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Green's function method for time-fractional diffusion equation on the star graph with equal bonds
Z. A. Sobirov1'2, K.U. Rakhimov2, R. E. Ergashov2 University of Geological Sciences, Olimlar str., 49, 100041, Tashkent, Uzbekistan 2National University of Uzbekistan, Universitet str., 4, 100174, Tashkent, Uzbekistan z.sobirov@nuu.uz, kamoliddin_ru@inbox.ru
PACS 02.30.Jr, 02.30.Em DOI 10.17586/2220-8054-2021-12-3-271-278
This work devoted to construction of the matrix-Green's functions of initial-boundary value problems for the time-fractional diffusion equation on the metric star graph with equal bonds. In the case of Dirichlet and mixed boundary conditions we constructed Green's functions explicitly. The uniqueness of the solutions of the considered problems were proved by the method of energy integrals. Some possible applications in branched nanostructures were discussed.
Keywords: Time-fractional diffusion equation, IBVP, PDE on metric graphs, Green's function. Received: 5 May 2021 Revised: 9 June 2021
1. Introduction
In this work we consider initial-boundary value problem (IBVP) for time fractional diffusion equation
d2
Do>(x,t) = dx2u(x,t) - f (x,t), on the bounded star graph. Here D0 tg(t) is the fractional derivative defined by:
t
D^ tg(t) = F7T--rd i , g(!), 0 < a < 1,
nt w r(1 - a) dt J \t - £\a
n
where r(x) is the Gamma function.
It is known that the diffusion equation is widely used in many fields of science including physics, biology, mechanics, chemistry and others. In recent years, the theory of fractional calculus has been studied with great interest. In particular, in the work [1] Bartlomiej Dybiec and Ewa Gudowska-Nowak considered a long-time, scaling limit for the anomalous diffusion composed of the subordinated Levy-Wiener process. In [2] T. A.M. Langlands and B. I. Henry introduced mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms. In [3], the fractional Fokker-Planck equation for subdiffusion in a general space and time-dependent force field was studied. In [4], the authors solved an asympotic boundary value problem without initial conditions for diffusion-wave equation with time-fractional derivative and gave some applications in fractional electrodynamics. Igor Goychuk in [5], considered an alternative continuous time random walk and fractional Fokker-Planck equation description.
Naturally, such a powerful tool as a fractional derivative cannot stay away from nanoscience. In particular, this can be seen from the results in [6,7]. In [6] authors obtained solutions for the magnetohydrodynamic mixed convection problem of Maxwell fractional nanofluid. In [7] authors studied models with non-integer order derivatives to describe dynamics in nanofluids. In particular, it is noticed that the rate of heat transfer increases with increasing nanoparticle volume fraction and order of the time-fractional derivative.
It is known that the Green's function method is a powerful technique for solving boundary value problems. The Green's function method for fractional order equations was investigated by A. V. Pskhu in [8]. In [9] Green's function method was used to find numerical solution of boundary value problems for stationary and non-stationary differential equations in different dimensions.
Nowadays, differential equations are being investigated with great interest on the metric graphs. It is mostly due to the fact, that majority of physical systems demonstrate flows, e.g., flows through the branched nanotubes [10]. The Schrodinger equation on the metric graph are well studied (see [11-13] and references in them). In [14] the Schrodinger operator on the graph with varying edges was investigated. The Schrodinger equation on the metric graphs was also explored with Fokas unified transformation method in [15]. In [11] it was considered as scattering
problem on the quantum graph, which consists ring with two attached semi-infinite leads. In [16] authors proposed the model of time-dependent geometric graph for description of the dynamical Casimir effect. In [17], the authors used metric graph approximation for investigation of strong variation of the viscosity and density in cylindrical domains of the small radius. And paper [18] devoted to an explicitly solvable model for periodic chain of coupled disks. In [19-21] IBVP for time-fractional Airy equation on the star graphs are considered via method of potentials.
2. Formulation of the problems
In this paper we construct Green's function of two IBVPs on the metric star graph for the time-fractional subdiffusion equation. Motivation for the study of the fractional diffusion equations on metric graphs comes from such practically important problems as anomalous heat transport in mesoscopic networks, subdiffusion processes in nanoscale network structures, molecular wires, different lattices and discrete structures. In such discrete structures one has memory effect which leads, in particular, to time fractional equations. These problems, in particular, describe transport of nanofluids in branched structures [6,7].
We consider the star graph which has m = k + l bonds connected in one point O (Fig. 1). We define coordinates in the graph's edges by isometric mapping this bonds to the interval from 0 to L, such that in each edge the coordinate 0 corresponds to the vertex O. The bonds of the graph are denoted by Bj , j = 1, m.
Fig. 1. Star graph with m bonds. We considered two cases of the boundary condition (BC): (a) Dirichlet BC at the all boundary vertices; (b) Dirichlet BC at the end points of the blue bonds and Neumann BC at the end points of the green bonds
On each bond Bj of the graph, we consider the time-fractional subdiffusion equation
d2
D0>j (x,t) = ^-^uj (x,t) - fj (x,t), 0 < x < L, 0 <t<T,j = 1 ,m, (1)
with the following initial conditions
lim D'àt Uj(x,t) = ipj(x), 0 < x < L, j = 1, m. (2)
At the inner vertex of the graph, we use the following gluing (Kirchhoff) conditions
Ui(0,t)= U2(0,t) = ... = Um(0,t), (3) ( m d \
!im0g dxUi (x,t^=0, (4)
for all t g [0, T]. These conditions ensure the local flow conservation at the branching point of the graph. At the boundary vertices, we will use Dirichlet or Neumann boundary conditions (BC) given by
Ui(L,t) = (t), i = 1, m, (5)
Ui(L,t)= Yi(t), i =lk, (6) d _
—Uj(L,t) = Yj(t), j = k + 1,m. (7)
We suppose that the functions fj(x, t), j = 1, m, initial and boundary data are smooth enough on the closure of their domains and the compatibility conditions on the boundary and branching points hold.
Problem 1. Find regular solutions of equations (1) which satisfy conditions (2) — (5). Problem 2. Find regular solutions of equations (1) which satisfy conditions (2) — (4), (6), (7).
3. Uniqueness of solution
We put u = (u1, u2,..., um)T, f = (f1,..., fm)T, f = (^1,..., fm)T. We define the norm of a vector function v(x) = (v1(x),...,vm(x)) by IMI2 = E 1=1 f0L vi (x)dx.
t
= It-rf-a o <.< i,
0
is the time-fractional integral [22].
Lemma 1. The problem 1 has at most one solution. Furthermore, if ^¿(t) = 0, i = 1, m, then the solutions satisfy the following a-priori estimate
||D071u\\2 < Ea(ta) • \\f\\ + r(a)Ea'a(ta)D0-a\\D071f\\2, (8)
where Ea(z) = ^zn/r(an + 1) and Ea,M(z) = ^zn/r(an + are the Mittag-Leffler functions.
Proof. Let ^j(t) = 0, i = 1, m. We put w(x, t) = DJ ° 1u(x, t). It is clear, that the vector function w(x, t) = (w1(x, t), w2(x, t),..., wm(x,t))T satisfies the following equation:
doa01w = Wxx + D071f (x,t), 0 < x < L, 0 < t < T, (9)
with Caputo time-fractional derivative:
t
tg(t) = ^-r f , r (e) de, 0 < a < i.
n>tyw r(i - a) j |t - eia
n
From [22], we have:
L L
j WjDatwjdx > ^Da tJ w2dx, j = 1, 2, ...,m. (10)
o o
Taking into account inequality (10), we multiply (9) by wT from the left side, integrate the resulting relation with respect to x from 0 to L and use Cauchy's inequality:
da t||w||2 + 2||wx||2 < ||w||2 + ||D07/1|2.
Hence,
Do,t||w||2 < ||w||2 + ||Doa-/1|2. (11)
From the analog of Gronwall-Bellman lemma (see [22]), in the case of Caputo type fractional derivative and the relation (11), we get:
||w(-,i)||2 < Ea(0||w(-,0)||2 +r(a)Ea ,a(nD-«||D071 /(,i)||. So, we proved a-priory estimate (8). Uniqueness of the solution follows from this estimate.
Lemma 2. The problem 2 has at most one solution. Furthermore, if Yj(t) = 0, i = 1, m, then the solutions satisfy the following a-priori estimate
\\D071u\\2 < Ea(ta) • \\f\\ + r(a)Ea , a(ta)D°aWD^ 01f\\2, (12)
Proof. The proof is similar with the proof of the Lemma 1.
4. Green's function of Problem 1
The following theorem directly follows from the theorem 4.3.1. (see [8]). Let E = {(x,t): 0 < x < L,
0 < t < T} (where T is positive real number) and Et = {(£, t): 0 < £ < L, 0 < t < t}.
Theorem 1. Let the m x m matrix function V = V(x, t; £, t) satisfy following conditions:
1. The matrix function V is the solution for equation
% (x, t; £,t ) - V (x, t; £,t )=0
on the fixed point (x, t) g E;
2. For every m x 1 vector function h(x) g C[0, L] holds
L
Tim^ D^-V(x, t; £, t)h(£)d£ = h(x). (13)
0
3. The matrix functions V, V5, , D^-1V are continuous on the E x Et\{t = 0} and for all points (x, t) g E and (£, t) g Et holds inequality
|Vij(x, t;£,t)| < (t - t)a/2-1. If the function u(x, t) is the solution of the equation (1) and satisfies the condition (2), then for all (x, t) is hold,
that:
t
i(x, t) = j(V(x, t; L, t)uç(L, t) — V(x, t; 0, t)uç(0, t) — VÇ(x, t; L, t)u(L, t)+
t L
Vç(x, t; 0, t)u(0, t))dT — j V(x,t; £, 0)^(£)d£ + j j V(x,t; £,t)f (£, t)d£dT.
0 0 0 Proof. The proof is similar to the proof of the theorem 4.3.1. [8].
Taking into account the above theorem, we look for solution of Problem 1 in the following form
t
0
u(x, t) = j(G(x, t; L, t)uç(L, t) — G(x, t; 0, t)uç(0, t) — Gç(x, t; L, t)u(L, t) +
l t L
Gç(x, t; 0, t)u(0, t))dT — J ^(t)G(x,t; £, t)d£ — J J G(x,t; £,t)f (£, t)d£dT, (14)
0 0
where:
G=
V
G11 G12 . G1m
G21 G22 . G2m
Gm1 Gm2 Gmm
is a matrix-Green's function. Green's function satisfies the equation:
G55 - d£t g = 0,
for all £ = x, 0 < t < t.
We look for Green's function in the following form
G = ^ (A„r(x — £ + 2nL,t — t ) + B„r(x + £ + 2nL,t — t )), (15)
where An and Bn are constant matrices of dimension m x m and r(s, t) is:
r(s,t) = V/^eî'^f— . v ' > 2 W2 ^ t«/2 )
2 t° We have to find the matrices An and Bn.
Taking into account (14) and the conditions (2)—(4) we get the following conditions for matrix-Green's function:
G|ç=L = 0, (16)
L
n= —oo
G |ç=o = G |ç=o = ... = Gim|ç=o, i = i,m,
Gç1|ç=o + G|Vo + ... + Gç"|ç=o = 0, i = i,
Let
/ Af1 Af2 ... Afm \
An — Af1 Af2 ... A2m ... An
n n
\ Ami An m2 An mm ... An /
and Bn
( bu Bf2
R21 R22
n n
.. Bnm N
B2m
Dml Dm2 Dmm I
\ Bn Bn ... Bn )
From (16), we have: From (17), we get: From (18), we obtain:
An = - B
n-1.
/til /tim Dim oil An - An = Bn - Bn .
^Ajj =£ Bj, i = 1, m.
j=i j=i
We rewrite the relations (20) and (21) in the following matrix form Bn = AnM, where:
(2- m 2 ...
M =1
m
Combining (19), (20) and (21), we get:
And we find
2 2 - m ... \ 2 2 ...
Bn = AnM An+i = -Bn
An+i = -AnM.
2 2
2-m
From the condition (13) follows that A0 = I. So, we have
An = (-1)nMn, Bn = (-1)nMn+1. Substituting (23) into (15) we get matrix Green's function of the Problem 1:
+w
G = ^ (-1)nMn (r(x - £ + 2nL, t - t) + Mr(x + £ + 2nL, t - t)). Taking into account Lemma 1 we get following theorem.
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
t L
Theorem 2. Let ^(t),fi(t) G C [0,T] (i = 1,m,T > 0), and f (x,t) G C 0'1{(x,t) : 0 < x < L, 0 < t < T} Then the Problem 1 has unique solution in the form of:
t L
u(x,t) = —J G^ (x,t; L, t)u
00
where G(x, t; £, t) is given by (24)
(x,t) = -J Gç (x,t; l,t )u(L,T )dr -J ^(£)G(x,t; £,t )d£ -J J G(x,t; £,t )f (£, t )d£dr,
o o o o
5. Green's function of Problem 2
In this case, the Green's function has a form of:
Gi = ^ (Cnr(x - £ + 2nL,t - t) + D„r(x + £ + 2nL,t - T)),
(25)
where Cn and Dn are unknown constant matrices of dimension m x m. We have to find the matrices Cn and Dn. Taking into account (13) and conditions (2)—(4), (6), (7), we get following conditions for Green's function:
Gj |i=L = 0,
Gj l«=L = 0,
1, k, j = 1, m,
k + 1, m, j = 1, m
(26)
G11 |ç=c = GlVo = ... = G1n|Ç=o, i = 1, m, Gil |ç=o + Gi||5=o + ... + Gl™|5=o, i = I~m.
Let
G„ —
/ C11
cn1
c
ml
r<12
Cn
c22
C m2
cnm \
C 2m
mm Cn
( D,
11
and Dn
/
Combining (26), (27) and (28), respectively, we get:
D1
12
D21 D22
n "^i
D
m1
D
m2
c n = - Dij n-
Cij = n Dn- 1
/^vi 1 Cn im - Cn =
]Tcij ^ Dnj,
. i = 1, m. j=i j=i
And we rewrite the relations (30) and (31) on the following matrix form:
Cn
From this relation, we find Dn = CnM where: M given at the form (22). We find
C„+i =
where
-4 0 0 I
From the condition (13), we have that C0 = I. (29) can be written as:
C„ = $nMn, Dn = $nMn+1
where:
(-1)nifc 0
0 h
Substituting (32) into (25) we get matrix Green's function of the Problem 2:
D 1m Dn D2m Dn
Dmm Dn
/ 1 11 ... 1 1 \ / -1 -1 -1 .. . -1 1 \
-1 00 ... 0 1 1 0 0 .. .0 1
0 -1 0 ... 0 1 = Dn 0 1 0 .. .0 1
0 00 ... 0 1 0 0 0 .. .0 1
V 0 00 ... -1 1 / \ 0 0 0 .. .1 1 /
(27)
(28)
(29)
(30)
(31)
(32)
G1 = yj $nMn(r(x - £ + 2nL,t - t) + Mr(x + £ + 2nL,t - T)),
n=—oo
n=
1
1
n
n
As a result, we get following theorem.
Theorem 3. Let&(x) G C[0,L],7i(t) G C[0,T] and /¿(x,t) G C0'1{(x,t) : 0 < x < L,0 < t < T, }(i = 1, m, T > 0). Then the Problem 2 has a unique solution in the form of:
i(x, t) = J i )(x,t; L,r)
where
(£, r)
g1d) =
( G11 G12
Gk1 Gk2
01 01
.0.. 0
«=L
Glm \
G&m
0
— G^) (x, L; t, t(L, r) ] dr—
t L
j ^(e)Gi(x,t; e,r )de — j j Gi(x,t; e,r)/(e,r )dedr,
0 0
GlN) =
0 0
G1G1
V Gf1
Gm2
= (u1,..., , 0,..., 0)T, = (0,..., 0, uk+1,..., um)T and G1 is on the form (33)
G
0 0
1
gmm
Conclusion
In this paper, we gave Green's function approach for IBVP to time-fractional diffusion equation with Neumann and Dirichlet boundary conditions. We constructed Green's functions of the considered problems in the form of matrix series. We notice that Green's functions for IBVP for time-fractional diffusive equation on metric graphs were firstly constructed in the present paper. Green's function on the case of line-interval for different IBVPs are constructed in [4].
It is well-known that Green's functions on metric graphs are closely related to scattering problem at the branching points [23]. In our case, the component Gj, (i = j) describes, for example, heat flow conduction from ¿-th bond to j-th bond, while Gu describes heat flow reflection (thermal reflection) on ¿-th bond. Therefore, one can conclude that Green's functions constructed in the present paper can be considered as a powerful tool to investigate the conductivity (scattering) properties in sub-diffusive processes in nano-sized thin branched structures.
L
0
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