Research article Special Issues

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

  • 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

    Related Papers:

  • 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.



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    [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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