- Research
- Open Access
- Published:
Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense
Advances in Difference Equations volume 2021, Article number: 162 (2021)
Abstract
We study a conformable fractional nonlocal thermistor problem on time scales. Under an appropriate nonrestrictive condition on the resistivity function, we establish existence and uniqueness results. The proof is based on the use of Schauder’s point fixed theorem.
1 Introduction
Fractional calculus has aroused keen considerable attention of several researchers with many emerging applications, including memory effects and future dependence, in mathematical physics, biology, dynamical systems, chemistry, population dynamics, and recently in epidemic diseases [2, 3, 6, 9–11, 15, 16, 21, 23, 25, 26].
The theory of time scales is often used for describing models that cannot be considered as exclusively continuous or exclusively discrete processes. It was first introduced by Aulbach and Stefan Hilger [8] with many applications in calculus of variations, economics, quantum calculus [4]. To find out more about the fractional calculus on a time scale, excellent references are situated in the books of Miller and Ross [22] and Podlubny [24]. The monograph of Bohner and Peterson [14] and [4, 5] are considered as good bibliography for calculus on time scales.
In recent years, the notion of conformable fractional derivative, which is a natural extension of the classical derivative, was initiated in [20]. It was after extended to an arbitrary time scales in [13] generalizing the Hilger derivative, preserving the compatibility, and satisfying more standard formulas of the usual derivative. The main advantages and efficiency of the conformable calculus are described in a detailed way in [30]. A nabla conformable fractional calculus was introduced in [12] and more generally in [34] on arbitrary time scales.
A thermistor sensor is a temperature sensing element made of a sintered semiconductor material that is characterized by large changes in resistance proportional to small changes in temperature. Thermistors are made from a combination of metals and metal oxide materials. Once combined, the materials are formed and fired into the desired shape. The thermistors can then be used as disc-shaped thermistors or can be formed and assembled with conductive wires and coatings to form ball-shaped thermistors.
The main qualities of these sensors are: accuracy, nominal value for a given temperature (at 25 Celsius), response time (in seconds), sensitivity or temperature coefficient (variation of resistance according to temperature), extent or measurement range (min. and max. temperature of use), lifespan, stability (variation of the various parameters over time), dimensions, and low cost [17].
Due to its various applications, a great interest is given by researchers to the study of integer order, fractional order, or time scales thermistors models via different approaches [7, 28, 30, 31]. In [28], the question of existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional order derivatives was investigated. Global existence of solutions for a fractional nonlocal thermistor problem in the Caputo sense was addressed in [27]. Existence and uniqueness results for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary nonempty closed subsets of the real numbers were studied in [29], Dynamics and stability results for Hilfer fractional derivative, which interpolate both the Riemann–Liouville and the Caputo fractional derivative, thermistor problems were investigated in [33]. Existence of a particular tube solution of a nonlocal thermistor problem for fractional derivative in the conformable sense was proved in [32]. In the present work, we are concerned with a conformable fractional thermistor problem and with existence and uniqueness results on time scales.
In order that the paper be complete for beginning researchers wishing to infiltrate the field, we recall in Sect. 2 preliminary basic time scales notation, definitions, and lemmas that are used later. As an illustration, in Sect. 3, existence and uniqueness for a fractional nonlocal thermistor problem on time scales are addressed. The undermentioned model is formulated in the conformable sense. We employ essentially Banach and Schauder’s fixed point theorems.
2 Preliminaries
In the beginning we present the following auxiliary theorems that play a crucial role in reaching our existence and uniqueness results.
Theorem 2.1
([18] Banach fixed point theorem)
Let \((X,d)\) be a complete metric space, and let \(F :X \rightarrow X\) be a contraction mapping. Then F admits a unique fixed point in X.
Theorem 2.2
([18] Schauder fixed point theorem)
Let E be a Banach space. Assume that \(F : E \rightarrow E\) is a completely continuous operator and the set \(V = \{ u \in E : u = \mu F u, 0 \leq \mu \leq 1 \}\) is bounded. Then F has a fixed point in E.
For convenience, we secondly give some important notations, definitions, and basic results on a fractional calculus on a time scale generalizing the Hilger derivative which can be found, for example, in [14]. A nonempty closed subset \(\mathbb{T}\) of \(\mathbb{R}\) is called a time scale.
The forward jump operator \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\) is defined by
while the backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) is defined by
with \(\inf \emptyset =\sup \mathbb{T}\) (i.e., \(\sigma (M)=M\) if \(\mathbb{T}\) has a maximum M) and \(\sup \emptyset =\inf \mathbb{T}\) (i.e., \(\rho (m)=m\) if \(\mathbb{T}\) has a minimum m).
A point \(t\in \mathbb{T}\) is called right-dense, right-scattered, left-dense, or left-scattered if \(\sigma (t)=t\), \(\sigma (t)>t\), \(\rho (t)=t\), or \(\rho (t)< t\), respectively.
The (forward) graininess function \(\mu :\mathbb{T}\rightarrow [0,\infty )\) and the backward graininess function \(\nu :\mathbb{T}\rightarrow [0,\infty )\) are defined by
respectively.
A closed interval on \(\mathbb{T}\) is denoted by \([a, b]_{\mathbb{T}}= \{ t \in \mathbb{T}; a \leq t \leq b \}\), where a and b are fixed points of \(\mathbb{T}\) with \(a < b\). If \(\mathbb{T}\) has left-scattered maximum m, then \(\mathbb{T}^{k}= \mathbb{T}-\{m\}\). Otherwise, we set \(\mathbb{T}^{k}=\mathbb{T}\). For the usual definitions of Δ-derivative, Δ-integral of a function, we refer the reader to [14].
Let \(h: \mathbb{T} \rightarrow \mathbb{R}\) be a real-valued function on a time scale \(\mathbb{T}, t\in \mathbb{T}^{k}\) and \(\alpha \in (0, 1]\). In [13], the authors define the conformable fractional(CF) derivative of a function h of order α at t, denoted by \(T_{\alpha }(h)(t)\), the number provided it exists such that, for all \(\varepsilon >0\), there exists a neighborhood \(V_{t} \subseteq \mathbb{T}\) of t such that, for all \(s \in V_{t}\),
We remind the following properties as given in [13].
Lemma 2.3
Let \(\alpha \in (0,1), t\in \mathbb{T}^{k}\), and \(h: \mathbb{T} \rightarrow \mathbb{R}\) be a function. The following properties hold:
-
(a)
If h is CF differentiable of order α at \(t>0\), then h is continuous at t.
-
(b)
If h is continuous at a right-scattered t, then h is CF differentiable of order α at t with
$$\begin{aligned} T_{\alpha }(h) (t)= \frac{h(\sigma (t))-h(t)}{\mu (t)}t^{1-\alpha }. \end{aligned}$$ -
(c)
If t is right-dense, then h is CF differentiable of order α at t if and only if \(\lim_{t \rightarrow s}\frac{h(t)-h(s)}{t-s}t^{1-\alpha }\) exists as a finite number. We then have
$$\begin{aligned} T_{\alpha }(h) (t) =\lim_{t \rightarrow s}\frac{h(t)-h(s)}{t-s}t^{1- \alpha }. \end{aligned}$$ -
(d)
If h is CF differentiable of order α at t, then \(h(\sigma (t))=h(t)+ \mu (t) t^{1-\alpha }T_{\alpha }(h)(t)\).
Let h be a regulated function on \(\mathbb{T}\). The indefinite α-CF integral of h is defined by
Finally, we define the Cauchy or definite α-CF integral of h by
Other fundamental properties and features of the CF derivative as linearity of conformable fractional derivatives, conformable chain rule, conformable fractional derivative of product, quotient and composite of functions are given in [19].
The following result of the calculus on time scales is also useful [13].
Proposition 2.4
Let \([a, b] \subseteq \mathbb{T}\) and h be an increasing continuous function on the time-scale interval \([a, b]\). If H is the extension of h to the real interval \([a, b]\) defined by
then
The above inequality can be extended to the case of double integral as follows:
3 Main results
In this section, we are concerned with the conformable fractional nonlocal thermistor problem governed by
with the boundary value conditions
where \(0< \alpha <1\), \(T_{\alpha }\) is the CF derivative. λ is a dimensionless real parameter. In case \(\alpha =1\), problem (1) describes the diffusion of heat through a thermistor. \(f(u)\) is the temperature-dependent resistivity of the conductor; and β is a heat transfer coefficient verifying \(0< \beta < 1\). \(\eta \in (0,T)_{\mathbb{T}}\), T is real fixed. Many existence results for different forms of thermistor problem were investigated using various fixed point theorems [28, 30]. Different to our earlier contributions, we consider the nonlocal thermistor problem in the conformable sense on a time scale. Further, we establish the main results under the following nonrestrictive hypothesis:
- \((H_{1})\):
-
\(f: \mathbb{T} \times \mathbb{R} \rightarrow \mathbb{R}^{+*}\) is a continuous function with respect to the first variable and a locally Lipschitzian function with respect to the second variable.
\(\mathbb{C}(\mathbb{T}, \mathbb{R})\) will denote the Banach space of all continuous real-valued functions defined on the time scale \(\mathbb{T}\) with the supremum norm.
Lemma 3.1
Assume that hypothesis \((H1)\) is verified. Then u is a solution to (1)–(2) if and only if u is a solution of the representation integral equation
where
Proof
Suppose that \(u(t)\) verifies (1), then after taking the α-CF integral of this equation from \(\rho (0)\) to s, we get
Using the property of inverse integral, we have
The boundary condition implies
we then get
We can then substitute in (3) and have
By taking the α-CF integral
Thus
which together with the second boundary value condition and (4) yields
It follows that
Therefore, we have the intended result. □
Theorem 3.2
There exists a unique solution for a well-chosen parameter λ.
Proof
Let \(S \subseteq \mathbb{C}(\mathbb{T}, \mathbb{R})\). Define the subset S(r) as
where r is described below. First, it is clear that the set \(S(r)\) is a nonempty, closed, convex, and bounded set. Define the operator F by
It is clear that \(u(t)\) is a solution of (1)–(2) if it is a fixed point of the operator Fu. Since f is a continuous function, there exist two positive constants such that \(M_{1} \leq f(t, u(t)) \leq M_{2}\). Then
Setting \(u(\rho (0)) = \mu \). Then, by using Proposition 2.4, we get
With the equality \(\mu + \frac{ M_{3}}{1-\beta } \frac{T^{2\alpha }}{\alpha ^{2}}=r\), we conclude that F is an operator from \(S(r)\) to \(S(r)\). We prove that F is contractive, we can easily see that
On the other hand, we have
Using that f is locally Lipschitzian, we have
Furthermore, we have
Then
Hence
By the assumption that \(\frac{\lambda }{1-\beta } (M_{4} +M_{5}) (\frac{T}{\alpha } )^{2} < 1\), we conclude that F is contractive. It implies that (1)–(2) has a unique solution by the Banach fixed point theorem. □
Theorem 3.3
Under hypothesis \((H1)\), (1)–(2) has a solution on \(\mathbb{T}\).
Step 1
We prove that F is continuous. Let \(u_{n}\) be a sequence which converges to u. In a similar way as above, one can have, for all \(t\in \mathbb{T}\),
Then
The above inequality permits to conclude that whenever \(u_{n} \rightarrow u\), then \(Fu_{n} \rightarrow Fu\). This proves the continuity of F.
Step 2
The map F sends bounded sets into bounded sets in \(\mathbb{C}(\mathbb{T}, \mathbb{R})\). Furthermore, we obtain for all \(u \in S(r)\) any ball centered in the origin
Step 3
To prove that \(F(S(r))\) resides in a relatively compact subset of \(\mathbb{C}(\mathbb{T}, \mathbb{R})\). Let \(t_{1} < t_{2}\), we have
This implies that F is a bounded and uniformly Cauchy subset of \(\mathbb{C}(\mathbb{T}, \mathbb{R})\). Thus, by virtue of the Arzela–Ascoli’s theorem, we conclude that F is relatively compact.
Step 4
Now, it remains to show that the set \(\Omega = \{ u \in \mathbb{C}(\mathbb{T}, \mathbb{R}): u(t) = \nu F(u(t)); 0 < \nu < 1 \}\) is bounded. Let \(u \in \Omega \), Thus, for each \(t \in \mathbb{T}\), we have
By Schauder’s fixed point theorem, we can conclude that F has a fixed point, which represents a solution for problem (1)–(2).
4 Conclusion
In the last decade, fractional thermistor problems have attracted the interest of several researchers with flourishing applications. As an additional stone in this subject, existence and uniqueness results for a fractional nonlocal thermistor problem in the conformable sense are examined. The proof of the main result is based on the employment of Schauder’s fixed point and Arzelà –Ascoli theorems.
Availability of data and materials
Not applicable.
References
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Agarwal, P., Choi, J.: Fractional calculus operators and their image formulas. J. Korean Math. Soc. 53(5), 1183–1210 (2016)
Agarwal, P., Deniz, S., Jain, S., Alderremy, A.A., Aly, S.: A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Phys. A, Stat. Mech. Appl. 542, 122769 (2020)
Agarwal, R.F., Bohner, M.: Basic calculus on time scales and some of its applications. Results Math. 35(1–2), 3–22 (1999)
Agarwal, R.F., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141(1–2), 1–26 (2002)
Alderremy, A.A., Saad, K.M., Agarwal, P., Aly, S., Jain, S.: Certain new models of the multi space-fractional Gardner equation. Phys. A, Stat. Mech. Appl. 545, 123806 (2020)
Antontsev, S.N., Chipot, M.: The thermistor problem: existence, smoothness, uniqueness, blowup. SIAM J. Math. Anal. 25(4), 1128–1156 (1994)
Aulbach, B., Hilger, S.: A unified approach to continuous and discrete dynamics. In: Bolyai, J. (ed.) Qualitative Theory of Differential Equations. Colloquia Mathematica Societatis, pp. 37–56. North-Holland, Amsterdam (1990)
Baleanu, D., Asad, J.H., Petras, I., Elagan, S., Bilgen, A.: Fractional Euler–Lagrange equation of Caldirola–Kanai oscillator. Rom. Rep. Phys. 64(4 Suppl), 1171–1177 (2012)
Baleanu, D., Petras, I., Asad, J.H., Velasco, M.P.: Fractional Pais–Uhlenbeck Oscillator: Int. J. Theor. Phys. 51(4), 1253–1258 (2012)
Baltaeva, U., Agarwal, P.: Boundary-value problems for the third–order loaded equation with noncharacteristic type-change boundaries. Math. Methods Appl. Sci. 41(9), 3307–3315 (2018)
Bendouma, B., Hammoudi, A.: A nabla conformable fractional calculus on time scales. Electron. J. Math. Anal. Appl. 7(1), 202–216 (2019)
Benkhettou, N., Hassani, S., Torres, D.F.M.: A conformable fractional calculus on arbitrary time scales. J. King Saud Univ., Sci. 28, 93–98 (2016)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001)
Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003)
El-Sayed, A.A., Agarwal, P.: Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math. Methods Appl. Sci. 42(11), 3978–3991 (2019)
Fowler, A.C., Frigaard, I., Howison, S.D.: Temperature surges in current-limiting circuit devices. SIAM J. Appl. Math. 52(4), 998–1011 (1992)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003)
Gülşen, T., lmaz Emrah, Y., Kemaloğlu, H.: Conformable fractional Sturm–Liouville equation and some existence results on time scales. Turk. J. Math. 42(3), 1348–1360 (2018)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.I.: Theory and Applications of Fractional Differential Equations, 1st edn. Elsevier, London (2006)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, London (1974)
Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)
Rekhviashvili, R., Pskhu, A.R., Agarwal, P., Jain, S.: Application of the fractional oscillator model to describe damped vibrations. Turk. J. Phys. 43(3), 236–242 (2019)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordan and Breach Science Publisher, New York (1993)
Sidi Ammi, M.R., Jamiai, I., Torres, D.F.M.: Global existence of solutions for a fractional Caputo nonlocal thermistor problem. Adv. Differ. Equ. 2017, 363 (2017). arXiv:1711.00143
Sidi Ammi, M.R., Torres, D.F.M.: Existence of positive solutions for non local p-Laplacian thermistor problems on time scales. JIPAM. J. Inequal. Pure Appl. Math. 8(3), Article IDÂ 69 (2007). arXiv:0709.0415
Sidi Ammi, M.R., Torres, D.F.M.: Existence and uniqueness results for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary time scales. J. King Saud Univ., Sci. 30(3), 381–385 (2018). arXiv:1703.05439
Sidi Ammi, M.R., Torres, D.F.M.: Existence of solution to a nonlocal conformable fractional thermistor problem. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(1), 1061–1072 (2019)
Sidi Ammi, M.R., Torres, D.F.M.: Analysis of fractional integro-differential equations of thermistor type. In: Kochubei, A., Luchko, Y. (eds.) Basic Theory, pp. 327–346. de Gruyter, Berlin (2019). https://doi.org/10.1515/9783110571622-013
Sidi Ammi, M.R., Torres, D.F.M.: Existence of solution to a nonlocal conformable fractional thermistor problem. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(1), 1061–1072 (2019)
Vivek, D., Kanagarajan, K., Sivasundaram, S.: Dynamics and stability results for Hilfer fractional type thermistor problem. Fractal Fract. 1, 5 (2017)
Wang, Y., Zhou, J., Li, Y.: Fractional Sobolev’s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales. Adv. Math. Phys. 2016, Article ID 9636491 (2016)
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions. Praveen Agarwal was very thankful to the SERB (project TAR/2018/000001), DST(project DST/INT/DAAD/P-21/2019, and INT/RUS/RFBR/308), and NBHM (DAE)(project 02011/12/2020 NBHM(R.P)/R&D II/7867) for their necessary support.
Funding
Jihad Asad is thankful to Palestine Technical University- PTUK, Tulkarm, Palestine for their financial support.
Author information
Authors and Affiliations
Contributions
We confirm that all authors contributed equally. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Agarwal, P., Sidi Ammi, M.R. & Asad, J. Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense. Adv Differ Equ 2021, 162 (2021). https://doi.org/10.1186/s13662-021-03319-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03319-7
MSC
- 34N05
- 26A33
- 34A12
- 47H10
Keywords
- Time scale calculus
- Conformable fractional derivative
- Existence and uniqueness
- Fixed point theorem