Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method Текст научной статьи по специальности «Математика»

NANOSYSTEMS: Bekakra Y., et al. Nanosystems:

PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2024,15 (1), 5-15.

http://nanojournal.ifmo.ru

Original article DOI 10.17586/2220-8054-2024-15-1-5-15

Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method

Y. Bekakra1", A. Bouziani1,2,6

1ICOSI Laboratory, Abbes Laghrour University, Khenchela, 04000, Algeria 2L'arbiBen M'hidiUniversity, Oum El Bouagui, 04000, Algeria

[email protected], [email protected]

Corresponding author: Y. Bekakra, [email protected]

PACS 02.60-x, 47.11.Bc, 02.30.Jr

Abstract This paper aims to study the existence and uniqueness of a weak solution for the boundary value problem of a time fractional equation involving the Caputo fractional derivative with an integral operator. By utilizing the discretization method, we first derive some a priori estimates for the approximate solutions at the points (x,tj). We then evaluate the accuracy of the proposed method to demonstrate that the implemented sequence of a-Rothe functions converges in a certain sense, and its limit is the solution (in a weak sense) of our problem. It must be pointed out that the constructed L1 scheme is designed to approximate the Caputo fractional derivative mentioned in the problem.

Keywords weak solution, a priori estimates, Fractional diffusion equation, Rothe's method Acknowledgements The authors would like to thank the reviewer for his valuable comments and suggestions.

For citation Bekakra Y., BouzianiA. Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method. Nanosystems: Phys. Chem. Math., 2024,15 (1), 5-15.

1. Introduction

In the beginning, fractional calculus was developed as a pure mathematical concept. In recent decades, its use has expanded into a variety of different fields of science, such as physics, mechanics, economics, and engineering (see, for instance, [1-4]).

The time-fractional-order diffusion equations (TFDEs) have attracted many scholars' attention, it is worth mentioning that the crucial importance of this kind of fractional equations is due to its wide use in many real-world applications in the fields of biology [5], chemistry [6], engineering [7]. It is pointed out that this kind of equation also appears, especially in nanofluid [8-10], nanotechnology [11,12] and nanophysics [13-15].

In [16], authors studied the following time-fractional diffusion equation:

CDXx, t) = Au(x, t) + f (x, t), (x, t) e Q

where CDJ* is the Caputo fractional derivative of order a(0 < a < 1). They prove the existence and uniqueness of the solution by using the Lax-Milgram Lemma in some suitable Sobolev spaces. Paper [17] dealt with a time fractional equation on the metric star graph. The authors applied the method of energy integrals to construct Green's matrix-function and discussed applications to nanostructures.

Accordingly, various types of numerical methods have been applied to study this concept. One such method that has drawn the attention of many scholars is the Rothe method. Researchers have exploited and enhanced this technique for study of various differential fractional equations [18-20]. Yang [21] presented a difference scheme for a kind of linear space-time fractional convection-diffusion equation using a finite difference method. Du et al. [22] apply the Rothe method to establish existence, uniqueness, and a priori estimate for a strong solution to an approximate fractional problem.

Assume that y = y (x, t) is a Lipschitz function with respect to t and let G be a bounded domain in multiply connected Q with a Lipschitz boundary r, we consider two-dimensional time fractional equation with an integral operator of the following form:

dgty(x, t) - Ay =[ K(x,s,y(x,s))ds + f (x) in Q = G x (0, T), 0 <a< 1, (1)

jo

y(x, 0) = 0,

B1y = 0 on r x (0,T),

© Bekakra Y., Bouziani A., 2024

where d®tu(x,t) denotes the Caputo fractional derivative of order - (0 < - < 1), f e L2(G), and Bi is a given linear operator. Additionally, it is assumed that the kernel k(x, t, u) satisfying the following conditions:

||k (x,t2,y2) - K (x,ti,yi)|| < C ^|i2 - til + h11 11 y 2 - yi||) , (2)

yi,y2 e L2(G), ti,t2 e I =[0,T].

u e $ ^ k e l2(G). (3)

Here $ is the space defined as follows

$ = { v; v e W2(k)(G), Biv = 0, in the sense of traces}. (4)

In the context of nanomaterials, equation (1) could model the movement of a particle or the diffusion of heat or matter through a medium with non- standard diffusion properties. Details for nanomaterials could be as follows: Memory and non-locality: Nanomaterials can exhibit memory effects due to their interaction with a non-homogeneous environment, where the past influences the present movement.

Heterogeneous Diffusion: In heterogeneous media, such as nanopores, diffusion is not uniform and is affected by the structure of the material. The function k(x, s, y(x, s)) could represent how the heterogeneous medium affects diffusion. It is worth mentioning that the boundary conditions considered in our problem may reflect the interactions of the particle energy with the boundaries of the nanomaterial, which could be reactive surfaces or interfaces with other materials. First, we introduce the Caputo finite difference formula to discretize the time-fractional derivative of order a [23]:

day (x,tj) h—a 4—\

-dta = r(2 _ ) (z4—fc (x) - z4—fc—i(x)) afc 0 <a< 1 (5)

( a) k=0

where

afc = [(k +1)i—a - ki—a] k = 0,1,...,

T

while zo(x) denotes the numerical approximation to the exact solution y(x, t,-), t,- = jh, 0 < j < p, where h = — is

p

the time step. It is straightforward to check that

<7j > 0, j =0,1,... ,p,

1 = o"o > i > ...0"p.

The organization of the paper is as follows: In Section 2, some preliminary facts regarding fractional calculus and some notations are presented. In Section 3, we obtain a priori estimates. Sections 4 and 5 are devoted to discussing the existence and uniqueness of the weak solution, respectively.

2. Preliminaries

In the first part of this section, we recall some basic background of fractional calculus that we will need in the sequel.

Definition 1. [24] The Riemann-Liouville fractional integration of order a > 0 of function y(t) is defined as:

1 rt

Io,t¥>(t) = ^-t (t - s)a—V(s)ds (a > 0,t > 0),

r(a) ./o

where r is the well-known Euler Gamma-function.

Definition 2. [24] The Caputo derivative of fractional order a e]0; 1[ of function u(t) is defined as:

1 f* '

' 1(1 - a) J0

Theorem 2.1. [25] If x(t) e C0 [0, T] for T > 0 and a > 0, then

I0>(0) = 0.

Theorem 2.2. [25] If x(t) e Ci [0, T] and 0 < a < 1, then

aa tol 0,tI0,t

öoai/oaix(t) = x(t).

i-1

Lemma2.1 (Gronwall's lemma [26]). Lei m1,...,mj be nonnegative numbers satisfying n1 ^ A, m ^ A+Bh^^nfc, ^i

k=1

2,..., j, where A, B, and h are positive constants. Then

m < A exp [B(i - 1)h], i = 1, 2 ...,j.

Theorem 2.3. If a sequence {xn} in a Hilbert space H is a weakly convergent to x e H, then the following statements hold:

d0Vn ^ d^jx in H, (6)

Io, txn ^ I^x «« H. (7)

For the proof of our theorem, we need the following lemma.

Lemma 2.2. [26] A sequence {xn} in Hilbert space H converges weakly to x e H implying that for any bounded linear functional g defined on H we have

g(xn) ^ g(x).

Proof of Theorem 2.3 (6) The caputo derivative is a bounded linear functional. Consequently, from it follows according to the preceding lemma that

d0 , txnk ^ d0 , tx-

Finally, the strong convergence implies weak convergence and this proof is completed. (7) The proof is the same as for the previous case.

Lemma 2.3. [27] Any absolutely continuous function v (t) on [0, T] satisfies the inequality

Moa,tv) > 1 da, t ||v||2 , 0 < a < 1.

Lemma 2.4. [27] Let a nonnegative absolutely continuous function y(t) satisfy the inequality

do,ty(t) < ciy(t) + c2(t), 0 < a < 1, for a. a. t in [0, T], where c1 > 0 and c2 (t) is an integrable nonnegative function on [0, T]. Then

y(t) < y(0)Ea (cita) +r(a)Ea , a (cita) DotaC2(t)

where Ea(z) = V^ —--r(an + 1) and Ea „(z) = V^ —-- are the Mittag-Leffler functions.

^ ilan +1) ^ ilan + u)

n=0 v ' n=0 v

The following assertions are presented at the end of this section. Let ||.|| and (.) denote L2(G)-normand L2(G)-inner product, respectively. By W2(k) (G) we denote the usual Sobolev space. We denote the bilinear form corresponding to the operator - Aw by (.,.):

6z (u-v) = £ X dlxi dx

Definition 3. [26] The form (v, u) ¿s called $ — elliptic ¡fa constant n can be found such that

be(v,v) > nlMlW«(G) Vv g (8)

In L2 (I, $), the scalar product is provided by:

(yi,y2)L2(/j h) = J (yi(t),y2(t))H dt,

and, consequently, the norm is given by:

IMi ¿2(1 , 1 n'f WII^

2 = I ii».(+)ii2 dt. (9)

3. Weak Formulation and a priori estimates Substituting (5) into (1), we get

-a j-i

h-

r(2 -—) ^J (zj-k(x) - zj_fc_i(x)) afc - Azj(x) = h (k 0 + «1 + .. .Kj-i) + f, 0 < a < 1. (10)

( a) k=0

With simplification by omitting the dependence of Zj (x) on x, (10) can be rewritten as follows

-r(2 - a)Azj + = - Zj-k -h:j-k-i)^

+r(2 - a)h (k 0 + K1 + ... Kj-1) + r(2 - a)f, 0 < a < 1,

where

Kj(x) = k (x, jh, Zj(x)), j = 1, .. . ,p.

(11)

Hence, the integral identities i.e. the corresponding weak formulation has the following form:

r(2 - -)6C(zo,v) + h^ (zj - zj—i,v)

( 1 j—i A

= ( - ha (Zj—k - Zj—k—i) wfc + r(2 - a)h (ko + Ki + ... Kj—i) + r(2 - a)f, v I Vv e 0 < a < 1.

Thus, we can construct the a-Rothe function yi(t) = yi(x, t) defined in the intervals Ij = [tj—i, tj], j = 1,... ,p, by:

yi(t)= Zj—i + Zj - i (t - tj—i)a 0 <-< 1.

In the same way, we get, for the divisions dn, n = 2,3 ..., with

h T

hn =

2n—i p the a-Rothe sequence

{yn(t)} (12)

of functions:

yn(t) = zj—i + j(x) -aZj—i(x) (t - tn— i)° 0 < a < 1. (13)

hn

Now, we are in a position to establish some a priori estimates.

Lemma 3.1. There exist two constants ci, c2 independent of a and j such that

l|zj|| < ci, j = 1,...,p, (14)

I|Zj|| < C2, j = 1,...,p, (15)

with

Z = zj(x) - zj—i(x) j ( T h a . Proof. First, since y belongs to the Lipschitz class with respect to t and because ofz0 = 0, it produces

llzj ll < llzil + ||Z2 - zil + ... + ||zj - Zj—il < jh < IT = ci,

where l is the Lipschitz constant.

Next, putting v = Zi in the first integral identity (11), we get

r(2 - -)6C(zi, Zi) + h^ (zi, Zi) = r(2 - a) (hKo + f, Zi), 0 < a < 1.

Hence, due to the Schwartz inequality and the fact that 6z(zi, Zi) = 6z(zi, zi) > 0 we obtain that

||Zi|| < r(2 - -)(h|ko| + |f ||). (16)

Subtracting the integral identities (11) written for j = 1 from that written for j = 2, we obtain

r(2 - -)6z(z2 - zi,v) + (Z2 - Zi,v) = (-aiZi + r(2 - -)hKi + f, v) Vv e Putting v = Z2, we get

||Z2|| < (ao - ai) ||Zi|| + r(2 - -)h ||ki| . (17)

Similarly, one obtains

||Zs| < (ao - ai) ||Z2| + (ai - a2) ||Zi || +r(2 - -)h ||k || . Following this procedure, one comes to the estimation

j—i

llZj || < ^ (ak—i - ak) |Zj—k || + r(2 - a)h ||kj-—i| . (18)

k=i

Adding up (16)-(18) yields

llZj|| < r(2 - a)^o If || +r(2 - a)h(Vo ||ko|| + Vi ||ki| + ... + ji ||Kj—1||) (19)

with

j — (m+2)

^rn = (ao - ai)j—(m+i)+ (ao - ai)j—k—(m+2)(ak - ak—i)(j - k - (m + 1)),

k=i

for all m = 0,1,..., j - 1.

Now, we can easily proof that < < ... < = 1, so (19) yields

||Zj|| < r(2 - a) [||f || + h (|K0| + ||ki| + ... + ||Kj-i||)] . (20)

In the other hand, by using the fact that z0 = 0, we obtain from (2) that

||ki| < |M| + ||«i - K0||

< |K0| + C ( h + |N

ha-i

= ||Ko|| + Ch (1 + ||Zi|).

Then,

|M < ||ko|| + |K2 — Ko|

|N

< ||Ko| + C 2h +

ha-i

^11 II ||z2 - zi|| , || zi |

< M + Ch +-hO-+ "h^

= ||K0|| + Ch (2+ ||Zi| + ||Z2||) = |K0| + Ch(2+ ||Zi| + ||Z2|). Hence, using the analogous arguments, we can conclude that

^j-J < KH + |Kj-1 - K0^

< |K0| + Ch (j - 1 + ||Zi|| + ||Z2| + ... + ||Zj-i||). (21)

Inserting these results into (20), we obtain

|Zj|| < r(2 - a) [||f || + jh |K0| + Ch2 (1 + 2 + ... + j - 1)] + Ch2 [(j - 1) ||Zi|| + (j - 2) ||Z2|| + ... + |Zj-i|]. Now, (j - 1) h < ph = T. Thus, we have

||Zj|| < r(2 - a) [||f || + T |K0| + CT2 (1 + 2 + ... + j - 1)] + CTh (||Zi|| + ||Z2| + ... + ||Zj-i||) j = 1, 2,.. .p. Using Lemma 2.1, we get, finally

||Zj|| < r(2 - a) (||f || + T |K0| + CT2) eCTj-1)h

< r(2 - a) (||f || + T |K0| + CT2) eCT2, j = 1, 2,.. .p. Then (15) holds with c2 = r(2 - a) (||f || + T ||K0|| + CT2) eCT2, and so the proof is complete. □

Remark 1. The estimates (14), (15) are independent of h, consequently, they remain valid for any division dn,n = 1 , 2, . . .. Thus, we have

llzjll < ci, (22)

< c2, (23)

for all n = 1, 2,..., and 1 < j < 2n-1p, where

zn(x) - Zj-i(x)

Zn(x)= ^ 1 J 1 '. (24)

hn

Recall that the space $ is determined by (4). Taking into account (22) and using (8), we arrive at the following corollary:

Corollary 3.1. There exists a constant c3 such that

< C3, (25)

for all n =1, 2,..., and 1 < j < 2n-1p.

Lemma 3.2. ||k"|| is uniformly bounded with respect to j and n as well, where k"(x) = K(x,jhn,z" (x)). Proof. For the division dn, due to (21), one obtains

K?ll < ||«nH + Chn j + ||zn| + ||zn|| +... + ||zn||).

Kj || ^ ||K0 || + Chn j + ||Z1 || + ||Z2 || + ... + ||Zj

Taking into account (23), we come to the inequality

K|| < KH + CT + CTC2 = C4. (26)

□

4. Existence of weak solution

This section is devoted to proving the existence of a weak solution to our problem. For this purpose, let's start with the following lemma:

Lemma 4.1. The a-Rothe sequence (12) admits a subsequence {ynfc (t)} weakly convergent in L2(I, $) to function y e L2 (I, $), we write

ynfc ^ y in ¿2(I,^). (27)

Proof. Since L2(I, $) is a Hilbert space it is sufficient to show that the a-Rothe sequence (12) is bounded in this space. Keeping in mind that

' t " tn—i'

0 <

< 1 in j

and by (13), (25), we obtain, for arbitrary t e I,

IMi)IU =

<

j-1

1 -

1 -

-J^H , ..„ (t - j-i)

ha hn

^ - t"-l)

n

j-1/ I zn

+ zn

+

ha hn

^ - tn-i)

n

J-1' zn

< A (t - tn-^\ + (t - tn-^

< 11--^- C3 +---C3 = C3.

ha hn

ha

From (9), we get, for n = 1, 2, . . .

llyn(t)||2(/,^) = £ |yn(t)!i dt < c|T. Thus, the a-Rothe sequence (12) is bounded in L2(I, $), and the proof is finished. Remark 2. [26] Note that besides (27) it holds that

ynfc ^ y in C(I, ¿2 (G).

□

(28)

Now, taking into account (23) and using the same reasoning as in the proof of the preceding corollary, we arrive at the following corollary:

Corollary 4.1. There exists a subsequence {Y0fc (t)} and a function Y a such that

Ya

Ya in L2 (/,L2(G)):

(29)

where defined by

Lemma 4.2.

Yn (t) =

Ya (t) : I ^ L2(G), n =1, 2,...

r(a +1)Zn'a for t = 0,

r(a + 1)j a for t G 7n = (tn-i,tn], j = 1,..., 2n-1p.

50ity(t) = Y a (t) in ¿2 (I, ¿2(G)) for a. a. t e I. Proof. It follows from (13) and (30) that

r(a)

/ (t - s)a-1Ynfc(t)(s)ds = ynfc(t). j0

(30)

(31)

(32)

Recalling the definition of the Riemann-Liouville integral, the left-hand side of (32) is nothing but IO^U^ (t), hence (32) becomes

I"^ (t)= ynfc (t).

Applying the Caputo derivative d" t to the both sides of (33), we get by Theorem 2.2

öo>nfc (t)= Ynfc (t).

Applying now (6) for (27), we obtain in view of (29), (34), and by the uniqueness of the weak limit that

doty(t) = Ya(t) in ¿2 (G) for a.a. t e I.

(33)

(34)

The proof is finished.

□

IX

n

_V

1

Rothe method for a time fractional problem 11 Corollary 4.2. The function y(t) verifies

y(0) = 0 in C (I, L2(G)), (35)

y e AC (i,L2(G)) . (36)

Proof. From (7), (27), and (33), following the same reasoning as in the proof of (31), we get

10^)=y(t).

Hence, by Theorem 2.1, we complete the proof. □

Lemma 4.3. [26] If

ynfc ^y in L2(I,$),

then also

ynfc ^y in L2(I,$), (37)

where {yn(t)} is the sequence defined by:

f zi1 fort = 0

2/n(t) = { 1 (38)

( z™ in j ^j-ij j = 1,...,p.

Let

w(t) = w(x,t) = k(x, t, y(x, t)). (39)

Since y e $ for a. a. t e I, we have, by (3) w e L2 (G) for a.a. t e I. Consequently, we obtain the following corollary.

Corollary 4.3.

lim Wnfc (t) = w(t) in L2(G) for a.a. t e I, (40)

nfc ^TO

where

Wn(t) : I ^ L2(G), n = 1, 2,...

is defined by

k" for t = 0,

w

(t) =

Kj-1 in ^ = (j-1'j

L" = (t"_i,t"], j = 1,..., 2n-1p.

Proof. Let £ > 0 be given. We have to show that

||w(t) - wnk(t)|| < £ fora.a. t G I if nk > n0(£). (41)

By (2) we obtain for fixed division d„fc and for arbitrary t G Irj+1

||w(t) - W„fc (t)|| = ||«;(x,t,y(t)) - k (x, jh„fc,z"fc) ||

< C (jt - jh„fc | + ||y(t) - j ||) < C [h„fc + ||y(t) - y (jh„fc )|| +

+ ||y (jh„fc) - j ||] .

£

Choosing n = ^, we get the desired result due to (28) and (36). □

3C

Remark 3. (40), (41) imply that the function w(t) is Bochner integrable as an abstract function from I into L2(G). Consequently, we can define the following functions

rt

W(t) = Í w(s)ds, (42)

0

W„(t)^ Wn(s)ds. (43)

0

The proof of the following corollary is essentially the same as that in [26]. For the sake of convenience we give the proof. Corollary 4.4. Define

f hnk0n for t = 0,

Wn(t) = { n (0 ) „ (44)

( hn (k" + ... + k"-0 in Ijn,

then

lim Wnfc = W in L2(I,L2(G)). (45)

Proof. First, from (41) we immediately obtain

lim W„fc = W in ¿2(I,L2(G)). (46)

"fc^TO

From (43), (44), we get for t e 7":

3

W"(t) - W„(i)^ W„(s)ds + ... + / w7„(s)ds+

jo J(j—2)h„

+ / W„(s)ds - Wn(t) = h„ (k" + ... + k"—2) + J(3—i)hn

+ [t - (j - 1)h„] k"—i - hn (k" + ... + k"—J = (t - jhn) k"—r Hence, for t e I73" it holds that

Wn(t) - Wn(t)

3—11

Combining (46), (26), we complete the proof of Corollary 3.4. □

We're now prepared to demonstrate in which senses the function y(t) satisfies the given equation (1). In view of (24), we can rewrite the integral identities (11) for the division d„ in the following way:

r(2 - -)6C (z» + (joZ—k^vj (47)

= r(2 - a) (h„ (k" + k" + ...k"— i) +r(2 - a)f,v) Vv e tf, 0 <-< 1.

Let v(t) be a fixed abstract function from L2(I, $). Using (30), (38) and (44) and defining the abstract function f (t) from I into L2 (G) by f (t) = f Vt e I, we transform (47) to the form

r(2 - -)6C(7", v) + r(-1+ 1) (EYa,^ = r(2 - a) +r(2 - a) (f, v) for a.a. t e I.

■a)&C (7",v) + r(-1+1^ ^ ^ Y" ,v ) = r(2 - a) (W„^| +r(2 - a) (f, v) for a.a. t e I. (48)

Taking into consideration the indices nk from (27) only and integrating (48) with n replaced by nk between the limits

t = 0 and t = T, we get

!-T 1 3— i T

1 ^ ^ ' '"/a

I 1 I

r(2 - a)yo bC(7"fc ,v)dt +r(a + 1) XO ^ (Y"fc, v) dt

(49)

= r(2 - a) J (W„t ,v)dt + r(2 - a) J (f,v) dt.

Hence,

v e ¿2(I,0) ^ v e ¿2(I,L2(G)).

Thus, (29), (31) imply

1 /"T /3 —i \ 1 3 —i rT

lim rT-T^ Y"fc, v dt = r7—^ E (doty, v) dt.

r(a 0 r(a +1T i=0 ^

Similarly, by (45), one obtains

lim r(2 - a) / fW„fc, v) dt = r(2 - a) / (W, v) dt, 7o v ' jo

where W(t) = / k(x, s, y(x, s))ds, by (39), (42). Finally, for v e L2(I, tf) fixed,

jo

/ bc(y, v)dt ■Jo

is a bounded linear functional in L2 (I, $). Thus,

,T ,T

lim r(2 - a)/ 6C (7"fc ,v)dt = r(2 - a)/ 6C (y,v)dt jo jo

holds by (37). Consequently, for nk ^ to, (49) gives one

/t 1 3—i T T T

6c(y, v)dt + r(a + 1) ^ JQ (d0°ty,v) dt = r(2 - a) ^ (W,v) dt + r(2 - a) ^ (f,v) dt. (50)

"

Due to the fact that function v e L2(I, $) is chosen arbitrarily, (50) holds for every v e L2 (I, $). In this weak sense, equation (1) is fulfilled. Thus, the function y(t) satisfies

y e L2(I,$) n AC (I,L2(G)), (51)

d(0ty(t) e L2 (I, ¿2(G)), (52)

y(0) = 0 in C (I, ¿2(G)), (53)

,T 1 T

1 /"T

(y,v)dt + r( + 1) a' / (doty,v) dt (a + ) i=o ■Jo

= r(2 - a) f (W, v) dt + r(2 - a) f (f, v) dt Vv G L2(/,tf). Jo Jo

(54)

Definition 4. Function y(t) satisfying (51) - (54) is called weak solution to our problem. 5. Uniqueness

To establish uniqueness, we assume the existence of two distinct weak solutions y1 and y2 and come to a contradiction. Denote their difference by

y = yi - y2.

Therefore, to prove the uniqueness, it suffices to show that y* = 0. By (51)-(54), y*(t) satisfies

y* g ¿2(1,0),

y* G AC (Z,L2(G)),

d0>* G L2 (I,L2(G)), y*(0) = 0 in C(I,L2(G)), fT /J-1

r(2 - a) ^ 6C(y*,v)dt + Jo ^a^y*,^ dt

= r(2 - a) ^ ^ [k (x, s, y2(s)) - k (x,s,yi(s))]ds,^ dt Vv G L2(/,tf).

(55)

According to (2), we have

ds

/ [k (x, s,y2(s)) - K (x, s,yi(s))] / 0

ft

< / Il k (x, s,y2(s)) - k (x, s,yi(s))|| ds Jo

10

11y2(s) - yi(s)|| ds = ¡^i 04 ||y*(s)||ds < C J4 ||y*(s)||ds. (56)

Let us divide the interval I into a finite number of subintervals of lengths l. The function y(t) belongs to C (I, L2(G)) hence the function ||y(t) || is continuous in the interval [0, l]. Consequently, ||y(t) || attains its maximum on this interval at a certain point t1 e [0,1]. Let

max ||y*(t)|| = ||y* (ti)| . (57)

te[0,i]

For v(t) in (55), let us choose the function

f y* (t) for t e [0,ti], v (t) = <

[ 0 for t e (ti,T].

Thus, we obtain

,-t, j-1 T

Z>ti J rl

r(2 - aW 6C (y*,y* )dt + V aj (y*, d0V) dt Jo l=1 Jo

t

= r(2 - a) J ^y*, J [k (x, s, y2(s)) - k (x, s, yi(s))] ds^ dt.

(58)

'o \ Jo 0-ellipticity of the form (v, y) gives us that

r(2 - a)/ 1 6C(y*,y*)dt > 0. (59)

Jo

Additionally, according to Lemma 2.3, one has

j-1

d§<ty dt

v^i-1

EJ=1 ^

dot I

I2 = C5Ö0at I|y(t1)|2

(60)

i = i

where c5 is a constant. (56) yields

r(2 - a) [k (x, s,y2(s)) - k (x, s, y1(s))] ds^ dt

< r(2 - aW ||y*(t1)

[k (x, s, y2(s)) - K (x, s, y1 (s))]

ds

(61)

(62)

< r(2 - a) ^ ||y*(ti)|| ^C J0 I|y*(s)|dsj < C12 ||y*(ti)H2 . Combining (58) - (61), we can get

dot l|y*(ti)l|2 < c6 ||y*(ti)|2 , where c6 is a constant. Using Lemma 2.4, we obtain with regard to (35),

l|y*(ti)ll = 0.

Formula (57) then implies

y*(t) = 0 in l2(G) vt e [0,1]. Performing the same consideration in the interval [1, 21] with the function

i y* (t) for t e [0,t2],

v (t) = <

[ 0 fort e (t2,T],

where t2 is a point at which ||y(t)|| attains its maximum in the interval [1, 21], and using the just obtained result (62), we obtain that

y*(t) = 0 in L2(G) Vt e [1, 21]. After a finite number of steps we thus come to the conclusion that

y*(t) = 0 in l2(G) vt e I.

Hence, the uniqueness is proved.

6. Conclusion and future scope

Two objectives have been successfully reached in this study. Firstly, a new function has been constructed using the method of discretization for a time fractional equation. Secondly, the proposed method has been validated by demonstrating the convergence of the a-Rothe sequence to the unique weak solution of our problem. As we've seen, TFDE is closely linked to various nanoscience phenomena, consequently, the authors are optimistic that the results obtained can be extended to a wide range of nanoscience applications, thus offering a promising avenue for further exploration (Non-Fourier conduction, confinement effects, Phonon-Phonon and Phonon-Defect Interactions...).

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Submitted 29 August 2023; revised 12 January 2024; accepted 14 January 2024

Information about the authors:

Y. Bekakra - Department of Mathematics and Informatics, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 04000, Algeria; ORCID 0000-0001-5672-4348; [email protected]

A. Bouziani - Department of Mathematics and Informatics, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 04000, Algeria; Department of Mathematics, L'arbi Ben M'hidi University, Oum El Bouagui, 04000, Algeria; ORCID 0000-0002-1216-8033; [email protected]

Conflict of interest: the authors declare no conflict of interest.