On the initial value problem for fractional Volterra integrodifferential equations with a Caputo–Fabrizio derivative

Nguyen Huy Tuan¹ ; Nguyen Anh Tuan² ; Donal O’Regan³ ; Vo Viet Tri²

¹ Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
² Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam.
³ School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 18.

Résumé

In this paper, a time-fractional integrodifferential equation with the Caputo–Fabrizio type derivative will be considered. The Banach fixed point theorem is the main tool used to extend the results of a recent paper of Tuan and Zhou [J. Comput. Appl. Math. 375 (2020) 112811]. In the case of a globally Lipschitz source terms, thanks to the Lp − Lq estimate method, we establish global in time well-posed results for mild solution. For the case of locally Lipschitz terms, we present existence and uniqueness results. Also, we show that our solution will blow up at a finite time. Finally, we present some numerical examples to illustrate the regularity and continuation of the solution based on the time variable.

Détail

DOI : 10.1051/mmnp/2021010

Bibliographie

[1] K. Aissania, M. Benchohrab, J.J. Nieto Controllability for impulsive fractional evolution inclusions with state-dependent delay Adv. Theory Nonlinear Anal. Appl 2019 18 34

[2] N. Al-Salti, E. Karimov, K. Sadarangani On a differential equation with Caputo–Fabrizio fractional derivative of order 1 β ≤ 2 and application to mass-spring-damper system Progr. Fract. Differ. Appl 2017 257 263

[3] B. Andrade, A. Viana Abstract Volterra integrodifferential equations with applications to parabolic models with memory Math. Ann 2017 1131 1175

[4] B. Andrade, A. Viana Integrodifferential equations with applications to a plate equation with memory Math. Nachr 2016 2159 2172

[5] D. Baleanu, S. Etemad, S. Rezapour A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions Bound. Value Probl 2020 64

[6] D. Baleanu, S. Etemad, S. Pourrazi, S. Rezapour On the new fractional hybrid boundary value problems with three-point integral hybrid conditions Adv. Differ. Equ 2019 473

[7] D. Baleanu, H. Mohammadi, S. Rezapour Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative Adv Differ Equ 2020 71

[8] D. Baleanu, A. Mousalou, S. Rezapour The extended fractional Caputo-Fabrizio derivative of order 0 ≤ σ 1 on Cℝ[0,1] and the existence of solutions for two higher-order series-type differential equations Adv. Differ. Equ 2018 255

[9] D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative Chaos, Solitons Fractals 2020 109705

[10] D. Baleanu, A Jajarmi, S.S. Sajjadi, J.H. Asad The fractional features of a harmonic oscillator with position-dependent mass Commun. Theor. Phys 2020 055002

[11] D. Baleanu, S. Rezapour, Z. Saberpour On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation Bound. Value Probl 2019 79

[12] D. Baleanu, S. Rezapour, H. Mohammadi Some existence results on nonlinear fractional differential equations Philos. Trans. R.Soc. Lond. Ser. A Math. Phys. Eng. Sci 2013 20120144

[13] E.G. Bazhlekova Subordination principle for fractional evolution equations Fract. Calc. Appl. Anal 2000 213 230

[14] Y. Cao, J. Yin, C. Wang Cauchy problems of semilinear pseudo-parabolic equations J. Differ. Equ 2009 4568 4590

[15] M. Caputo, M. Fabrizio applications of new time and spatial fractional derivatives with exponential kernels Progr. Fract. Differ. Appl 2016 1 11

[16] M. Caputo, M. Fabrizzio A new definition of fractional derivative without singular Kernel Prog. Fract. Differ. Appl 2015 73 85

[17] T. Caraballo, J. Real Attractors for 2D-Navier-Stokes models with delays J. Differ. Equ. 2004 271 297

[18] T. Caraballo, J. Real Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci 2003 3181 3194

[19] Y. Chen, H. Gao, M. Garrido–Atienza, B. Schmalfuß Pathwise solutions of SPDEs driven by Hö lder-continuous integrators with exponent larger than 1/2 and random dynamical systems Discr. Continu. Dyn. Syst. Series A 2014 79 98

[20] M. Conti, E. Marchini, V. Pata Reaction-diffusion with memory in the minimal state framework Trans. Amer. Math. Soc 2014 4969 4986

[21] M. Conti, E. Marchini, V. Pata A well posedness result for nonlinear viscoelastic equations with memory Nonlinear Anal 2014 206 216

[22] M. D’Abbicco The influence of a nonlinear memory on the damped wave equation Nonlinear Anal 2014 130 145

[23] J.E.L. Delgado, J.E.S. Perez, J.F.G. Aguilar, R.F.E. Jiménez A new fractional-order mask for image edge detection based on Caputo–Fabrizio fractional-order derivative without singular kernel Circuits Syst. Signal Process 2020 1419 1448

[24] M. Fabrizio, S. Polidoro asymptotic decay for some differential systems with fading memory Appl. Anal 2002 1245 1264

[25] E. Franc, D. Goufo Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation Math. Model. Anal 2016 188 198

[26] A. Jajarmi, D. Baleanu A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems Front. Phys 2020 220

[27] A. Jajarmi, D. Baleanu On the fractional optimal control problems with a general derivative operator Asian J. Control 2021 1062 1071

[28] T.H. Kaddoura, A. Hakema Sufficient conditions of non global solution for fractional damped wave equations with non-linear memory Adv. Theory Nonlinear Anal. Appl. 2018 224 237

[29] H. Khan, J.F.G. Aguilar, T. Abdeljawad, A. Khan Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation Fractals 2020 9

[30] A. Lunardi on the linear heat equation with fading memory SIAM J. Math. Anal 1990 1213 1224

[31] E. Meerschaert, E. Nane, P. Vellaisamy Fractional Cauchy problems on bounded domains Ann. Probab 2009 979 1007

[32] F. Mohammadi, L. Moradi, D. Baleanu, A Jajarmi A hybrid functions numerical scheme for fractional optimal control problems: application to non-analytic dynamical systems J. Vib. Control 2018 5030 5043

[33] D. Mozyrska, D.F.M. Torres, M. Wyrwas Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales Nonlinear Anal. Hybrid Syst 2019 168 176

[34] J.E. Muñoz Rivera, L.H. Fatori Smoothing effect and propagations of singularities for viscoelastic plates J. Math. Anal. Appl 1997 397 427

[35] A. Nabti Life span of blowing-up solutions to the Cauchy problem for a time-space fractional diffusion equation Comput. Math. Appl 2019 1302 1316

[36] S. Sajjadi, D. Baleanu, A. Jajarmi, H.M. Pirouz A new adaptive synchronization and hyperchaos control of a biological snap oscillator Chaos Solitons Fractals 2020 109919

[37] S. Samko, J.J. Trujillo On the existence of blow up solutions for a class of fractional differential equations Fract. Calc. Appl. Anal 2015 281 283

[38] N.H. Tuan, M. Hakimeh, S. Rezapour A mathematical model for COVID-19 transmission by using the Caputo fractional derivative Chaos Solitons Fractals 2020 110107

[39] N.H. Tuan, Z. Yong Well-posedness of an initial value problem for fractional diffusion equation with Caputo-Fabrizio derivative J. Comput. Appl. Math 2020 112811

[40] A. Viana Local well-posedness for a Lotka-Volterra system in Besov spaces Comput. Math. Appl 2015 667 674