THE TIME-FRACTIONAL AIRY EQUATION ON THE METRIC GRAPH Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-376-388 УДК 517.95

The Time-fractional Airy Equation on the Metric Graph

Kamoladdin Rakhimov*

National University of Uzbekistan Tashkent, Uzbekistan

Zarifboy Sobirov^

University of Geological Sciences Tashkent, Uzbekistan

Nasridin Jabborov*

National University of Uzbekistan Tashkent, Uzbekistan

Received 10.09.2020, received in revised form 10.12.2020, accepted 20.02.2021 Abstract. Initial boundary value problem for the time-fractional Airy equation on a graph with finite bonds is considered in the paper. Properties of potentials for this equation are studied. Using these properties the solutions of the considered problem were found. The uniqueness theorem is proved using the analogue of Gronwall-Bellman inequality and a-priory estimate.

Keywords: time-fractional Airy equation, IBVP, PDE on metric graphs, fundamental solutions, integral representation.

Citation: K. Rakhimov, Z. Sobirov, N. Jabborov, The Time-fractional Airy Equation on the Metric Graph, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 376-388. DOI: 10.17516/1997-1397-2021-14-3-376-388.

Introduction

In recent years noticeable interest has been shown in the study of initial and initial-boundary value problems for equations of fractional order. This is due to the fact that fractional-integral calculus have applications in the study of diffusion and dispersion processes in various fields of science (see [1-5]).

The Schrodinger equation on metric graphs was studied (see [6,7] and references therein). Such graphs sometimes called quantum graphs. The Schrodinger equation on the metric graph was also studied with Fokas unified transformation method [8].

The Airy equation on an interval was studied with Fokas unified transform method [9] and [10]. The potential theory for solutions of this equation was developed [11] and [12]. The linearised Airy equation on metric graphs was considered in [13-16] and [17]. M. Cavalcante considered non linearised KdV equation [18].

A. Pskhu studied properties of the Airy equation with time-fractional derivative. Fundamental solution of the equation was found and properties of potentials were studied (see [19]). Later, second fundamental solution was found and the properties of the some additional potentials were

* [email protected] [email protected] [email protected] © Siberian Federal University. All rights reserved

studied [20,21]. Using this results solutions of initial and some IBVPs over infinite and finite intervals were found.

In this paper we consider the initial boundary value problem (IBVP) on a closed star graph with finite bonds. The solutions are found with the use of the potential method developed in [19-21].

1. Basic concepts

The operator

-Dx^rr-ay{dn, 0<a<1 (1)

is called fractional derivative (Caputo derivative) (see [22]), where r(x) is the Gamma function. Inverse of this operator is called operator of fractional integration

r(a)L \t — d1

JnAt) = FTTT I u n 1-a (2)

It is easy to show that

na „/-A _ T,

Function

c Dltg(t) = c D%t-V g(t). (3)

4>(\,K z) := , N > A> — 1,V e c

nW(An + v)

(4)

is called Wright function (see [23]). Wright function can be represented as

2JHa a^

where the integral is taken along the Hankel contour (see [22]). We have following estimate (see [19])

^ (-A,v; z)\ < ^exp — \z\, C = C (A,v,v), (5)

where v < (1 — A) Acos-\—g—1, — n < \arg z\ ^ n. The value of integral of this

1 — A 2

function is (see [19])

f1

az)dz = —^——r. (6)

J o + A)

2. Formulation of the problem

The Cauchy problem for time-fractional Airy equation on a metric graph with infinite bonds was considered ([21]). Now we consider a graph with k incoming and m outgoing bonds. In the incoming bonds coordinates are set from Lj (Lj < 0, j = 1,k) to 0, and on the outgoing bonds the coordinates are set from 0 to Li (Lj > 0,i = k + 1, k + m). The bonds of the graph are denoted by bj, j = 1, k + m (Fig. 1).

On each bond bj (j = 1, k + m) of the graph, we consider the Airy equation with a fractional time derivative

d 3

C Dau (x,t) — Uj (x,t) = fj (x,t), 0 <t < T. (7)

Fig. 1. Star-shaped graph

Let 0 ^ t ^ T, and x G bj, j = 1, k + m. We need to impose the following initial conditions

vertex conditions

u(x, 0) = uo(x),

Au(0,t) = 0,

d d -u+m = B-u-m,

(8)

(9) (10)

where u = (ux,u2,..., uk)T , u+ = (uk+i,uk+2,uk+m)T,

( 1 -a2 0 1 0 a3

A=

1

00

\

100

0 0

0

ak+m J

and B is the constant m-by-k matrix.

We need impose the following conditions which are sometimes called the Kirchhoff conditions or the condition of conservation of flow rate at the vertex of the graph

__d2 u (x, t) I = d2u+(x,t)

where C- = ( —, —— ),C + =

a2 ak Boundary conditions are

dx2 x=0 ( 1

dx2

x=0

(11)

ak + i

ak+i

- ), ai = 1 and aj = 0 for j = 2, k + m.

,(L,t) = y(t),

du (x,t)

dx

= m,

(12)

where y = (yi,y2,..., Pk+m)T and $ = ($1,$k)T.

A regular solution of equation (7) is constructed on the graph defined above that satisfies conditions (8)-(12).

+

u

1

L

x

2.1. Uniqueness of solution

Theorem 1. Let BTB — Ik be negative defined matrix. Then problem (7), (8)-(12) has at most one solution.

Proof. Let us consider the following inequality [24]

I vcD0'tvdx > 1 cDo,t I v2dx ■J a 2 J a

. Using the Cauchy inequality and conditions (8)-(9), we have

cDZtNlo < (u-)T(BTB — Ik)(u-) + 2\\u\\0\\f jjo < 2\\u\\0\\f ||0 < \\u\\20 + \\f HO,

k+m

iiuM0 = JB u°dx'

where

k+m

lo - I j j=iJ B>

U = (u1,U2,...,Uk+m)-

Using the analogue of Gronwall's inequality [24], we obtain from the last inequality the following a priori estimate

\\u\\0 < \\uo\\2Ea(2ta) + r(a)Ea,a(2ta)cD^tf \\0- (13)

The proof of the theorem follows from (13). □

2.2. Fundamental solutions

We construct the solution of the problem with the use of the potential method. To begin with, we need to obtain a special solution of equation (7) that is called fundamental solution. A fundamental solution of the equation was found in the following form [19]

i f — a/3,2a/3;X), x< 0,

eia/3(x,t) = 3Jo3 M r , ta,) x ^ (14)

, x > 0.

3t1-2a/3 \ — 2Re\e2ni/3^ — a/3, 2a/3; e2^i/3-O^) Using results from [21], second fundamental solution can be written in the following form

VOa/3(x,t) = ^ Im [e2ni/3^ — a/3, 2a/3; e2ni/3 O^)}, x> 0. (15)

These functions have the following properties (see [19])

c D5ttG2(x,t) = G2-v (x,t), d^ G«(x,t) = (x,t) (16)

with estimate

\cDv0Ga(x,t)\ < Cx-et»+e°/3-1, (17)

where

e > i 0, —) e N0,

1, (—v) e No.

Using these functions we define functions that are called potentials

wi(x,t) = i G2aa/3(x - a, t - n)ri(n)dn, wi(x,t) = f V2a/3(x - a,t - n)T2(n)dn, J 0 J 0

ft d2 f1 d2 w3(x,t) = J dx2 Ga2a/3(x - a,t - n)T3(n)dn, w4(x,t) = J d^i Vl°'/3(x - a,t - n)T4(n)dn,

f b f t f b w5(x,t) = Gla/3(x - z,t)r5(Z)d£ and w6(x,t) = / G2aa/3(x - £,t - v)f(^,v)d^dv.

•J a J 0 J a

Let us show some properties of these functions in the following lemmas.

Lemma 1. Let functions Tk(t), k = 1, 2 are continuous and bounded on (0;+ro). Then 1. Functions wi(x,t) and w2(x,t) are solutions of the equation

cD0tUj (x,t) -

d3Uj (x, t) dx3

0;

2. Functions wi(x,t) and w2(x,t) satisfy conditions

lim wk (x, t) = 0,k = 1, 2.

t^o

Lemma 2. Let r3(n),r4(n) G CVL(0,h). Then

1 2

lim w3(x,t) = — T3(t), lim w3(x,t) = - — T3(t), lim w4(x,t)=0.

x^a — O 3 x^a+O 3 x^a+O

The proofs of these lemmas can be found in [21].

Lemma 3. Let T5(x) G C[a,b]. Then function w5(x,t) is the fundamental solution of equation (7) and

lim cD0— 1w5(x,t) = T5(x).

Proof. Let us show that function w5(x,t) is the fundamental solution of equation (7). Using relations (16), we obtain

/• b /• b

CD%w(x,t)= I CD0tG20a/3(x - £,t)T5(№ = G—a/3(x - £,t)T5(£)d£

aa

and

d 3 dx3

w5

/b d3 i'b

Gia/3(x - Ç,t)n(№ = J G—a/3(x - Ç,t)T5(Odt

Comparing these equalities, we obtain that function w5(x,t) is the fundamental solution of equation (7). Let us find

bb CDa-— lw5(x,t)= CDa-— 1G2aa/3(x - t,t)M№ = G—/3(x - Ç,t)M№.

aa

Using inequality (17), we have the following estimate

Ic Da-— 1w5(x,t)\

f b fb

G10Ta/3(x - £,t)T5(£)d£ < max t5(x) C\x - £\—et(1—e)ad£ Ja a^x^b Ja

x — n

where 1 > e > 0. It shows that the integral form converges. Replacing ta/3 with y and taking into account that

/w /*w

ga(y)dy = ta/3Gl-a/3(yta/3,t)dy =

-w J -w

/0 1 r nw -|

w a 1 — t- y>d'y — 'M°/3Re [2'i/31 3O33a.1 — 1 * (41 — a-y) dy — 2Re № £ * (rP — 1^) d

3 J_ry-, V 3 3

2Re

(

e2ni/3 1 I ' ( a - a

3 J0 T V 3'~ 3

e2ni/3_

3\F(1 — a/3 + a/3) e2ni/3r(1 — a/3 + a/3)_

we obtain

r b

lino cDO-- W (x, t) = lino J GO-a/3 (x — n, t)T5 (nd =

= lim I ^ ta/3Gi-a/3(yta/3,t)T5(x — ta/3y)dy = ^ f+W ga(y)dy = T5(x).

tn0J 3 J-w

ta/3

The lemma is proved. □

d3

Lemma 4. The equation cDotu(x,t) — dx$u(x,t) = f (x,t) with initial condition

c DO- 1u(x,t)\t=o = 0

has a solution in the form

w6(x,t)= f dn t G2aa/3(x — n,t — n)f (€,v)d£-

■J0 J a

Proof. Using the results given in [19], it is easy to show that solution of the Cauchy problem for

d3

the homogeneous equation cDOtv(x,t) — v(x,t) = 0 with initial condition v(x, 0) = v0(x) is

' dx3

f b

(x,t) = c DO-1 G2aa/3(x — n,t)vo(№.

a

Let us determine the derivatives of function w6(x,t)

ft /■ b

d t b

c DO, t,w6(x,t) = - J dn J c DO- lGiO/3(x — n,t — n)f (£,n)d£ =

/b rt rb d

c DO- 1G2a°/3(x — n,t — n)f (n,n)dn + JQ dn J ^ c DO- 1G2a°/3(x — £,t — n)f (£,n)d£.

Taking into account (3) and relation (16), we obtain

bb / cDO- 1G2aO/3(x — n,t — n)f(n,n)dn = cD0-G2O/3(x — n,t — n)f(n,nd =

aa

r b

= Gi-O/3(x — n,t — n)f (£,n)d£.

a

1

v

x — Z

It follows from relation (5) that integral I\ converges uniformly. Substituting-^^ for y in

(t - n)a/

this integral and taking into account (6), we obtain

/• b /• b

I-1 = lim / CDa-- 1G2aa/3(x - Z,t - n)f (Z, nd = lim / Gl-a/3(x - Z,t - n)f (Z, n)dZ =

n J a n J a

lim f G1~a/3((t - n)a/3y,t - n)f (x - (t - n)a/3y,n)(t - n)a/3dy n t J

(t-n)a/3

(t-n)a/3 /3 /3

& I -2f " - ''^ - n) Re (^i-f. 1 - f; ^ dy+

0

0

+nm / ,(t^n)a/3* (-f,1 - f;y)f(x - (t - n)a/3y,n)(t - n)a/3dy

x-b W - n)

(t-n)a/3

X "a/3

- 2 lim | Re (e2ni/3* (-f, 1 - f; e2ni/3y)) f (x - (t - n)a/3y, n)dy+ 0

0

+ 3 lim J * {-a, 1 - f; y) f (x - (t - n)a/3y, n)dy =

(t--a/3

= -2R< f e2"/S* (-f ■ 1 - I; e2n'/3y^ f x,») + 3 f_ J (-3,1 - f„) f (x

=- 3Re <ce2"/3 e^mo)f xt>+1 ^-m)f =f xt)-

d3

Now we have I\ = f (x,t). Furthermore we show that I2 = u(x,t). We begin with

dx3

' b d

/t r d

dn J Jt cDa- 1Gla/3(x - Z,t - n)f (Z, n)dZ =

/t r b d

dn ja JtCD0— G2a/3(x - Z,t - n)f (Z,n)dZ = (18)

/t fb d

dn J dtG1-a/3 (x - Z,t - n)f (Z,n)dZ-

d 3

To determine u(x,t) we use relation (16). So, we have

dx3

d3 d3 Ct i'b

dx3 u (x,t) = J dn J G2aa/3(x - Z,t - n)f (Z,n)dZ =

/t rb d)3 ft fb

dn J G2aa/3(x - Z,t - n)f(Z,n)dZ = J dn J G2^/3-a(x - Z,t - n)f(Z,n)dZ = (19) i't C b d3

= J0 dn J dx3 G-a/3(x - Z,t - n)f (Z,n)dZ-

d 3

Comparing (18) and (19), we obtain I2 = tt^tu(x,t). The Lemma is proved.

dx3

2.3. Existence of solutions

Let

F- = (Fu...,Fk )T, F+ = (Fk+1

a- = (ai,. .., ak)T,

ß- = (ßi,...,ßk )T, T

ß+

(ak+i, (ßk+i,

. . , Fk+m ) , T

..,ak+m) ,

. . , ßk+m)T,

Y =(Yi,...,Yk) , Y + = (Yk+i,...,Yk+m)

P =(Pi,P2,...,Pk)T, P+ = (Pk + i

T

. , Pk+m) ,

and b

,ß

(Lj ;0), j = l,k

ß-ß+

F

F-F+

(0; Lj), j = k + l,k + m ' Let us find solutions in the form

uj (x,t) = i G2aa/3 (x — Lj,t — t) aj(t)dr + f V2a/3 (x — Lj,t — t) /3j(t)dr+

Jo Jo

+ f G0a/3 (x — 0,t — t) Yj (t)dT + [ V0aj3 (x — 0,t — t) pj (t)dT + Fj(x,t), j = l,k + m, oo

where functions aj, Yj (j = l,k + m), 3j (j = l, k), pj (j = k + l,k + m) are unknown functions, pj(t) =0, (j = l, k); 3i(t) = 0, i = k + l,k + m and

Fj(x,t) = f uoj(£)cD0t 1Gla'3(x — tt — 0)dt + Î [ G2aa/3(x — Ç,t — 0)fj(Ç,T)dÇdT. Jbj Jo Jbj

It follows from Lemma 4 and the results given in [19] that these functions are the solutions of equation (7) and they satisfy initial conditions (8). Taking into account condition (9), we have

aj i G2aa/3(—Lj,t — t)aj(t)dT + j ( Vlaj3(—Lj,t — t)3j(t)dT+ Jo Jo

+ aj f G2aa/3 (0,t - r ) Yj (t )dr + aj f V2aa/3 (0,t - r ) pj (r )dr + aj Fj (0,t) Jo Jo

= !tG2aa/3 (-Li,t - r) ai(r)dr + f tV2a/3 (-Li,t - r) ßi(r)dr+ oo

+ f C2aa/3 (0,t - r) Yi(r)dr + Fi (0,t), j = 2,k + m.

o

Furthermore

f (Gla/3 (-Li, t - r ) ai(r ) + V2a/3 (-Li,t - r ) ßi(r )) dr+

+

a

a

Y

a

Y

P

+

+

a

Y

r* <t>(-a, ^;0) + / + F1 (0,t)

lo 3(t - t)1-2a/3

f Gla/3 (-Lj, t - t ) aj (t )dT + a J ' V2a/3 (-Lj, t - t ) j (t )dT+ Jo Jo

ft 4>(-3, ;0) , I aj V 3 3'2 /3 Yj (t )dT + Im j Jo 3(t - t)1-2a/3 j

e2ni/30(-33, ;0)

aj I -" „ — Pn(t)dT

lo 3(t - t)1-2a/3

~Pj(t)dT

+ ajFj (0,t).

So, we have

rt V3a

Pj(t) - ajYj(t) + yi(t)

aj Fj (0, t) - Fi (0, t) = 2 -' dT -

jjK'> Jo 3r (f ) (t - t )1-2a/3

jj

lo 31 J _

-a, i G2a/3 (-Lj,t - t) aj(t)dT - a^ V^a/3 (-Lj,t - t) j(t)dT+

Jo Jo

+ / G2aa/3 (-L1,t - t ) a1(T )dT + / V2a/3 (-L1,t - t ) ^(t )dT.

Jo Jo

Y1(t ) - aj Yj (t ) + pj (t ) = 3c (aj F0 (0,t) - F1 (0,t)) -

-3ajc DlT^f Gla/3 (-Lj, t - t ) aj (t )dT + £ V^a/3 (-Lj ,t - t ) jj (t )d^j -

-3c d 2T(J* G2a'/3 (-L1,t - T ) a1(T )dT + £ V^a/3 (-Lut - t ) t )d^j . From above relation we obtain

Y1(t ) - aj Yj (t ) + ^ Pj (t ) = 3c DIf3 (aj Fj (0,t) - F1 (0,t)) -

-3(J ' G0a (-L1, t - t ) a.1 (t )dT + ^ ' V0 (-L1, t - t ) j1(T )d^j +

+ 3aj^ G0a (-Lj,t - t) aj(t)dT + ^ Va° (-Lj,t - t) j(t)d^j , j = 2,k + m. (20) In a similar manner, we obtain from condition (10) that

By-(t) - Y+(t) +^P+(t) = 3cDtf (F+(0,t) - BF-(0,t)) --3J (BG%/3 (L-,t - T) a-(t) + BV^/3 (-L-,t - t) j-(T)) dT + (21)

+ 3^' G/3 (-L+ ,t - t) a+(T)) dT.

Taking into account condition (11) and using Lemmas given above, we have

d2 /■ t

C-Y-(t) + 2C + Y+(t) = 3C lim ^ G2aa/3(x - L,t - t)a(T)dT+

x->■ 0 dx2 J0

d2 t

+ 3C lim 0x2 J V2a/3(x - L,t - T)j(T)dT + 3CFXx(0,t),

and

t

where C = (—C-,C+).

Using conditions (12), we have

T2a/3

Jo,t i a

(^j (t) + f Pj (t)^ + 0 GlO/3 (Lj ,t — T) Yj (T )dT+

+ \tVlO/ (Lj ,t — t) pj (T)dT + Fj (Lj, t) = Vj (t), j = (1,k + m). ■Jo

Applying the properties of fractional operators, we obtain

a3 (t) + ^ Pj (t) = c DOT (W (t) — Fj (Lj ,t)) —

^2u/3 t

t

/0

— c D20"/3(J0 GlO/3 (Lj ,t — T) Yj (T )dT + Jo /3 (Lj ,t — t ) pj (t )d^j , j = 1,k + m

Equations given above can be written in the following form

a(t) + ^p(t) = —( G°a(L, t — t )y(t )dT —i Vo(L,t — T)p(T)dT+ 2

+cD20<t/3 (w(t) — F(L,t)).

(23)

In a similar manner, we have from condition (12) that

a-(t) — ^P-(t) = O GO(L-,t — t)y-(t)dT + £ VO(L-,t — t)p-(t)dT+

(24)

+cDtf (*(t) — F-(L-,t)) .

We obtain the following system of integral equations (20)-(24) with respect to unknowns A(t)

QA(t)+ i K(t — t)A(t)dT = H, Jo

(25)

where A is the unknown functions, Q is the (3k + 3m)-by-(3k + 3m) matrix, K is the matrix of potentials. Using above system, the matrices can be written in the form

/ —3Ac D20"/3F (0,t) \ 3cDO/3 (F+(0,t) — BF-(0,t)) H = 3CFXX (0,t)

cDlO/3 (V(t) — F(L,t)) v cDOf (*(t) — F-(L-,t)) y

where M is the matrix on the form

( 1(k-i)xi —diag(a,2,...,ak)

A

a

P Y

p

, Q =

0M Q1 0

M

\

V3

1(m)yA —diag(ak+1,...,ak+m)^r diag(ak+1 ,...,ak+m)

B

1 1

\ a1 ak

-T

—m

22

V3 j 2 Tm 0

ak + 1 ak+m

0

Qi

Ik 0

Ik

0 %'k

In

0

^ r

— T" Ik /

and K

Ki = 3

Ki 0 0 Ko

f —AG0a(—L)

—BG0a(L-) | GK—L+)

where

—AVo (—L)

Vo (—L-)

do / d2 /

Clim^ o dxO G2 (x — L) —C- lim^ o d^ôV<°a/3(x — L)

Ko =

—Ga (L) —Vo(L) —G°a(L-) | 0 Vo(L+)

It is obvious that det(Q) = 0 and elements of matrix K(t,T) are bounded and continuous functions on (0,T). It was proved that detM = 0 [21]. So, matrix integral equation (25) has unique solution in (C[0,t])2k+m.

So, we arrive at the following theorem.

Theorem 2. Let BTB — Ik be negative defined matrix, functions uj,o(x) G C(bj), fj (x,t) G Co,i(bj x [0, T]) for j = l,k + m, p(t) and 4>(t) are differentiable functions on [0, T]. Then problem (7)-(12) has unique solution on 0 ^ t ^ T.

0

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Уравнение Эйри с дробной производной по времени на метрическом графе

Камоладдин Рахимов

Национальный университет Узбекистана Ташкент, Узбекистан

Зарифбой Собиров

Университет геологических наук Ташкент, Узбекистан

Насридин ^Кабборов

Национальный университет Узбекистана Ташкент, Узбекистан

Аннотация. Мы рассматриваем задачу Коши и начально-краевую задачу для уравнении Эйри с дробной производной по времени на метрическом графе с ограниченными и с неограниченными ветвями. Мы изучали свойства потенциалов для этого уравнения и, используя эти свойства, нашли решения рассматриваемой задачи. Теорема единственности была доказана с помощью аналога неравенства Гронуолла-Беллмана и априорной оценки.

Ключевые слова: уравнение Эйри с дробной производной по времени, начально-краевая задача, уравнения в частных производных на метрическом графе, фундаментальные решения, интегральное представление.