R E S E A R C H
Open Access
Global existence of solutions for a
fractional Caputo nonlocal thermistor
problem
Moulay Rchid Sidi Ammi
1*, Ismail Jamiai
1and Delfim F. M. Torres
2*Correspondence: [email protected] 1Department of Mathematics, AMNEA Group, Faculty of Sciences and Techniques, Moulay Ismail University, B.P. 509, Errachidia, Morocco
Full list of author information is available at the end of the article
Abstract
We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions.
MSC: 26A33; 35A01; 58J20
Keywords: fractional derivatives; fractional PDE; local existence; global existence;
fixed point theorem
1 Introduction
Fractional calculus is acknowledged as an important research tool that opens up many horizons in the field of dynamical systems []. According to Professor Katsuyuki Nishi-moto, ‘the fractional calculus is the calculus of the XXIst century’ []. This opinion is strengthened by a huge increase of interest in this research tool, expressed by an increase in the number of theoretical developments and basic theory on this subject; see, e.g., [–]. Recently, it has also been proved that fractional differential equations are significant and essential tools when applied in the study of nonlocal or time-dependent processes and in the modeling of many applications, including chaotic dynamics, material sciences, me-chanic of fractal and complex media, quantum meme-chanics, physical kinetics, chemistry, biology, economics and control theory []. For instance, a fractional generalization of the Newtonian equation to describe the dynamics of complex phenomena, in both science and engineering, has been proposed in []; a fractional Langevin equation, with applications in polymer layers, has been investigated in []. One can say that real-world problems require the definitions of fractional derivatives for initial and boundary value problems [, ]. Fractional mathematical models describing natural phenomena, like shallow wa-ter waves and ion acoustic waves in plasma and vibration of large membranes, as well as personal and interpersonal realities, like smoking, romantic relationships and marriages, can be found in [, ] and [, ], respectively. Details of the geometric and physical interpretation of fractional differentiation can be found in [].
Thermistor is a thermo-electric device constructed from a ceramic material whose elec-trical conductivity depends strongly on the temperature. This makes thermistor problems highly nonlinear []. They can be used as a switching device in many electronic circuits.
©The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
A broad application spectrum of thermistor problems in heating processes and current flow can be found in several areas of electronics and its related industries []. Generally, there are two kinds of thermistors: the first have an electrical conductivity that decreases with the increasing of temperature; the second have an electrical conductivity that in-creases with the increasing of temperature [, ]. Here we consider a prototype of elec-trical conductivity that depends strongly in both time and temperature. Our goal consists in proving global existence of solutions for a fractional Caputo nonlocal thermistor prob-lem. The results are obtained via Schauder’s fixed point theorem. Precisely, we consider the following fractional order initial value problem:
CDα,tu(t) =
λf(t, u(t))
(tf(x, u(x)) dx), t∈ (, ∞),
u(t)|t== u,
()
whereCDα,tis the fractional Caputo derivative operator of order α with < α < a real
parameter. The function u denotes the temperature and λ is a positive real. We shall as-sume the following hypotheses:
(H) f :R+× R+→ R+is a Lipschitz continuous function with Lipschitz constant Lf with
respect to the second variable such that c≤ f (s, u) ≤ cwith c and ctwo positive
constants;
(H) there exists a positive constant M such that f (s, u)≤ Ms;
(H) |f (s, u) – f (s, v)| ≤ s|u – v| or, in a more general manner, there exists a constant ω ≥
such that|f (s, u) – f (s, v)| ≤ sω|u – v|.
In the literature, questions involving the existence and uniqueness of solution for frac-tional differential equations (FDEs) have been intensely studied by many mathematicians [, , , ]. However, much of published papers have been concerned with existence-uniqueness of solutions for FDEs on a finite interval. Since continuation theorems for FDEs are not well developed, results as regards global existence-uniqueness of the so-lution of FDEs on the half axis [, +∞), by using directly the results from local existence, have only recently flourished [, ].
In contrast with our previous work [–] on fractional nonlocal thermistor problems, which was focused on local existence and numerical methods, here we are concerned with continuation theorems and global existence for the steady state fractional Caputo nonlocal thermistor problem. The paper is organized as follows. In Section , we collect some back-ground material and necessary results from fractional calculus. Then we are concerned in Section with local existence on a finite interval for () (Theorem .). Section is de-voted to the (non-)continuation (Theorem .) associated with problem (), which allows one to generalize the main result of Section . Our proofs rely on Schauder’s fixed point theorem and some extensions of the continuation theorems for ordinary differential equa-tions (ODEs) to the fractional order case. One of the main difficulties lies in handling the nonlocal term λf(t,u(t))
(tf(x,u(x)) dx), representing a heat source and that depends continuously on
time; another one in the fact that electrical conductivity depends on both time and tem-perature. Based on the results of Section , in Section we prove existence of a global solution for (): see Theorems . and .. We end with Section presenting conclusions.
2 Preliminaries and basic results
In this section, we collect from the literature [, , , –] some background material and basic results that will be used in the remainder of the paper.
Let C[a, b] be the Banach space of all real valued continuous functions on [a, b] en-dowed with the normx[a,b]= maxt∈[a,b]|x(t)|. According to the Riemann-Liouville
ap-proach to fractional calculus, we introduce the fractional integral of order α, α > , as follows.
Definition . The Riemann-Liouville integral of a function g with order α > is defined by RLD–α,tg(t) = (α) t (t – s)α–g(s) ds, t> , where is the Euler gamma function given by
(α) = ∞
tα–e–tdt,
α> .
The natural next step, after the notion of fractional integral has been introduced, is to define the fractional derivative of order α, α > .
Definition . The Riemann-Liouville derivative of the function g with order α > is defined by RLDα,tg(t) = (n – α) dn dtn t (t – s)n–α–g(s) ds, t> , where n – < α < n∈ Z+.
Note the remarkable fact that, in the Riemann-Liouville sense, the fractional derivative of the constant function is not zero. We now give an alternative and more restrictive def-inition of fractional derivative, first introduced by Caputo in the end of the s [, ] and then adopted by Caputo and Mainardi in [, ]. In Caputo sense, the fractional derivative of a constant is zero.
Definition . The Caputo derivative of the function g(t) with order α > is defined by
CDα,tg(t) = (n – α) t (t – s)α–g(n)(s) ds, t> , where n – < α < n∈ Z+.
For proving our main results, we make use of the following auxiliary lemmas.
Lemma .(See []) Let M be a subset of C([, T]). Then M is precompact if and only if
. {u(t) : u ∈ M} is uniformly bounded, . {u(t) : u ∈ M} is equicontinuous on [, T].
Lemma .(Schauder fixed point theorem []) Let U be a closed bounded convex subset
of a Banach space X. If T : U→ U is completely continuous, then T has a fixed point in U. Finally we recall a generalization of Gronwall’s lemma, which is essential for the proof of our Theorem ..
Lemma .(Generalized Gronwall inequality [, ]) Let v : [, b]→ [, +∞) be a real
function and w(·) be a nonnegative, locally integrable function on [, b]. Suppose that there exist a> and < α < such that
v(t)≤ w(t) + a t
v(s) (t – s)αds.
Then there exists a constant k= k(α) such that
v(t)≤ w(t) + ka t w(s) (t – s)αds for t∈ [, b].
3 Local existence theorem
In this section, a local existence theorem of solutions for () is obtained by applying Schauder’s fixed point theorem. In order to transform () into a fixed point problem, we give in the following lemma an equivalent integral form of ().
Lemma . Suppose that(H)-(H) holds. Then the initial value problem () is equivalent
to u(t) = u+ λ (α) t (t – s)α– f(s, u(s)) (tf(x, u) dx)ds. ()
Proof It is a simple exercise to see that u is a solution of the integral equation () if and only if it is also a solution of the IVP ().
Theorem . Suppose that conditions(H)-(H) are verified. Then () has at least one
solution u∈ C[, h] for some T ≥ h > . Proof Let
E=u∈ C[, T] : u – uC[,T]= sup
≤t≤T|u – u| ≤ b
,
where b is a positive constant. Further, put
Dh=
u: u∈ C[, h], u – uC[,h]≤ b
,
where h= min b λM (α + )c –α , T
and < α <. It is clear that h≤ T. Note also that Dhis a nonempty, bounded, closed, and
convex subset of C[, h]. In order to apply Schauder’s fixed point theorem, we define the following operator A: (Au)(t) = u+ λ (α) t (t – s)α– f(s, u(s)) (tf(x, u) dx)ds, t∈ [, h]. ()
It is clear that all solutions of () are fixed points of (). Then, by assumptions (H) and
(H), we have for any u∈ C[, h]
(Au)(t) – u ≤ λ (α) (ct) t (t – s)α–f s, u(s)ds ≤ λ (α) M c t (t – s)α–ds ≤ λM α(α) c hα = λM (α + ) c hα ≤ b.
It yields ADh ⊂ Dh. Our next step, in order to prove Theorem ., is to show that the
following lemma holds.
Lemma . The operator A is continuous.
Proof Let un, u∈ Dhbe such thatun– uC[,h]→ as n → +∞. One has
Aun(t) – Au(t) ≤ λ (α) t (t – s)α– f(s, un(s)) (tf(x, un) dx) – f(s, u(s)) (tf(x, u) dx) ds ≤ λ (α) t (t – s)α– (tf(x, un) dx) f s, un(s) – f s, u(s) + f s, u(s) (tf(x, un) dx) – (tf(x, u) dx) ds ≤ λ (α) t (t – s)α– (tf(x, un) dx) f s, un(s) – f s, u(s) ds + λ (α) t (t – s)α– f s, u(s) (tf(x, un) dx) – (tf(x, u) dx) ≤ I+ I. ()
We now focus on both right hand terms separately. By hypotheses (H) and (H), we have I≤ λ (ct)(α) t (t – s)α– f s, un(s) – f s, u(s) ds ≤ λLf c(α) t (t – s)α– un(s) – u(s) ds ≤ λLf c(α) un– uC[,h] t (t – s)α–ds ≤ λLf c(α) un– uC[,h] t (t – s)α–ds. Then I≤ λhαLf c (α + ) un– uC[,h]. ()
Concerning the second term, we have
I≤ λ (α) t (t – s)α–|f (s, u(s))| (tf(x, un) dx)( t f(x, u) dx) × t f(x, un) dx – t f(x, u) dx ds ≤ λc (ct)(α) t (t – s)α– t f(x, un) dx – t f(x, u) dx ds ≤ λc (ct)(α) t (t – s)α– × t f(x, un) – f (x, u) dx t f(x, un) + f (x, u) dx ds ≤ λct (ct)(α) t (t – s)α– t f(x, un) – f (x, u) dxds ≤ λctLf (ct)(α) t (t – s)α– t un(x) – u(x) dx ds ≤ λctLf (ct)(α)un – uC[,h] t (t – s)α–ds ≤ λcLf c(α) un– uC[,h] t (t – s)α–ds ≤ λchαLf c (α + ) un– uC[,h]. It follows that I≤ λchαLf c (α + ) un– uC[,h]. ()
Collecting inequalities () and () together, and inserting into (), we have Aun(t) – Au(t) ≤I+ I ≤ λhαL f c(α + )+ λc hαLf c(α + ) un– uC[,h]. Therefore, Aun– AuC[,h]≤ λhαL f c(α + )+ λc hαLf c(α + ) un– uC[,h]. ()
Consequently,Aun– AuC[,h]→ as n → +∞, which proves that A is continuous. This
ends the proof of Lemma .. To finish the proof of Theorem ., it remains to show the following.
Lemma . The operator ADhis continuous.
Proof Let u∈ Dhand ≤ t≤ t≤ h. Then
(Au)(t) – (Au)(t) ≤ λ (α) t (t– s)α– f(s, u(s)) (t f(x, u) dx) ds– t (t– s)α– f(s, u(s)) (t f(x, u) dx) ds ≤ λ (α) t (t– s)α–– (t– s)α– f(s, u(s)) (t f(x, u) dx) ds + t (t– s)α– f(s, u(s)) (t f(x, u) dx) ds– t (t– s)α– f(s, u(s)) (t f(x, u) dx) ds ≤ λ (α) t (t– s)α–– (t– s)α– f(s, u(s)) (t f(x, u) dx) ds + t (t– s)α–f s, u(s) (t f(x, u) dx) – (t f(x, u) dx) ds – t t (t– s)α– f(s, u(s)) (t f(x, u) dx) ds ≤ I+ I+ I, ()
where we have, by direct calculation,
I≤ λc (ct)(α) t (t– s)α–– (t– s)α– ds ≤ λc (ct)(α + ) tα– tα+ (t– t)α , I≤ λ (α) ctα– (ct)(ct) t t f(x, u) dx – t f(x, u) dx ds ≤ λ (α) ctα– (ct)(ct) ()
× t tf(x, u) dx + t f(x, u) dx t f(x, u) dx – t f(x, u) dx ds ≤ λ (α) c (t+ t)tα– (ct)(ct) t t f s, u(s)ds ≤ λ (α) c (t+ t)tα– (ct)(ct) |t – t|, and I≤ λc (ct)(α) t t (t– s)α–ds≤ λc (ct)(α + ) (t– t)α. ()
The right hand side of inequalities () and () do not depend on u and converge to zero as t→ t. Then{(Au)(t) : u ∈ Dh} is equicontinuous and Lemma . is proved.
Taking into account that ADh⊂ Dh, we infer that ADhis precompact. This implies that
A is completely continuous. As a consequence of Schauder’s fixed point theorem and Lemma ., we conclude that problem () has a local solution. This ends the proof of
The-orem ..
4 Continuation results
Our main contribution of this section is to prove a continuation theorem for the fractional Caputo nonlocal thermistor problem (). First, we present the definition of noncontinu-able solution.
Definition .(See []) Let u(t) on (, β) and ˜u(t) on (, ˜β) be both solutions of (). If β < ˜β and u(t) =˜u(t) for t ∈ (, β), then we say that ˜u(t) can be continued to (, ˜β). A solution u(t) is noncontinuable if it has no continuation. The existing interval of the noncontinuable solution u(t) is called the maximum existing interval of u(t).
Theorem . Assume that conditions(H)-(H) are satisfied. Then u = u(t), t∈ (, β),
is noncontinuable if and if only for some η∈ (,β) and any bounded closed subset S⊂ [η, +∞) × R there exists a t∗∈ [η, β) such that (t∗, u(t∗)) /∈ S.
Proof Suppose that there exists a compact subset S⊂ [η, +∞) × R such that
t, u(t): t∈ [η, β)⊂ S.
The compactness of S implies β < +∞. The remainder of the proof is given in two lemmas.
Lemma . The limit limt→β–u(t) exists.
Proof Let t, t∈ [η, β) such that t< t. From (), we have
I= I,+ I,, where I,= λ (α) η (t– s)α–– (t– s)α– f(s, u(s)) (t f(x, u) dx) ds
and I,= λ (α) t η (t– s)α–– (t– s)α– f(s, u(s)) (t f(x, u) dx) ds.
Under assumptions (H)-(H), there exists a positive constant M,such that
I,≤ M, η (t– s)α–– (t– s)α– ≤ M, (t– η)α– (t– η)α+ tα – tα .
Moreover, there exists a positive constant M,such that
I,≤ M, t η (t– s)α–– (t– s)α– ≤ M, (t– t)α+ (t– η)α– (t– η)α
and we also have I= I,+ I,, where
I,= η (t– s)α–f s, u(s) (t f(x, u) dx) – (t f(x, u) dx) ds and I,= t η (t– s)α–f s, u(s) (t f(x, u) dx) – (t f(x, u) dx) ds.
In the same manner as in the proof of Lemma ., there exists positive constants M,and
M,such that
I,≤ M,|t– t|,
I,≤ M,|t– t|.
We have already proved in (), for some positive constant M, that
I≤ M|t– t|α.
Therefore, we conclude that all Ii, i = , , , converge to zero when t→ t. Thus, from
Cauchy’s convergence criterion, it yields limt→β–u(t) = u∗. This finishes the proof of
Lemma ..
The second step of the proof of Theorem . consists in showing the following result.
Lemma . Function u(t) is continuable.
Proof As S is a closed subset, we can say that (β, u∗)∈ S. Define u(β) = u∗. Hence, u(t)∈
C[, β]. Then we define the operator K by (Kv)(t) = u+ λ (α) t β (t – s)α– f(s, v(s)) (tf(x, v) dx)ds,
where u= u+ λ (α) β (t – s)α– f(s, v(s)) (tf(x, v) dx)ds, v∈ C [β, β + ], t∈ [β, β + ]. Set Eb= (t, v) : β≤ t ≤ β + , |v| ≤ max β≤t≤β+ u(t) + b and Eh= v∈ C[β, β + ] : max t∈[β,β+h] v(t) – u(t) ≤b, v(β) = u(β) , where h = min{(b( λM (α+)c)
–)α, }. Analogously to the proof of Theorem ., we prove
that K is completely continuous on Eb. Indeed, let {vn} ⊆ C[β, β + h]. Then vn –
v[β,β+h]→ as n → +∞ and similar arguments to the one above for (), allow us to
declare that there exists a positive constant chdepending on h such that
Kvn– KvC[β,β+h]≤ chvn– vC[β,β+h].
Hence,(Kvn)(t) – (Kv)(t)[β,β+h]→ as n → +∞, which shows that the operator K is
continuous. We show that KEhis equicontinuous. For all v∈ Eh, we have (Kv)(β) = u(β)
and, in view of the choice of h, it follows from hypotheses (H) and (H) that
(Kv)(t) – u = (α)λ βt(t – s)α– f(s, v(s)) (tf(x, v) dx)ds ≤ λM (α + )ch α≤ b.
Therefore, we get KEh⊂ Eh. Furthermore, for any v∈ Ehand β≤ t≤ t≤ β + h, we have
(Kv)(t) – (Kv)(t) = I+ I.
By a calculation analogous to the earlier one, there exists a positive constant Msuch that
I= λ (α) β (t– s)α– f(s, v(s)) (t f(x, v) dx) ds – λ (α) β (t– s)α– f(s, v(s)) (t f(x, v) dx) ds ≤ λ (α) β (t– s)α–– (t– s)α– f(s, v(s)) (t f(x, v) dx) ds + λ (α) β (t– s)α– f(s, v(s)) (t f(x, v) dx) ds– f(s, v(s)) (t f(x, v) dx) ds ≤ M (t– β)α– (t– β)α+ tα – tα +|t– t| . ()
An analogous treatment as in ()-() yields the existence of a positive constant Msuch that I = λ (α) t β (t– s)α– f(s, v(s)) (t f(x, v) dx) ds – λ (α) t β (t– s)α– f(s, v(s)) (t f(x, v) dx) ds ≤ M (t– β)α– (t– β)α +|t– t| + |t– t|α . ()
Since the right side of inequalities () and () go to zero as t→ t, we deduce that
{(Kv)(t) : v ∈ Eh} is equicontinuous. Consequently, K is completely continuous. Then
Schauder’s fixed point theorem can be applied to see that the operator K has a fixed point ˜u(t) ∈ Eh. On other words, we have
˜u(t) = u+ λ (α) t β (t – s)α– f(s,˜u(s)) (tf(x,˜u(x)) dx)ds = u+ λ (α) t (t – s)α– f(s,˜u(s)) (tf(x,˜u(x)) dx)ds, t∈ [β, β + h], where ˜u(t) = ⎧ ⎨ ⎩ u(t), t∈ (, β], ˜u(t), t ∈ [β, β + h]. It follows that ˜u(t) ∈ C[, β + h] and
˜u(t) = u+ λ (α) t (t – s)α– f(s,˜u(s)) (tf(x,˜u(x)) dx)ds.
Therefore, according to Lemma ., ˜u(t) is a solution of () on (, β + h]. This is absurd because u(t) is noncontinuable. This completes the proof of Lemma .. Theorem . follows from Lemmas . and ..
Remark . Uniqueness of solution to problem () is easily derived from the proof of Theorem . for a well chosen λ.
5 Global existence
Now we provide two sets of sufficient conditions for the existence of a global solution for () (Theorems . and .). We begin with an auxiliary lemma.
Lemma . Suppose that conditions(H)-(H) hold. Let u(t) be a solution of () on (, β).
If u(t) is bounded on [τ , β) for some τ > , then β = +∞.
Theorem . Suppose that conditions (H)-(H) hold. Then () has a solution in
C([, +∞)).
Proof The existence of a local solution u(t) of () is ensured thanks to Theorem .. We already know, by Lemma ., that u(t) is a also a solution to the integral equation
u(t) = u+ λ (α) t (t – s)α– f(s, u(s)) (tf(x, u(x)) dx)ds.
Suppose that the existing interval of the noncontinuable solution u(t) is (, β), β < +∞. Then u(t) = u+ λ (α) t (t – s)α– f(s, u(s)) (tf(x, u(x)) dx)ds ≤ |u| + λ (α) (ct) t (t – s)α– f s, u(s) ds ≤ |u| + λ (α) c t |u(s)| (t – s)–αds.
By Lemma ., there exists a constant k(α) such that, for t∈ (, β), we have u(t) ≤|u| + k|u| λ (α) c t (t – s)α–ds,
which is bounded on (, β). Thus, by Lemma ., problem () has a solution u(t) on
(, +∞).
Next we give another sufficient condition ensuring global existence for ().
Theorem . Suppose that there exist positive constants c, c and c such that c ≤
|f (s, x)| ≤ c|x| + c. Then () has a solution in C([, +∞)).
Proof Suppose that the maximum existing interval of u(t) is (, β), β < +∞. We claim that
u(t) is bounded on [τ , β) for any τ∈ (, β). Indeed, we have u(t) = u+ λ (α) t (t – s)α– f(s, u(s)) (tf(x, u(x)) dx)ds ≤ |u| + λ (α) (cτ) t (t – s)α– f s, u(s) ds ≤ |u| + λ (α) c (cτ) t (t – s)α–ds+ λ (α) c (cτ) t |u(s)| (t – s)–αds. If we take w(t) =|u| + λ (α) c (cτ) t (t – s)α–ds, which is bounded, and
a= λc
(α) (cβ)
it follows, in accordance with Lemma ., that v(t) =|u(t)| is bounded. Thus, by Lemma ., () has a solution u(t) on (, +∞).
6 Conclusion
In our paper we consider a prototype of electrical conductivity that depends strongly in both time and temperature. The model relates to modern developments of thermistors, where fractional PDEs have a crucial role. It turns out that available computational meth-ods are not theoretically sound in the sense they rely on results of local existence. The main novelty of our paper is that we prove global existence for a nonlocal thermistor problem with fractional differentiation in the Caputo sense. Moreover, we extend some results of continuation and global existence to the fractional order initial value thermistor problem. The proofs rely on Schauder’s fixed point theorem. We trust that our results will have a positive impact on the application of computer mathematics to fractional thermistor de-vices.
Acknowledgements
The authors were supported by the Center for Research and Development in Mathematics and Applications (CIDMA) of University of Aveiro, through Fundação para a Ciência e a Tecnologia (FCT), within project UID/MAT/04106/2013. They are grateful to two anonymous referees, for several comments and suggestions of improvement.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All three authors contributed equally to this work. They all read and approved the final version of the manuscript. Author details
1Department of Mathematics, AMNEA Group, Faculty of Sciences and Techniques, Moulay Ismail University, B.P. 509, Errachidia, Morocco.2Department of Mathematics, R&D Unit CIDMA, University of Aveiro, Aveiro, 3810-193, Portugal.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 21 July 2017 Accepted: 31 October 2017
References
1. Almeida, R, Pooseh, S, Torres, DFM: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015)
2. Nishimoto, K: An Essence of Nishimoto’s Fractional Calculus. Descartes Press, Koriyama (1991)
3. Aubin, J-P, Lygeros, J, Quincampoix, M, Sastry, S, Seube, N: Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Autom. Control 47(1), 2-20 (2002)
4. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
5. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
6. Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)
7. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives. Gordon & Breach, Yverdon (1993). Translated from the 1987 Russian original
8. Machado, JT, Kiryakova, V, Mainardi, F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140-1153 (2011)
9. Baleanu, D, Golmankhaneh, AK, Golmankhaneh, AK, Nigmatullin, RR: Newtonian law with memory. Nonlinear Dyn. 60(1-2), 81-86 (2010)
10. Baleanu, D, Golmankhaneh, AK, Nigmatullin, R, Golmankhaneh, AK: Fractional Newtonian mechanics. Cent. Eur. J. Phys. 8(1), 120-125 (2010)
11. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus. Series on Complexity, Nonlinearity and Chaos, vol. 5 World Scientific, Hackensack (2017)
12. Ortigueira, MD: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, vol. 84. Springer, Dordrecht (2011)
13. Kumar, D, Singh, J, Baleanu, D: A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math. Methods Appl. Sci. 40, 5642-5653 (2017)
14. Srivastava, HM, Kumar, D, Singh, J: An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45, 192-204 (2017)
15. Singh, J, Kumar, D, Qurashi, MA, Baleanu, D: A new fractional model for giving up smoking dynamics. Adv. Differ. Equ. 2017, 88 (2017)
16. Singh, J, Kumar, D, Qurashi, MA, Baleanu, D: A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships. Entropy 19(7), 375 (2017)
17. Sidi Ammi, MR, Torres, DFM: Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem. Math. Comput. Simul. 77, 291-300 (2008)
18. Sidi Ammi, MR, Torres, DFM: Optimal control of nonlocal thermistor equations. Int. J. Control 85(11), 1789-1801 (2012) 19. Kwok, K: Complete Guide to Semiconductor Devices. McGraw-Hill, New york (1995)
20. Maclen, ED: Thermistors. Electrochemical publication, Glasgow (1979)
21. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000)
22. Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)
23. Arara, A, Benchohra, M, Hamidi, N, Nieto, JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 72(2), 580-586 (2010)
24. Li, C, Sarwar, S: Existence and continuation of solutions for Caputo type fractional differential equations. Electron. J. Differ. Equ. 2016, 207 (2016)
25. Sidi Ammi, MR, Torres, DFM: Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives. Differ. Equ. Appl. 4(2), 267-276 (2012)
26. Sidi Ammi, MR, Torres, DFM: Galerkin spectral method for the fractional nonlocal thermistor problem. Comput. Math. Appl. 73(6), 1077-1086 (2017)
27. Sidi Ammi, MR, Torres, DFM: Existence and uniqueness results for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary time scales. J. King Saud Univ., Sci. (2017) in press. doi:10.1016/j.jksus.2017.03.004
28. Li, Y, Chen, Y, Podlubny, I: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965-1969 (2009)
29. Li, C, Deng, W: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777-784 (2007)
30. Li, CP, Zeng, FH: Numerical Methods for Fractional Differential Calculus. Chapman & Hall, Boca Raton (2015) 31. Caputo, M: Linear models of dissipation whose Q is almost frequency independent. II. Geophys. J. R. Astron. Soc.
13(5), 529-539 (1967)
32. Caputo, M: Elasticità e Dissipazione. Zanichelli, Bologna (1969)
33. Caputo, M, Mainardi, F: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II) 1, 161-198 (1971) 34. Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and
Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lect., vol. 378. Springer, Vienna (1997) 35. Henderson, J, Ouahab, A: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59(3), 1191-1226
(2010)
36. Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075-1081 (2007)
37. Kou, C, Zhou, H, Li, C: Existence and continuation theorems of Riemann-Liouville type fractional differential equations. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22(4), 1250077 (2012)
