1 t , l1 . On the half-line z ∈ (−∞, l1 ], we introduce the finite-difference grid with nodes + ∗ ∗ zj = −jh + z0 , j = 0, 1, 2, . . . , where h = λ+ 1 t and l1 − 2λ1 t < z0 ≤ l1 . We introduce the notation DIFFERENTIAL EQUATIONS

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+ yj = f1+ (zj ), gj = g1+ (zj ), j = 2, 3, . . . , y1 = Φ+ 1 (z1 ), and y0 = Φ1 (z0 ). Note that the values y0 and y1 are given. Then the recursion relation (18) at the grid points acquires the form

yj − Cyj−1 + Ayj−2 = gj ,

y0 = Φ+ 1 (z0 ) ,

j = 2, 3, . . . ,

y1 = Φ+ 1 (z1 ) ,

0 1 2 0 where C = k1→1 k1→1 + k2→2 k2→2 and A = det Tg . By [9], the solution of this finite-difference problem has the form

1 X uj vn − un vj gn+1 , A n=1 un−1 vn − un vn−1 j−1

yj = Bj0 y0 + Bj1 y1 −

j = 0, 1, 2, . . .

(20)

By using the expression (20), one can represent the value of the unknown function f1+ (z) as follows. + ∗ Let z be an arbitrary value  in the domain z ≤ l1 − 2λ1 t . For this value of z, we choose z0 = z + ∗ z + N1+ λ+ ([·] is the integer part of the number). Then z is the grid point 1 t , where N1 = − + ∗ λ1 t zj for j = N1+ . Consequently, f1+ (z) = yN1+ , and, by (20) and the above-introduced notation, the function f1+ (z) admits the representation


0 + ∗ 1 + ∗ f1+ (z) = BN z + N1+ λ+ + BN z + N1+ − 1 λ+ + Φ1 + Φ1 1t 1t 1

1

N1+ −1

−



1 X uN1+ vn − un vN1+ + ∗ g1 z + N1+ − n − 1 λ+ 1t A n=1 un−1 vn − un vn−1

(21)

∗ for z ≤ l1 − 2λ+ 1t . − The solution of the recursion relations for f2+ (z) and f1,2 (z) can be constructed in a similar way; they can be found in [10]. In conclusion of the section, we note that analytical expressions of the form (21) lead to some conclusions concerning the behavior of the amplitudes of velocity waves on a graph of two synchronous edges in the course of time. 0,1 The expressions for BN contain q in the form q N . The parameter q is a root of the characteristic √ C ± C 2 − 4A equation corresponding to a second-order finite-difference equation. Here either q = 2 √ ± ± or q = A, and the parameter N takes one of the four values N . Since the values N infinitely 1,2 0,1 1,2 also infinitely grows in the course of grow with the time of wave propagation, it follows that BN ± time for |q| > 1, which, together with (21), implies an unbounded growth of the amplitudes f1,2 of the velocity waves. If |q| ≤ 1 for all roots of the characteristic equation and none of the functions defined at the boundary vertices is periodic with period Tk = t∗ /k, k = 1, 2, . . . , then the amplitudes of the velocity waves remain bounded in the course of their propagation. But if max q = 1 and at least one of the functions defined at the boundary vertices is periodic with period Tk = t∗ /k, k = 1, 2, . . . , then a resonance occurs. In this case, the amplitude of the velocity waves grows linearly in the course of time. Note also that since |q1 ||q2 | = |A|, we find that the condition |A| = |det Tg | > 1 is sufficient for an unbounded growth of the amplitudes of the velocity waves.

3. A SYSTEM OF DELAY FUNCTIONAL EQUATIONS 1. A Recursion Relation for a System of Homogeneous Functional Equations Note that the homogeneous relations (15) and (16), which appear in the case of a boundary value problem on a graph with two edges, are special cases of the homogeneous system of functional equations of the form T

T

(f1 (z), f2 (z), . . . , fn (z)) = T (f1 (z + a1 ) , f2 (z + a2 ) , . . . , fn (z + an )) , DIFFERENTIAL EQUATIONS

Vol. 40

No. 1

2004

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102

ASHMETKOV et al.

where T = (tij ) ∈ Rn×n , ai > 0, i = 1, . . . , n, are constants, and n is the dimension of the system of homogeneous functional equations. Let fi be functions defined on the half-lines (−∞, li ]. The following assertion was proved in [10]. Theorem. The functions fi (z) satisfying system (22) satisfy the recursion relations n X X ...im fi (z) = (−1)m+1 Mii11...i fi (z + ai1 + · · · + aim ) , i = 1, . . . , n, m m=1

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(i1 ,...,im ) 1≤i1