Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

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1 Al-Refai and Abdeljawad Advances in Difference Eqations :315 DOI /s R E S E A R C H Open Access Analysis of the fractional diffsion eqations with fractional derivative of non-singlar kernel Mohammed Al-Refai 1* and Thabet Abdeljawad 2 * Correspondence: m_alrefai@ae.ac.ae 1 Department of Mathematical Sciences, UAE University, P.O. Box 15551, Al Ain, UAE Fll list of athor information is available at the end of the article Abstract In this paper we stdy linear and nonlinear fractional diffsion eqations with the Capto fractional derivative of non-singlar kernel that has been lanched recently Capto and Fabrizio inprog. Fract. Differ. Appl.12:73-85,215. We first derive simple and strong maximm principles for the linear fractional eqation. We then implement these principles to establish niqeness and stability reslts for the linear and nonlinear fractional diffsion problems and to obtain a norm estimate of the soltion. In contrast with the previos reslts of the fractional diffsion eqations, the obtained maximm principles are analogos to the ones with the Capto fractional derivative; however, extra necessary conditions for the existence of a soltion of the linear and nonlinear fractional diffsion models are imposed. These conditions affect the norm estimate of the soltion as well. Keywords: fractional diffsion eqations; maximm principle; fractional derivatives 1 Introdction Fractional diffsion models FDM are generalization to the diffsion models with integer derivatives. In recent years there has been great interest in the stdy of FDM becase of their appearance in modeling varios applications in the physical sciences, medicine and biology; see, forinstance, [2 1]. Therefore, analytical and nmerical techniqes have been implemented to stdy these models. The maximm principle is one of the common tools to stdy partial differential eqations analytically, see [11, 12] for intensive srvey and reslts. In recent years, maximm principles have been developed to stdy varios types of fractional diffsion systems see [13 22], and we refer the reader to [23]forthe recent development on the theory of fractional differential eqations. In [13]and[14], two classes of eigenvale problems of Capto fractional order, 1< < 2, were considered. Maximm principles and the method of lower and pper soltions have been developed and sed to establish certain existence and niqeness reslts of the problems. In [19]and [21] Lchko has developed and implemented maximm principles to stdy the generalized fractional diffsion eqation of Capto fractional derivative. Existence and niqeness reslts were established by the new maximm principles obtained by estimating the fractional derivative of a fnction at its extreme points. Analogos reslts were obtained in [2, 22] for the fractional diffsion systems of mlti-term and distribted order frac- The Athors 217. This article is distribted nder the terms of the Creative Commons Attribtion 4. International License which permits nrestricted se, distribtion, and reprodction in any medim, provided yo give appropriate credit to the original athors and the sorce, provide a link to the Creative Commons license, and indicate if changes were made.

2 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 2 of 12 tional derivatives of Capto type. Another maximm principle for the linear mlti-term fractional differential eqations with the modified Riesz fractional derivative of Capto type was introdced and employed in [24]. The applicability of maximm principles for the linear and nonlinear fractional diffsion systems with the Riemann-Lioville fractional derivativewasdiscssedandprovedforthefirsttimebyal-refaiandlchkoin[15], where existence, niqeness and stability reslts were established. Analogos reslts for the fractional diffsion systems with the mlti-term and distribted order fractional derivatives of Riemann-Lioville type were obtained in [16] and[18]. In [25] and[26], a maximm principle was sed to analyze a type of fractional diffsion eqation withot an explicit formlation of this principle. Recently [1] Capto et al. have introdced a new type of fractional derivative with nonsinglar kernel. After then, the athors in [27 29] stdied their discrete versions and analyzed the monotonicity properties for the fractional difference operator. Insisting on the importance of having fractional operators with non-singlar kernels, later in [3 32]the athors introdced, explored and stdied fractional operators of Mittag-Leffler kernels together with their discrete versions. In this paper, we extend the reslts presented in [15] for the fractional diffsion eqations with the Capto fractional derivative of non-singlar kernel. The rest of this paper is organized as follows. First, we give the basic definitions and reslts abot fractional derivatives with exponential kernels. In Section 2, an estimate of the Capto fractional derivative of non-singlar kernel of a fnction at its extreme points is dedced in a form of certain ineqality. This ineqality is then employed to derive a weak and a strong maximm principles for the time-fractional diffsion eqation with the Capto fractional derivative of non-singlar kernel. We apply the obtained maximm principles to analyze the soltions of linear and nonlinear time-fractional diffsion models in Sections 3 and4,respectively. Uniqeness and stability reslts as well as norm estimates of soltions are obtained. Some illstrative examples are presented in Section 5. Finally, we close p with some conclding remarks in Section 6. Definition 1.1 [1, 27] Let f H 1 a, b, a < b, [, 1], the left Capto fractional derivative in the sense of Capto and Fabrizio is defined by D a f t= 1 t The associated fractional integral is defined by a f t x [ xe 1 ] dx. 1.1 CF Ia f t= 1 f t+ t a f s ds, 1.2 where > is a normalization fnction satisfying B = B1 = 1. Lemma 1.1 [27] For < <1,we have CF I a CFC D a f x=f x f a.

3 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 3 of 12 2 Maximm principles We start with estimating the fractional derivative of a fnction at its extreme points. These reslts are analogosto the ones obtained in [33] for the Capto fractional derivative. We then se these reslts to establish new maximm principles for linear fractional eqations with Capto fractional derivative of non-singlar kernel. Lemma 2.1 Let a fnction f H 1, T attain its maximm at a point t [, T] and < <1.Then the ineqality D f t 1 t f t f 2.1 holds tre. Proof We define the axiliary fnction gt =f t f t, t [, T]. Then it follows that gt, on [, T], gt =g t =and D gt = CFC D f t. Since g H1, T, then g is integrable, and integrating by parts with = e 1 t τ, and dv = g τ, yields D g t = 1 = 1 t g τe 1 t τ dτ e 1 t τ gτ t 1 t = gt e 1 t g gτe 1 t τ dτ t gτe 1 t τ dτ. 2.2 Since gt on[,t], theintegralinthelasteqationisnonnegative, andths D g t gt e 1 t g = 1 1 t g = 1 t f t f. 2.3 The last ineqality yields D f t 1 t f t f, which proves the reslt. Lemma 2.2 Let a fnction f H 1, T attain its maximm at a point t, T] and < <1.If f t is not the constant fnction on [, t ], then the ineqality D f t > 1 t f t f 2.4 holds tre.

4 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 4 of 12 Proof Since f t isnotconstant,thengt =f t f t and not identically zero on [, t ]. Ths t gτe 1 t τ dτ >. Then Eq. 2.3willleadto D g t < 1 t f t f, which proves ineqality 2.4. We now employ the previos reslts to derive weak and strong maximm principles for the time-fractional differential operator P of parabolic type defined by P x, t = L x, t D x, t t, < <1,t, T], x, where is the rectanglar region : {<x < l,<t < T}, L=ax, t xx + bx, t x,and ax, t >. The sides of are S 1 : {x =, t T}, S 2 : { x l, t =}, S 3 : {x = l, t T}, and S 4 : { x l, t = T}. We have the following weak maximm principle. Theorem 2.1 Weak maximm principle Sppose that x, t C 2 [, l] H 1, T] satisfies the ineqality P in the rectanglar region T =,l, T], then max = max. S 1 S 2 S 3 Proof Assme by contradiction that the reslt is not tre, then attains a maximm at x, t T with x, t =M 1, and M 1 >max s1 S 2 S 3. Becase x x, t =, xx x, t andax, t >,wehave L x, t x,t=x,t = ax, t xx x, t +bx, t x x, t, and D x, t t 1 t x, t x, = 1 t M 1 x, >. 2.5 Ths, P x, t = L x, t D x, t t <, and a contradiction is reached.

5 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 5 of 12 Theorem 2.2 Strong maximm principle Sppose that x, t C 2 [, l] H 1, T] satisfies the ineqality P in the rectanglar region T =,l, T] and that attains a maximm at x, t T. Then x, t=x, t for all t t. Proof For < t t, assme by contradiction that the reslt is not correct. That is, x, t is not constant on, t ]. By ineqality 2.4wehave D x, t t >, and ths P x, t = L x, t D x, t t <, which contradicts the assmption of the theorem. Since x, t=x, t, for all < t t, and by the continity of the soltion x, t, we have x, t=x, t for all t t, which completes the proof. 3 Linear fractional diffsion problems In this section, we consider the time-fractional initial-bondary vale problem Q =L D t f x, t=, < <1,x, t, 3.1 x,=gx, x l, 3.2, t=h 1 t, l, t=h 2 t, t, T], 3.3 where f x, t is continos on =[,l] [, T]. We start with the following niqeness reslt. Theorem 3.1 The time-fractional initial-bondary vale problem has at most one soltion C 2 [, l] H 1, T]. Proof Assme that 1 x, tand 2 x, t are two soltions of , and let vx, t= 1 2. Then vx, tsatisfies P v=, < <1, x, t, vx,=, x l, v, t=vl, t=,t, T]. Applying the simple maximm principle will lead to vx, t, x, t.theabovestatements hold tre for vx, t, and ths vx, t. Hence vx, t =andtheresltisobtained. In the following we present essential reslts to garantee the existence of a soltion to the time-initial-bondary vale problem and to obtain analytical bond of the soltion.

6 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 6 of 12 Lemma 3.1 Consider the fractional initial vale problem D k t=f t, k = c, t >, 3.4 where f t C[,T]. The initial vale problem has the niqe soltion kt=c + 1 f t+ t f τ dτ, if and only if f =. Moreover, it holds that kt [,T] k T f t. 3.5 [,T] Proof By the help of Lemma 1.1,wehave kt k = CF I CFC D kt= CF I 1 f t= f t+ t f τ dτ, which proves the reslt. The reslt in Eq. 3.5 follows since t f τ dτ t f t [,T] [,T] dτ = f t [,T] t f t [,T] T. We have the following necessary condition for the existence of soltion to Eq Lemma 3.2 The time-fractional eqation 3.1 has a soltion only if L x,=f x,, where L=ax, t xx + bx, t x. Proof We have D t=l f x, t. Applying the fractional integral operator CF I yields x, t x,= CF I 1 L f = L f + t L f dτ. By the continity of the soltion, the above eqations hold tre at t =,whichyields L fx,=. Theorem 3.2 Let C 2 [, l] H 1, T] be a soltion to the time-fractional initialbondary vale problem with f x,=.let zt C[, T] be sch that f x, t zt, x, t, z =, and kt betheniqesoltionof D k t=zt, k =. 3.6

7 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 7 of 12 Then it holds that kt [,T] + max { gx [,l], h1 t [,T] + kt [,T], h2 t [,T] + kt [,T] }, 3.7 where kt [,T] T zt [,T]. Proof Since z=, applyingthe resltin Eq. 3.5, we have kt [,T] 1 1+ zt 1 T [,T]. Let vx, t =x, t kt, then P vx, t = L vx, t D vx, t t = L x, t D x, t t+ D k t = f x, t+zt. The initial and bondary conditions of vx, tare vx,=gx k = gx, x S 2, v, t=h 1 t kt, vl, t=h 2 t kt, t, T]. Applying the weak maximm principle, we have { vx, t max gx, h1 t kt, h 2 t kt }, x,t S 1 S 2 s 3 and ths { x, t kt+ max gx, h1 t kt, h 2 t kt }. 3.8 x,t S 1 S 2 s 3 For wx, t = x, t, we have P wx, t = f x, t. Let vx, t =wx, t kt, then vx, t satisfies P v=p w+ D k t= f x, t+zt, < <1,x, t, vx,= gx, x S 2, v, t= h 1 t kt, vl, t= h 2 t kt, t, t]. Applying the simple maximm principle will lead to { x, t=wx, t kt+ max gx, h1 t kt, h 2 t kt }. 3.9 x,t S 1 S 2 s 3

8 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 8 of 12 Combining Eqs. 3.8and3.9willleadto kt [,T] + max { gx [,l], h1 t+kt [,T], h2 t+kt [,T] }, which proves the reslt. Corollary 3.1 Let C 2 [, l] H 1, T] be a soltion to the time-fractional initialbondary vale problem with f x, t=.then it holds that max { gx [,l], h 1 t [,T], h 2 t [,T] }. Proof Since f x, t =, wechoose zt =, andthsthefractionalinitialvaleproblem D k t=, k =, has the niqe soltion kt =, which proves the reslt. We have the following stability reslt. Theorem 3.3 Let 1 x, t and 2 x, t be two soltions of the time-fractional diffsion eqation 3.1 that satisfy the same bondary condition 3.2 and the initial conditions 1 x,=g 1 x, 2 x,=g 2 x, x l. Then it holds that 1 x, t 2 x, t g1 x g 2 x [,l]. Proof Let vx, t= 1 2.Thenvx, tsatisfies P v=, < <1,x, t, vx,=g 1 x g 2 x, x S 2, v, t=vl, t=, t, T]. Applying the weak maximm principle, we have vx, t g 1 x g 2 x. Applying analogos statements for vx, twillleadto vx, t g 2 x g 1 x. Ths, vx, t g1 x g 2 x, x, t, and the reslt is obtained. 4 Nonlinear fractional diffsion problems We consider the nonlinear time-fractional diffsion eqation of the form N = D t L Fx, t, =, < <1,x, t, 4.1 sbject to the initial and bondary conditions , where Fx, t, isasmoothfnction. We start with the following niqeness reslt.

9 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 9 of 12 Theorem 4.1 If Fx, t, is nonincreasing with respect to, then the nonlinear timefractional diffsion eqation 4.1 sbject to the initial and bondary conditions possesses at most one soltion C 2 [, l] H 1, T]. Proof Assme that 1 x, t and 2 x, t are two soltions of 4.1 sbject to the initial and bondary conditions , and let vx, t= 1 2.Thenvx, tsatisfies P vx, t = L vx, t D vx, t t=fx, t, 2 Fx, t, 1, x, t, vx,=, x l, v, t=vl, t=, t, T]. Applying the mean vale theorem to Fx, t, yields Fx, t, 2 Fx, t, 1 = F 2 1 = F v, where =1 μ 1 + μ 2 for some μ 1. Ths, P vx, t = L vx, t D vx, t t= F v. 4.2 Assme by contradiction that v is not identically zero. Then v has either a positive maximm or a negative minimm. At a positive maximm x, t T,wehave M = vx, t >, v x x, t =, v xx x, t. Becase ax, t >,wehavelvx, t x,t=x,t, which together with D vx, t t = 1 t vx, t vx, = 1 t M > will lead to L vx, t x,t=x,t D vx, t t <. 4.3 Becase F is nonincreasing, we have F, and ths F vx, t = F M. 4.4 The reslts in Eqs. 4.3and4.4 contradict Eq Analogos steps will lead to a contradiction if v has a negative minimm. Ths, v =and 1 = 2. Theorem 4.2 Let 1 x, t and 2 x, t be two soltions of the time-fractional diffsion eqation 4.1 that satisfy the same bondary condition 3.2 and the initial conditions 1 x, = g 1 x, 2 x, = g 2 x, x l. If Fx, t, is nonincreasing with respect to, then it holds that 1 x, t 2 x, t g 1 x g 2 x [,l].

10 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 1 of 12 Proof Let vx, t= 1 2.Thenvx, tsatisfies,seeeq.4.2, P vx, t = L vx, t D vx, t t= F v, x, t, vx,=g 1 x g 2 x, x l, 4.5 v, t=vl, t=, t, T]. Let M a = g 1 x g 2 x [,l], and assme by contradiction that the reslt of the theorem is not tre. That is, vx, t M a. Then v either has a positive maximm at a point x, t with vx, t =M 1 > M a,or it has a negative minimm at a point x, t with vx, t =M 2 < M a.ifvx, t = M 1 > M a, sing the initial and bondary conditions of v, wehavex, t T. Applying analogos steps in the proof of the previos theorem, we have and L vx, t x,t=x,t D vx, t t < 4.6 F vx, t = F M a >, 4.7 which contradicts Eq Applying analogos steps will lead to a contradiction for the case of vx, t =M 2 < M a.ths, vx, t M a, which proves the reslt. By applying analogossteps in the proofof Lemma 3.2, we have the following necessary condition for the existence of soltion of 4.1. Lemma 4.1 The nonlinear time-fractional eqation 4.1 has a soltion only if L x,=f x,,x,, where L=ax, t xx + bx, t x. 5 Illstrated examples Example 5.1 Consider the time-fractional initial-bondary vale problem D t= 1 e t xx te x, <x <1,t >, sbject to the initial condition x,= gx, < x < 1,and the homogeneos bondary conditions, t=1, t=,t >.Now, f x, t = te x t,<x <1,andthswechoose zt=t in Eq We have kt [,T] T zt 1 [,T] 1+ 1 T T.

11 Al-Refai and Abdeljawad Advances in Difference Eqations :315 Page 11 of 12 For T =1,thelasteqationyields kt [,T] 1. Applying the reslt in 3.7, we have 1 { } gx + max [,1], 1. Example 5.2 Consider the nonlinear time-fractional diffsion eqation N = D t xx e =, <x <1,t >. Since Fx, t, = e satisfies F = e, then Fx, t, = e is nonincreasing, and the above diffsion eqation has at most one soltion by virte of Theorem 4.1. By Lemma 3.2 a necessary condition for the existence of soltion of the above diffsion eqation is xx x,=e x,. 6 Conclding remarks We have considered linear and nonlinear fractional diffsion eqations with Capto fractional derivative of non-singlar kernel. We have obtained an estimate of the Capto fractional derivative of non-singlar kernel of a fnction at its extreme points. We then have derived a weak and a strong maximm principles for the linear time-fractional diffsion eqation. We have analyzed the soltions of the linear and nonlinear time-fractional diffsion models sing the obtained maximm principles. Some examples are presented to illstrate the applicability of the obtained reslts. The obtained reslts will lead to better nderstanding of the time-fractional diffsion models with Capto fractional derivative of non-singlar kernel. In contrast to previos stdies on fractional diffsion models with Capto and Riemann-Lioville fractional derivatives, it is noticed that extra conditions have been imposed to garantee the existence of soltions to the linear and nonlinear time-diffsion models. Acknowledgements The first athor acknowledges the spport of the United Arab Emirates University nder the Fnd No. 31S239. The second athor wold like to thank Prince Sltan University for fnding this work throgh research grop Nonlinear Analysis Methods in Applied Mathematics NAMAM grop nmber RG-DES Competing interests The athors declare that they have no competing interests. Athors contribtions All athors read and approved the final manscript. Athor details 1 Department of Mathematical Sciences, UAE University, P.O. Box 15551, Al Ain, UAE. 2 Department of Mathematics and General Sciences, Prince Sltan University, P.O. Box 66833, Riyadh, 11586, Sadi Arabia. Pblisher s Note Springer Natre remains netral with regard to jrisdictional claims in pblished maps and instittional affiliations. Received: 1 May 217 Accepted: 11 September 217

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