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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 15, Issue 1


Volume 13 (2015)

Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

Yuriy Golovaty / Volodymyr Flyud
Published Online: 2017-04-09 | DOI: https://doi.org/10.1515/math-2017-0030


We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

Keywords: Metric graph; Hyperbolic equation; Singular perturbed problem; Asymptotics; Boundary layer

MSC 2010: 35R02; 35L20

1 Introduction

The boundary value problems for ordinary and partial differential operators on metric graphs describe a wide variety of physical processes: vibration and diffusion in networks, wave propagation in waveguide networks, expansion of signals in neurons etc. Currently, there is increasing interest in models on graphs, in particular, as a reaction to a great deal of progress in fabricating graph-like structures of the semiconductor materials, (see the survey [1] for details). The idea to investigate the quantum dynamics of particles confined to metric graphs originated with the study of free electron models of organic molecules [24]. Among the systems those were successfully modeled by graphs we also mention e.g., single-mode acoustic and electro-magnetic waveguide networks [5], the Anderson transition [6], fraction excitations in fractal structures [7], and mesoscopic quantum systems [8]. This resulted into the significant intensification of development of ordinary differential equations as well as PDEs on the metric graphs for the last three decades and numerous publications respectively; we refer the reader to e.g. [913].

It is worth pointing out that the boundary value problems for hyperbolic operators of the second order on metric graphs as well as for hyperbolic systems of conservation laws on graphs describe a diversity of physical processes. Such problems arise for example in the modelling of transversal vibrations of networks, gas transportation networks, traffic flow on road networks, supply chain management, water flow in open canals etc. (see [1416]). The d’Alembert operator =t2Δ and the associated Klein-Gordon operator □ + m2 on metric graphs play an important role in relativistic quantum theories [17, 18].

Recently, there has also been a growing interest in the singularly perturbed problems on metric graphs. This is partly motivated by importance of such models for many applied problems in classical and quantum mechanics, theory of non-homogeneous media, scattering theory etc. The differential equations on graphs with small or large parameters in their coefficients represent the natural models of various complicated devices with the irregular “ramified” geometry and heterogeneous properties (see e.g. [19] and [20]). Also, the asymptotic analysis of Schrödinger operators with singular perturbed potentials is one of the most natural ways to define the Hamiltonians corresponding to point interactions supported by a discrete set [21, 22] as well as the Hamiltonians on quantum graphs with point interactions at vertices [2325].

This paper can be viewed as a natural continuation of our work [26], where the vibrations of star-shaped network of strings with vanishingly small stiffness were treated. In [26], the boundary value problem for hyperbolic equation containing a small parameter multiplying the second space derivative was studied within the framework of singular perturbation theory, and asymptotics of solutions were constructed. The main objective of the present paper is to describe the vibrations of networks with the essentially different physical properties, e.g., the stiffness coefficients of the strings have a different order of smallness as a small parameter goes to zero. We study the asymptotic behaviour of solutions to the initial boundary value problem for hyperbolic differential operator on a star-shaped metric graph with the Dirichlet type boundary conditions. In contrast to the previous paper, where the case of total degeneration of the elliptic part was treated, we consider here the partial degeneracy of a hyperbolic operator with different degeneracy factors on subgraphs. Since the coefficients of the operator depend on a small parameter in the singular way, we show that this leads to a relatively complicated behavior of solutions. We apply the boundary layer methods [2729] in order to construct and justify the asymptotics of solutions. Note these results are readily extended to the case of more general finite graphs. It is worth to mention that different models of vibrating systems with the singularly perturbed stiffness were studied in [3032].

The paper is organized as follows. In the next section, we recall the basic notions of metric graphs and PDEs on graphs, and introduce the main object of the paper–the hyperbolic problem on a star-like graph depending on a small parameter. In Sections 3 and 4 we construct the formal asymptotic expansion of a solution for the singularly perturbed problem. Finally, in the last section we justify the asymptotics constructed above.

2 Statement of problem

A non-oriented finite graph is a pair 𝒢 = (𝒱,𝓔), where 𝒱 is a finite set of vertices and 𝓔 is a finite set of edges. We consider a star-shaped planar graph 𝒢 with n edges, i.e., 𝒱 = {a, a1, …, an} and 𝓔 = {(a, a1), …, (a, an} The edge ej :=(a, aj)can be considered as a piece of line connecting two points a and aj in ℝ2. We will endow the graph with the metric structure. Any edge ej ∈ 𝓔 will be associated with an interval [0, ℓj] as follows. Set ℓj = ∥ aja∥ and suppose that the map πj:[0, ℓj] → ej is given by πj(τ)=a+τj(aja).

The map is a parametrization of ej by the arc length parameter τ such that πj(0) = a and πj(ℓj) = aj. Hence, there is a canonical distance function d(x, y), (x, y ∈ 𝒢)making the graph a metric space or a metric graph.

We call f : 𝒢 → ℝ the function on 𝒢. Here and subsequently, fe denotes the restriction of f to the edge e and fe(a)stands for the limit values limexafe(x). The function f is continuous on the star-shaped metric graph 𝒢 if it is continuous on each edge e ∈ 𝓔 and at the central vertex a. The continuity f at the vertex a means that all limit values fe(a)along edges e ∈ 𝓔 equal the value of f at the vertex. We also introduce the differentiation of functions along edges. Set fe(x)=ddτ(feπe)(τ),(1) where x = πe(τ). Let C(𝒢)be the space of continuous functions on 𝒢. Let Cs(e) denote the space of functions f such that f(j):e → ℝ is continuous on edge e for all j = 0, 1, …, s. We also define the spaces C˙s(G)={f:feCs(e)for alleE},Cs(G)={f:fC(G),feCs(e)for alleE}.

We note that the derivatives of f at the central vertex a can not be defined; we must be content with the set of limit values fe(a) along the edges. In other words, the first derivative f′ of a C1(𝒢)-function f is generally discontinuous at x = a. Remark that the value f′(a)does not matter in our considerations against the set {fe(a)}e ∈ 𝓔.

Let us divide the set 𝓔 of edges into non-empty disjoint subsets 𝓔0, 𝓔1, …, 𝓔k. Thereafter the graph 𝒢 breaks into k + 1 star subgraphs 𝒢0, 𝒢1, …, 𝒢k, as is shown in Fig. 1. We will denote by ∂ 𝒢 the set of vertices {a1, …, an}. Then G=i=0kGi, where ∂ 𝒢i is a collection of the vertices of 𝒢i minus a. Set also G=i=1kGi. Consequently, 𝒱* = (∂𝒢∖ ∂ 𝒢0)∪{a} and 𝓔* = 𝓔 ∖ 𝓔0.

The star graph 𝒢 and the subgraphs 𝒢i.
Fig. 1

The star graph 𝒢 and the subgraphs 𝒢i.

We consider the function bε:𝒢→ ℝ that is constant on each subgraph 𝒢i. Set bε(x) = ε2mi for x ∈ 𝒢i, where m0 = 0 and m1, …, mk are positive integers such that m1 < m2 < ⋯ < mk, and ε is a small positive parameter. The function bε can be considered as the stiffness coefficient of a bundle of strings, which possesses significantly different values on the subsystems 𝒢0, …, 𝒢k as ε → 0. Remark that the case of rational powers m1, …, mk can be reduced to the case under consideration by a suitable change of the small parameter εεα. Let I=Rt+ be the positive time half-line and 𝒬 = 𝒢 × 𝓘.

We study the asymptotic behaviour of solution uε:𝒬 → ℝ of the boundary value problem t2uεx(bεxuε)+quε=finQ,(2) uε=φ,tuε=ψonG×{0},(3) uε=μonG×I,(4) uε(,t)is continuous atx=afor alltI,(5) eEbeε(a)xueε(a,t)=0for alltI,(6) where f:𝒬 → ℝ, μ:∂𝒢 × 𝓘 → ℝ and q, φ, ψ:𝒢 → ℝ are given functions. By xj we mean the operator of differentiation along an edge as in (1). Since bε is constant on each subgraph 𝓔i, (2) is actually the set of differential equations t2ueεε2mix2ueε+qe(x)ueε=fe(x,t)ine×I, where e ∈ 𝓔i and i ∈{0, …, k}. Condition (6) is usually called the Kirchhoff vertex condition.

In order to obtain both a smooth enough solution uε of (2)(6) for each positive ε and the complete asymptotic expansion of uε, it is necessary to put some restrictions on the input data. We suppose that qC˙(G),fC˙(Q),φ,ψC(G),μC2(G×I).(7)

Moreover, the following compatibility conditions φ(aj)=μ(aj,0),ψ(aj)=tμ(aj,0)forajG,eEiφe(a)=0fori{0,,k}(8) hold.

Remark 2.1

It may be mentioned here that we can make assumptions upon the input data to ensure the existence of a C2(𝒬)solution uε for all ε ∈(0,1). Namely, (8) must be supplemented with the following conditions t2μ(aj,0)φ(aj)+q(aj)φ(aj)=f(aj,0)forajG0,t2μ(aj,0)+q(aj)φ(aj)=f(aj,0),φ(aj)=0forajGG0,eEiψe(a)=0fori{0,k},φe(a)=0foreE,(9) and the function q(x)φ(x)− f(x, 0)must be continuous at x = a. But these conditions are relatively restrictive. Therefore we assume below that only C1 compatibility conditions (8) hold.

It is well-known [33, Ch.7.2] that the singularities of solutions of hyperbolic equations propagate along characteristics. Since the initial data of (2)(6) are smooth enough, the single reason for discontinuity of uε and its derivatives is a mismatch of the initial conditions and boundary value ones at vertices of the graph. Suppose that the C1 compatibility conditions (8) are satisfied. If conditions (9) do not hold, but there exist finite values of all quantities t2μ(aj,0),φ(aj),f(aj,0), in these equalities, then the second derivatives of solution can possess only the jump discontinuities along characteristics starting at the vertices. Therefore the second derivatives exist almost everywhere in 𝒬 and remain bounded on each compact subset of 𝒬. We introduce the space U(Ω)={uC1(Ω):t2u,xtu,xU2Lloc(Ω)}.

From now on, by a solution of the hyperbolic equation we mean a weak solution that belongs to 𝒰(𝒬).

Conditions (8) together with continuity of the initial data φ and ψ at x = a ensure existence of a unique weak solution uε of (2)(6) [10, 11, 15]. Remark that all well-known results [3438] on the unique solvability for hyperbolic initial boundary value problems are still true for the problems on metric graphs.

3 Formal asymptotic expansion of solution: leading terms

3.1 Limit problem

We look for an approximation to the solution uε of (2)(6) for ε small enough, and begin by setting uε(x,t)=u(x,t)+o(1)asε0.

Recall that the function bε vanishes on the subgraph 𝒢* as ε goes to zero. Upon substituting ε = 0 into (2) we see that u must satisfy the following equations t2ux2u+qu=finQ0,t2u+qu=finQ, where 𝒬0 = 𝒢0 × 𝓘 and 𝒬* = 𝒢* × 𝓘. In addition, our sending ε → 0 in (6) yields the condition eE0xue=0on{a}×I.

Thus on the subgraph 𝒢* hyperbolic equation (2) degenerates into the ordinary differential equation with respect to t, depending on parameter x. Therefore u can not satisfy all of the boundary conditions (4)(6). It is reasonable to consider the problem t2ux2u+qu=finQ0,(10) t2u+qu=finQ,(11) u=φ,tu=ψonG×{0},u=μonG0×I,(12) the restriction ofu(,t)toG0is continuous atafor alltI,(13) eE0xue=0on{a}×I,(14) which is actually an uncoupled system of the initial boundary value problem for the hyperbolic equation on the graph 𝒢0 and ordinary differential equations on edges of 𝒢*.

Lemma 3.1

Under assumption (7), (8) there exists a unique weak solution of (10)(14).


First, the restriction u to 𝒬0 solves the hyperbolic boundary value problem on 𝒢0: t2ux2u+qu=finQ0,(15) u=φ,tu=ψonG0×{0},u=μonG0×I,(16) u(,t)is continuous atx=afor alltI,(17) eE0xue=0on{a}×I.(18)

The problem admits a unique weak solution, because the input data are smooth enough and the C1 compatibility conditions [15] φ(aj)=μ(aj,0),ψ(aj)=tμ(aj,0)forajG0,eE0φe(a)=0(19) hold, which follows from (7) and (8).

Next, we can find u on the rest of edges by solving the Cauchy problems for ordinary differential equations with respect to time t2u+qu=fine×I,u=φ,tu=ψone×{0}(20) for each e ∈ 𝓔* separately. All these problems can be solved explicitly: u(x,t)=φ(x)cosq(x)t+ψ(x)q(x)sinq(x)t+1q(x)0tf(x,τ)sinq(x)(tτ)dτ,ifq(x)0,(21) u(x,t)=φ(x)+tψ(x)+0t(tτ)f(x,τ)dτ,ifq(x)=0(22) for all x ∈ 𝒢* t ∈ 𝓘. Here we choose the single-valued branch of the root w=z such that w(1) = 1. It is easy to check that the solution u given by (21) and (22) is a real-valued C-function on each edge e ∈ 𝓔*, since the functions q, f, φ and ψ are smooth by (7). Therefore the restriction u to the subgraph 𝒬* belongs to Ċ(𝒬*). Returning now to the whole graph 𝒢, we see that u is a solution of (10)(14).  □

From now on, U0 stands for the solution of (15)(18) and u0 for the solution of (20). Thus u(x,t)=U0(x,t)if(x,t)Q0,u0(x,t)if(x,t)Q.(23)

Because we can not control in time the value of u0 both at the central vertex a and on the boundary of graph 𝒢*, the approximation (23) generally ignores continuity condition (5) as well as boundary conditions (4) on ∂𝒢*. This approximation is therefore not suitable for the whole graph 𝒢. We shall refine (23) by applying boundary layer approximations.

3.2 Boundary layers in a vicinity of the central vertex

First we will modify the approximation (23) in the area of degeneration of the hyperbolic equation in order to satisfy the continuity condition (5).

Given a subgraph 𝒢i, we introduce the new variables ξ = εmi(xa). In the auxiliary space ℝ2 with the variables ξ we consider the star graph Giε , which is the image of 𝒢i under the expanding mapping xξ. For all positive ε this image lies on the noncompact star graph Gi=({O},Ei), as shown in Fig. 2.

The graphs 
Giε $\mathcal G^\varepsilon_i$ 
Gi∞ $\mathcal G_i^{\infty}$ .
Fig. 2

The graphs Giε and Gi.

Therefore the star-like graph Gi has the same number of edges as 𝒢i, but its edges are noncompact, i.e., the edges are rays with the origin at point O of the auxiliary plane ℝ2. Let Ei be the set of edge-rays ej of Gi. All such edges ej possess the natural parametrisation πj:[0,+)ej,πj(τ)=ajaajaτ.

Since ∂ξ = εmix, in terms of the new variables the homogeneous PDE (2) on 𝒢i can be written as t2vεξ2vε+q(a+εmiξ)vε=0inGiε×I(24) for all i = 1, …, k.

The coefficient bε in (2) is infinitely small on 𝒢* as ε → 0, therefore we can use bε as a “small parameter” in this subgraph. Let us introduce the fast variables yε=(xa)/bε(x) on the whole graph 𝒢*, which coincide with ξ = εmi(xa) on each subgraph 𝒢i, and modify our approximation uε(x,t)U0(x,t)if (x,t)Q0,u0(x,t)+v0(yε,t)if (x,t)Q.(25) We define the function v0 as follows. Fix a number i∊ {1, 2,…, k} and an edge e∊ 𝓔i, and let e be the corresponding edge-ray of Ei in ℝ2 with the coordinates ξ = εmi(xa) (see Fig. 2). In view of (24) and the formal Taylor expansion qe(a+εmiξ)=qe(a)+εmiqe(a)ξ+12ε2miqe(a)ξ2+,(26) we assume that the restriction of v0 to the edge e solves the initial boundary value problem t2vξ2v+qe(a)v=0in e×I,v=0,tv=0on e×{0},v=U0(a,)u0e(a,)on {O}×I.(27)

The value U0(a, t) is uniquely defined, since U0 is continuous at the vertex a. Hence, the approximation (25) satisfies (5), since u0e(a, t) + v0e(O, t) = U0(a, t) for all e∊ 𝔼* and t ∊ 𝓘. Note that the initial data of (27) satisfy the compatibility conditions: limt0v(O,t)=U0(a,0)u0e(a,0)=φ(a)φ(a)=0,limt0tv(O,t)=tU0(a,0)tu0e(a,0)=ψ(a)ψ(a)=0, which follows from (16), (20) and continuity of φ and ψ.

Lemma 3.2

Let v0 be a solution of (27). Given T > 0, the composition Vε(x,t)=v0xabε(x),t represents a boundary layer function near the central vertex as ε→ 0, i. e., Vε is different from zero in the bε t-neighbourhood of the vertex a only.


Problem (27) is a standard initial boundary value problem for hyperbolic equation with constant coefficients in a quarter-plane [39, II.2] of the form t2vs2vϑv=0in {(s,t)R2:s>0,t>0},v(s,0)=α(s),tv(s,0)=β(s)for s>0,v(0,t)=ν(t)for t>0.(28)

It admits a unique solution, provided the C1 compatibility conditions ν(0)=α(0),ν(0)=β(0)(29) hold. Under the characteristic st = 0 passing through the origin, where the boundary value condition ν have no effect on the solution, we have v(s,t)=12(α(s+t)+α(st))+12sts+t(k(t,s,y)β(y)+tk(t,s,y)α(y))dy,(30) where k(t,s,y)=J0(ϑ((sy)2t2)) and J0 is the Bessel function (see [39, II.5]). The last formula is valid for all real ϑ, in particular, for ϑ = 0 it turns into d’Alembert’s formula.

Hence, for st > 0 the solution v(s, t) depends on the initial data α and β in the interval [st, s+t] ⊂ ℝ+ only. From this we deduce that a solution v0 of (27) is equal to zero for s > t, because it satisfies the homogeneous initial conditions. In other words, v0 can be different from zero in the t-neighbourhood of the origin 𝒪 only. Then Vε is nonzero on the set {(x,t):t>0,xa<bε(x)t} only, i.e., in the bε t-neighbourhood of the vertex a only. This region is small as ε → 0 uniformly on t∊ (0, T).  □

3.3 Improvement of the asymptotics near the boundary 𝒢*

We now modify asymptotics (25) near the boundary vertices aj∊ ∂ 𝒢*, in the same way as we did it near the central vertex a. Let e be the edge of subgraph 𝒢i connecting the vertices a and aj. We introduce the new variables z=εmi(xaj) for xe. Suppose that the function w0,e solves the initial boundary value problem in the quarter-plane t2wz2w+qe(aj)w=0 in {(z,t):z<0,t>0},w(z,0)=0,tw(z,0)=0 for z<0,w(0,t)=μe(t)u0e(aj,t) for t>0.(31) In view of (8) and (20), the C1 compatibility conditions for the problem hold. As in the earlier proof of Lemma 3.2, one likewise deduces that w0,e is nonzero above the characteristic z + t = 0 only.

We define w0ε :𝒬*→ ℝ as follows. For any ej = (a, aj) such that ej ∊ 𝓔i, i = 1,…, k, the restriction of w0ε to the set ej × 𝓘 coincides with the function w0,ej(εmi(xaj),t).(32) Hence, w0ε is a boundary layer function, which describes the singular behaviour of uε in a neighbourhood of boundary ∂ 𝒢*. Note the boundary conditions at z = 0 have been chosen in the manner that u0 + w0ε satisfies (4).

We will from now on assume that the time variable t belongs to a finite interval (0, T), and introduce the notation 𝓘T = (0, T), 𝒬T = 𝒢× 𝓘T, Q0T=G0×IT, and QT=G×IT. Then if ε is small enough and t∊(0, T), the functions v0(yε, ⋅) and w0ε are actually boundary layers localized near the vertex of 𝒢*(see the second plot in Fig. 3).

The boundary layers near the vertices a and aj on the edge ej of subgraph 𝒢*⋅ The function v0 is a solution of (27) and 
w0ε $w^\varepsilon_0$ 
is defined by (31), (32).
Fig. 3

The boundary layers near the vertices a and aj on the edge ej of subgraph 𝒢*⋅ The function v0 is a solution of (27) and w0ε is defined by (31), (32).

Hence, we set uε(x,t)U0(x,t)if (x,t)Q0T,u0(x,t)+v0(yε,t)+w0ε(x,t)if (x,t)QT.(33) We will refer to (33) as the leading behavior or leading terms of the asymptotics. At this point, this approximation produces the largest error of order εm1 in the Kirchhoff condition (6). We end the section by construction of the first correctors on 𝒢0, although we already solved the task declared in the title of this section. The problems for these correctors are slightly different from the problem for U0.

3.4 Correction of approximation error in the Kirchhoff condition

To improve the accuracy of the approximation in the Kirchhoff condition (6), we finally add new terms to the approximation on 𝒢0: uε(x,t)U0(x,t)+i=1kεmiU1(i)(x,t)if (x,t)Q0T,u0(x,t)+v0(yε,t)+w0ε(x,t)if (x,t)QT.(34) Our substituting (34) into (6) yields eE0xU0e(a,t)+i=1kεmieE0XU1e(i)(a,t)+i=1kεmiγEiξv0γ(O,)+i=1kε2miγEixu0e(a,t)0.(35)

The function w0ε is absent in the last formula, because it vanishes in a neighbourhood of the vertex a. The first sum in (35) is zero due to (18). The next two double sums have to eliminate each other. Therefore the function U1(i) must be a solution to the problem t2Ux2U+qU=0 in Q0,U=0,tU=0 on G0×{0},U=0 on G0×I,U(,t) is continuous at a for all tI,eE0xUe(a,)=γEiξv0γ(O,) on I,(36) where i ∊ {1, …, k}. The problem is a partial case of the hyperbolic boundary value problem on 𝒢0 with the nonhomogeneous Kirchhoff condition: t2ux2u+qu=0 in Q0,u=φ,tu=ψ on G0×{0},u=μ on G0×I,u(,t) is continuous at a for tI,eE0xue=ν for {a}×I.(37)

It is reasonably easy to see that the C1 compatibility conditions for the problem have the form: φ(ai)=μ(ai,0),ψ(ai)=tμ(ai,0) for aiG0,φ,ψ are continuous at x=a,eE0φe(a)=ν(0),(38) provided ν is a continuous function on 𝓘. And indeed these conditions hold for (36), since v0γ(y, 0) = 0 for all γEi . Problem (37) admits a unique solution u ∊ 𝒰(𝒬0)[26], see also the proof of Lemma 5.1.

By construction, approximation (34) satisfies conditions (3)-(5) exactly and (6) up to terms of order ε2m1, since m1 is the smallest number among the powers mi, i = 1, …, k. Besides, v0 and w0ε cause small errors in the right hand side of (2) which are localized in the boundary layer regions.

4 Formal asymptotic expansions: general terms

Taking into account our work in the previous section, we will look for a complete asymptotic expansion of uε in the form uε(x,t)U0(x,t)+s=1i=1kεsmiUs(i)(x,t) if (x,t)Q0T,(39) uε(x,t)s=0{bε(x)}s2(us(x,t)+vs(yε,t)+wsϵ(x,t)) if (x,t)QT.(40) We also assume that all vs and wsε are functions of the boundary layer type, i.e., for small ε the functions vs(yε, t) are different from zero in a small neighbourhood of the central vertex and wsε can possess non-zero values in a vicinity of the boundary ∂ 𝒢* only. The validity of these assumptions will be considered further when we will construct the asymptotics. Recall that {bε(x)}s2=εsmi for xGi.

This may happen that some terms in (39) as well as in (35) have the same order of smallness. This is because there are equal numbers among the integers smi. Let us introduce the sets Λ(p)={(n,i)N×{1,,k}:nmi=p} for each natural p. Of course, some of them are empty. All sets Λ(p) are finite, therefore the terms of the same order in (39) can be aggregated in the final form of approximation: s=1i=1kεsmiUs(i)=p=1εp(s,i)Λ(p)Us(i)

Substituting (39) and (40) into (2)(6) and collecting powers of ε give a sequence of problems for the terms of series. First, the functions Us(i) are solutions to the problems t2Us(i)x2Us(i)+qUs(i)=0in Q0,Us(i)=0,tUs(i)=0 on G0×{0},Us(i)=0 on G0×I,Us(i)(,t) is continuous at a for all tI,eE0xUs,e(i)=eEixus2,eγEiξvs1,γ(O,) on {a}×I(41) for s ≥ 2 and i ∊ {1, …, k}. In our approximation the terms Us(i) reduce errors in the Kirchhoff condition (6), but without modification of continuity condition (5) as well as the initial and boundary value conditions (3), (4). Recall that each subgraph 𝒢i = {𝒱i, 𝓔i} is associated with the non-compact graph Gi={Vi,Ei}, obtained by the dilatation ξ = εmi(xa) as ε → 0. The point O is the origin of plane ℝ2 with the variables ξ.

The functions us are terms of the regular asymptotics on subgraphs 𝒢1, …, 𝒢k. Since the hyperbolic equation (2) becomes degenerate for ε = 0, we can find us on each edge e ∊ 𝓔* by solving the Cauchy problems for ordinary differential equations with respect to time t2us+qus=x2us2 in e×I,us=0,tus=0 on e×{0}(42) for s ≥ 1, where u0 is a solution of (20) and u−1 = 0. The task of us is to exhaust the residual in the right-hand side of (2) on more flexible graphs 𝒢i, i ≥ 1.

Remark 4.1

Equation (42) is homogeneous for s = 1. Then u1 = 0 by uniqueness, and recursive calculations yield u3 = 0, u5 = 0, …. Hence all us = 0 with the odd index s are zero functions.

Next, the continuity condition (5) can be satisfied asymptotically by means of the boundary layer functions νs = νs(yε, t) which are localized about the central vertex. Given i ∈ {1, 2, …, k} and an edge e ∈ 𝓔i, we consider the corresponding edge-ray γ of Gi and define the restriction of νs to γ × 𝓘 as a solution of the problem t2vsξ2vs+qγ(a)vs=r=1s1r!qγ(r)(a)ξrvsr,γinγ×I,(43) vs=0,tvs=0onγ×{0},(44) vs=(r,l)Λ(smi)Ur(l)(a,)us,e(a,)on{O}×I(45)

for all s = 1, 2, ….

Finally, we define the map wsε:QR that is a collection of boundary layers on each edge e ∈ 𝓔*. These boundary layers are localized near the boundary ∂ 𝒢* and reduce the approximation error in boundary condition (4). For any i = 1, …, k and e ∈ 𝓔i, the restriction of wsε to e is given by ws,eε(x,t)=ws,e(εmi(xaj),t),

where ws, e solves the problem t2ws,eZ2ws,e+qe(aj)ws,e=r=1s1r!qe(r)(aj)zrwsr,einP,ws,e(z,0)=0,tws,e(z,0)=0forz<0,ws,e(0,t)=us,e(aj,t)fort>0(46)

for all s ∈ ℕ. Here P is the quarter-plane {(z, t):z < 0, t > 0} and z = εmi(xaj) is a new fast variable on the edge e.

Lemma 4.2

Suppose that the initial data of problem (2)(6) satisfy conditions (7) and (8). Then the coefficients Us(i),us,vsandwsε of formal series (39), (40) are unique and can be determined recursively up to arbitrary order s in the class of continuously differentiable functions which possess locally bounded derivatives of the second order. Furthermore, for ε small enough and t ∈ (0, T) the functions vs(yε,)andwsε are boundary layer corrections which are localized about the central vertex a and the boundary ∂ 𝒢* respectively.


The proof is by induction on s. Assume that we have already found the terms Ur(i),ur,vrandwrε for r < s with desired smoothness, where vr(yε,)andwrε are functions of boundary layer type as ε → 0. In order to show that the s-th step in the recursive process is solvable, we must analyze solvability of problems (41)(46) in appropriate spaces.

We start with problem (42) which admits the explicit solution: us(x, t) = 0 for odd s, and us(x,t)=1q(x)stx2us2(x,τ)sinq(x)(tτ)dτ,ifq(x)0,(47) us(x,t)=0t(tτ)x2us2(x,τ)dτ,ifq(x)=0(48)

for even s and (x, t) ∈ 𝒬*. As shown in the proof of Lemma 3.1, a solution u0 of (20) is a smooth function. Hence, all us are also Ċ(𝒬*)-functions due to the recursive procedure.

Hyperbolic problem (41) on graph 𝒢0 contains the non-homogeneous Kirchhoff condition, where the right hand side is known by induction hypothesis. The problem admits a unique solution Us(i)U(Q0), provided the C1 compatibility conditions (38) hold. In this case, the conditions take the form eEixus2,e(a,0)+γEiξvs1,γ(O,0)=0,

but the last equality is true, because for s > 2 the terms us−2 and vs−1 satisfy the homogeneous initial conditions at t = 0 by (42) and (44). For s = 2 we have eEixu0,e(a,0)=eEiφe(a)=0fori{0,...,k}

by the fitting conditions (8) and v1,γ(y, 0) = 0 on γ by (44).

As for problem (43)(45), we first of all note that all right hand sides are already defined. The problem is the classic mixed problem for hyperbolic equation on the half-line [35, 36]. There exists a unique solution vs,γ ∈ 𝒰(γ × 𝓘) for any edge γEi and i ∈{1, …, k if the compatibility conditions (29) hold, i.e., (r,l)Λ(smi)Ur(l)(a,0)=us,e(a,0),(r,l)Λ(smi)tUr(l)(a,0)=tus,e(a,0).

Both the equalities are true, since Ur(l) and us, e satisfy the homogeneous initial conditions at t = 0 as solutions of (41) and (42) respectively. Next, the right hand side of (43) is identically zero under the characteristic st = 0, by induction. Taking into account the homogeneous initial conditions (44) and integral representation (30) we conclude that vs,γ is equal to zero for s > t > 0.

Similar considerations can be also applied to problem (46) on the half-line, for which there exists a unique solution ws,e ∈ 𝒰(P). 

5 Justification of the asymptotic expansions

In this section we will prove that formal series (39), (40) actually solve the singularly perturbed problem (2)(6) in the sense of asymptotic approximation.

5.1 Estimation of remainder terms

We introduce the partial sums of (39), (40) uε,p(x,t)=U0(x,t)+s=1pi=1kεsmiUs(i)(x,t)if(x,t)Q0T,s=0p{bε(x)}s2(us(x,t)+vs(yε,t)+wsε(x,t))if(x,t)QT(49)

with all coefficients constructed in Sections 3 and 4. Substituting uε, p into singularly perturbed problem (2)(6), we can estimate the remainder terms in the equation and all conditions. A somewhat lengthy, but not complicated, computation shows that uε, p is a solution of the problem t2uε,px(bεxuε,p)+quε,p=f+hε,pinQT,(50) uε,p=φ,tuε,p=ψonG×{0},(51) uε,p=μonG×IT,(52) uε,p(,t)iscontinuousatx=aforalltIT,(53) eEbeε(a)xueε,p(a,t)=νε,p(t)foralltIT.(54)

By construction the function uε, p precisely satisfies the initial and boundary value conditions (51), (52) as well as the continuity condition (54). The term hε, p in right-hand side of (50) is different from zero in a neighbourhood of the central vertex a and the boundary ∂ 𝒢* only. Moreover, there exists a constant c1(T) such that |hε,p(x,t)|c1(T)ε(p+1)m1forall(x,t)QT.(55)

The remainder νε, p in the Kirchhoff condition (54) is a continuous function and |νε,p(t)|c2(T)ε(p+1)m1fortIT.(56)

Recall that m1 is the smallest number in the set {m1, …, mk}. In order to prove that the smallness of remainders hε, p and νε, p implies the asymptotic smallness of the difference between the exact solution uε of (2)(6) and the approximation uε, p as ε → 0 we need some estimates for solutions of hyperbolic problems on graphs.

5.2 A priori estimate

Let us consider the initial boundary value problem on graph 𝒢 t2ux(b(x)xu)+q(x)u=f(x,t)inQT,(57) u=φ,tu=ψonG×{0},(58) u=0onG×IT,(59) u(,t)iscontinuousatx=aforalltIT,(60) eEbe(a)xue(a,t)=ν(t)foralltIT.(61)

Throughout the section, W2l(Ω),l=0,1,..., stands for the Sobolev space of functions defined on a set Ω which belong to L2(Ω) together with their derivatives up to order l. In particular, we say that a function f belongs to the Sobolev space W2l(G) on graph 𝒢, if its restrictions fe belong to W2l(e) for all edges e ∈ 𝓔.

Lemma 5.1

(A priori estimate). Assume that the input data of (57)(61) satisfy conditions (7), (38) and the coefficient b is constant on each edge e ∈ 𝓔. If u is a solution of (57)(61), then uW22(QT)C(T)(φW21(G)+ψL2(G)+νL2(IT)+fL2(QT))(62)

for some constant C(T).


The main idea of proof is to decompose the problem on graph 𝒢 into n problems on edges, for which such estimate is a well-known result.

Let u be a solution of (57)(61) belonging to 𝒰(𝒬T). We will denote by σ the restriction of u to the set {a} × 𝓘T. Then for each edge e ∈ 𝓔 the restriction ue is a solution to the problem t2uebeη2ue+qeue=fein(0,j)×IT,(63) ue(η,0)=φe(η),tue(η,0)=ψe(η),η(0,j),(64) ue(0,t)=σ(t),ue(j,t)=0tIT,(65)

where e connects the vertices a and aj, and η is a point of the interval (0, ℓj). It should be stressed that σ is the same function for all edges e ∈ 𝓔, which is a consequence of continuity condition (60).

At the same time, the function σ is a solution of the Volterra integral equation of the second kind σ(t)0tK(t,τ)σ(τ)dτ=F(t)(66)

with the continuous kernel 𝒦 and right-hand F. This equation was derived and studied by authors in [26] and the explicit representation of F via the input data of (57)(61) was obtained. Next, for a solution of (63)(65) we have the estimate [37, IV.4] ueW22(e×IT)C1(T)(φW21(e)+ψL2(e)+σL2(IT)+fL2(e×IT)),(67)

where e ∈ 𝓔. On the other hand, the equation (66) admits a unique solution σ such that σL2(IT)C2(T)(φW21(e)+ψL2(e)+νL2(IT)+fL2(e×IT)),(68)

since the right-hand side F depends on the input data of (57)(61). Inequalities (67) and (68) combined give the estimate (62) after summation over e ∈ 𝓔.  □

5.3 Justification of asymptotics

We now have the desired result.

Theorem 5.2

Given T > 0 suppose that uε is a weak solution of problem (2)(6) in 𝒬T. Then uε admits the asymptotic expansion of the form uε(x,t)U0(x,t)+s=1i=1kεsmiUs(i)(x,t)if(x,t)Q0T,s=0{bε(x)}s2(uS(x,t)+vs(yε,t)+wSE(x,t))if(x,t)QT

with all coefficients constructed in Sections 3 and 4, namely for any p = 0, 1, … the solution uε and the approximation uε, p given by (49) satisfy the inequality uεuε,pW22(QT)Cp(T)ε(p+12)m1(69)

with constant Cp(T), being independent of ε.


Our proof starts with the observation that the difference uε, puε solves the problem (50)(54) with φ, ψ, μ and f replaced by zero functions. Therefore this difference can be estimated by the corresponding norms of remainder terms hε, p and νε, p using estimate (62). But one must be careful with this estimate, because the constant C(T) is inversely proportional to the ellipticity bound of (57). In the case of problem (50)(54) we have minx∈ 𝒢bε(x) = ε2mk for small ε, where mk = maxi{mi}. Hence, uεuε,pW22(QT)c1ε2mk(νε,pL2(IT)+hε,pL2(QT))c2ε(p+1)m12mk,

by (55) and (56). In order to improve the estimate, we consider the last inequality for bigger number p + r. Then uεuε,p+rW22(QT)c2ε(p+1)m1+(rm12mk)c3ε(p+1)m1,

provided rm1 ≥ 2mk. From this we readily deduce that uεuε,pW22(QT)uε,p+ruε,pW22(QT)c3ε(p+1)m1,

and hence that uεuε,pW22(QT)c3ε(p+1)m1+uε,p+ruε,pW22(QT).

It is worth remarking at this stage that the W22 norms of boundary layers νs(yε, ⋅) and wsε are infinite large as ε → 0. Moreover, the contributions in the norm from the second derivatives on x are the largest. Therefore uε,p+ruε,pW22(QT)c4((bε)p+12x2vp+1(yε,)L2(QT)+(bε)p+12X2wp+1εL2(QT)).

We will derive an estimate for the first norm in the right hand side of the last inequality. The second one can be bounded similarly. We have (bε)p+12x2vp+1(yε,)L2(QT)2=i=1kε2mi(p+1)γEix2vp+1,γ(yε,)L2(γ×IT)2c5i=1kε2mi(p+1)εmic6ε(2p+1)m1,

since x2vs,γ(yE,)L2(γ×IT)2=0Tγ|x2vs,γ(εmi(xaj),t)|2dγdt=ε2mi0T0εmi|ξ2vs,γ(εmiπγ(α),t)|2dαdt=εmi0T01|ξ2vs,γ(ξ,t)|2dξdtc7εmi

for any γ ∈ 𝓔i, where πγ:[0, ℓj]→ γ is the natural parametrization of the edge γ. Here we also used the main property of boundary layer vs(yε, ⋅) to be different from zero in the bεt -neighbourhood of vertex a only. Hence uε,p+ruε,pW22(QT)c8ε(p+12)m1.(70)

Combining the previous inequalities now yields (69).  □


The authors would like to thank Ruslan Andrusiak for helpful discussions. We are also greatly indebted to Referee for carefully reading the paper and suggesting some improvements.


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About the article

Received: 2016-10-21

Accepted: 2017-02-22

Published Online: 2017-04-09

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 404–419, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0030.

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© 2017 Golovaty and Flyud. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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