On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

Saima Rashid; Fahd Jarad; Khadijah M. Abualnaja; Saima Rashid; Fahd Jarad; Khadijah M. Abualnaja

doi:10.3934/math.2021635

AIMS Mathematics

2021, Volume 6, Issue 10: 10920-10946. doi: 10.3934/math.2021635

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On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

1.
Department of Mathematics, Government College University, Faisalabad, Pakistan
2.
Department of Mathematics, Çankaya University, Ankara, Turkey
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
4.
Department of Mathematics, Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 27 February 2021 Accepted: 12 July 2021 Published: 28 July 2021
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.
Citation: Saima Rashid, Fahd Jarad, Khadijah M. Abualnaja. On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative[J]. AIMS Mathematics, 2021, 6(10): 10920-10946. doi: 10.3934/math.2021635

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Abstract

This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.

References

[1]	A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination, Chaos Soliton. Fract., 136 (2020), 109860. doi: 10.1016/j.chaos.2020.109860
[2]	J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput, 316 (2018), 504–515.
[3]	J. Danane, K. Allali, Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Soliton. Fract., 136 (2020), 109787. doi: 10.1016/j.chaos.2020.109787
[4]	Y. M. Chu, S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, More new results on integral inequalities for generalized K-fractional conformable integral operators, DCDS-S, 14 (2021), 2119–2135. doi: 10.3934/dcdss.2021063
[5]	S. S. Zhou, S. Rashid, A. Rauf. F. Jarad, Y. S. Hamed, K. M. Abualnaja, Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Mathematics, 6 (2021), 8001–8029. doi: 10.3934/math.2021465
[6]	S. Rashid, S. Sultana, F. Jarad, H. Jafari, Y. S. Hamed, More efficient estimates via h-discrete fractional calculus theory and applications, Chaos Soliton. Fract., 147 (2021), 110981. doi: 10.1016/j.chaos.2021.110981
[7]	H. G. Jile, S. Rashid, F. B. Farooq, S. Sultana, Some inequalities for a new class of convex functions with applications via local fractional integral, J. Funct. Space., 2021 (2021), 6663971.
[8]	S. Rashid, S. Parveen, H. Ahmad, Y. M. Chu, New quantum integral inequalities for some new classes of generalized $\psi$-convex functions and their scope in physical systems, Open Phy., 19 (2021), 35–50.
[9]	A. A. El-Deeb, S. Rashid, On some new double dynamic inequalities associated with Leibniz integral rule on time scales, Adv. Differ. Equ., 2021 (2021), 125. doi: 10.1186/s13662-021-03282-3
[10]	S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Mathematics, 6 (2021), 4507–4525. doi: 10.3934/math.2021267
[11]	J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504–515.
[12]	K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Pheno., 13 (2018), DOI: 10.1051/mmnp/2018006.
[13]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus models and numerical methods, 2 Eds., Singapore: World Scientific, 2012.
[14]	S. B. Chen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, A new approach on fractional calculus and probability density function, AIMS Mathematics, 5 (2020), 7041–7054. doi: 10.3934/math.2020451
[15]	M. A. Qurashi, S. Rashid, S. Sultana, H. Ahmad, K. A. Gepreel, New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel, Math. Biosci. Eng., 18 (2021), 1794–1812. doi: 10.3934/mbe.2021093
[16]	Y. M. Chu, S. Rashid, J. Singh, A novel comprehensive analysis on generalized harmonically $\Psi$-convex with respect to Raina's function on fractal set with applications, Math. Method. Appl. Sci., 2021, DOI: 10.1002/mma.7346.
[17]	S. Rashid, F. Jarad, Z. Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, DCDS-S, 14 (2021), 3703–3718.
[18]	S. Rashid, S. I. Butt, S. Kanwal, H. Ahmad, M. K. Wang, Quantum integral inequalities with respect to Raina's function via coordinated generalized $\psi$-convex functions with applications, J. Funct. Space., 2021 (2021), 6631474.
[19]	S. Rashid, Y. M. Chu, J. Singh, D. Kumar, A unifying computational framework for novel estimates involving discrete fractional calculus approaches, Alex. Eng. J., 60 (2021), 2677–2685. doi: 10.1016/j.aej.2021.01.003
[20]	M. A. Qurashi, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu, Y. M. Chu, Achieving more precse bounds based on double and triple integral as proposed by generalized proportional fractional operators in the Hilfer sense, Fractals, 2021, 2140027.
[21]	M. K. Wang, S. Rashid, Y. Karaca, D. Baleanu, Y. M. Chu, New multi-functional approach for kth-order differentiability governed by fractional calculus via approximately generalized $(\psi, \hbar)$-convex functions in Hilbert space, Fractals, 2021, 2140019.
[22]	M. A. Qurashi, S. Rashid, A. Khalid, Y. Karaca, Y. M. Chu, New computations of ostrowski type inequality pertaining to fractal style with applications, Fractals, 2021, 2140026.
[23]	F. Jarad, T. Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457–3471. doi: 10.1140/epjst/e2018-00021-7
[24]	I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Differ. Equ., 2020 (2020), 329. doi: 10.1186/s13662-020-02792-w
[25]	C. J. Rozier, The one-dimensional heat equation, Cambridge University Press, 1984.
[26]	R. Ellahi, C. Fetecau, M. Sheikholeslami, Recent advances in the application of differential equations in mechanical engineering problems, Math. Probl. Eng., 2018 (2018), 1584920.
[27]	Y. M. Chu, Solution of differential equations with applications to engineering problems, Dynam. Syst.: Anal. Comput. Tech., 2017,233.
[28]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353.
[29]	S. S. L. Chang, L. Zadeh, On fuzzy mapping and control, IEEE T. Syst. Man Cy., 2 (1972), 30–34.
[30]	R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal.-Theor., 72 (2010), 2859–2862. doi: 10.1016/j.na.2009.11.029
[31]	M. Z. Ahmad, M. K. Hasan, B. De Baets, Analytical and numerical solutions of fuzzy differential equations, Inform. Sci., 236 (2013), 156–167. doi: 10.1016/j.ins.2013.02.026
[32]	Z. Alijani, D. Baleanu, B. Shiri, G. C. Wu, Spline collocation methods for systems of fuzzy fractional differential equations, Chaos Soliton. Fract., 131 (2020), 109510. doi: 10.1016/j.chaos.2019.109510
[33]	O. A. Arqub, M. AL-Smadi, S. Momani, T. Hayat, Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Soft Comput., 20 (2016), 3283–3302. doi: 10.1007/s00500-015-1707-4
[34]	O. A. Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput. Applic., 28 (2017), 1591–1610. doi: 10.1007/s00521-015-2110-x
[35]	O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52. doi: 10.5373/jaram.1447.051912
[36]	R. P. Agarwal, S. Arshad, D. O'Regan, V. Lupulescu, Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal., 15 (2012), 572–590. doi: 10.2478/s13540-012-0040-1
[37]	R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. M. Torres, Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math., 339 (2018), 3–29. doi: 10.1016/j.cam.2017.09.039
[38]	B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Set. Syst., 151 (2005), 581–599. doi: 10.1016/j.fss.2004.08.001
[39]	B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Set. Syst., 230 (2013), 119–141. doi: 10.1016/j.fss.2012.10.003
[40]	O. S. Fard, M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math., 271 (2014), 71–82. doi: 10.1016/j.cam.2014.03.019
[41]	N. V. Hoa, Existence results for extremal solutions of interval fractional functional integro-differential equations, Fuzzy Set. Syst., 347 (2018), 29–53. doi: 10.1016/j.fss.2017.09.006
[42]	N. V. Hoa, On the initial value problem for fuzzy differential equations of non-integer order $\alpha\in(1, 2)$, Soft Comput., 24 (2020), 935–954. doi: 10.1007/s00500-019-04619-7
[43]	N. V. Hoa, V. Lupulescu, D. O'Regan, A note on initial value problems for fractional fuzzy differential equations, Fuzzy Set. Syst., 347 (2018), 54–69. doi: 10.1016/j.fss.2017.10.002
[44]	T. Allahviranloo. A. Armand, Z. Gouyandeh, H. Ghadiri, Existence and uniqueness of solutions for fuzzy fractional Volterra-Fredholm integro-differential equations, J. Fuzzy. Set. Val. Anal., 2013 (2013), 1–9.
[45]	M. S. Shagari, S. Rashid, K. M. Abualnaja, M. Alansari, On nonlinear fuzzy set-valued $\Theta$-contractions with applications, AIMS Mathematics, 6 (2021), 10431–10448. doi: 10.3934/math.2021605
[46]	M. Mazandarani, M. Najariyan, Type-2 fuzzy fractional derivatives, Commun. Nonlinear. Sci., 19 (2014), 2354–2372. doi: 10.1016/j.cnsns.2013.11.003
[47]	D. S. Oliveira, E. C. de Oliveira, Hilfer-Katugampola fractional derivative, Comput. Appl. Math., 37 (2018), 3672–3690. doi: 10.1007/s40314-017-0536-8
[48]	L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Set. Syst., 161 (2010), 1564–1584. doi: 10.1016/j.fss.2009.06.009
[49]	L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal.-Theor., 71 (2009), 1311–1328. doi: 10.1016/j.na.2008.12.005
[50]	S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Anal.-Theor., 74 (2011), 85–93.
[51]	T. Allahviranloo, A. Armand, Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J. Intell. Fuzzy Syst., 26 (2014), 1481–1490. doi: 10.3233/IFS-130831
[52]	T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty, Soft Comput., 16 (2012), 297–302. doi: 10.1007/s00500-011-0743-y
[53]	N. V. Hoa, H. Vu, T. M. Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Set. Syst., 375 (2019), 70–99. doi: 10.1016/j.fss.2018.08.001
[54]	V. Lakshmikantham, R. N. Mohapatra, Theory of fuzzy differential equations and applications, London: CRC Press, 2003.

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