Properties of Certain Volterra type ABC Fractional Integral Equations

Year 2022, Volume: 6 Issue: 3, 339 - 346, 30.09.2022

Deepak Pachpatte , Juan Nieto

https://doi.org/10.31197/atnaa.1061019

Cited By: 2

Abstract

In this paper we study existence, uniqueness and other properties of solutions of Volterra type ABC fractional integral equations. We have used Banach fixed point theorem with Bielecki type norm and Gronwall inequality in the frame of ABC fractional integral for proving our results.

Keywords

Volterra Equation, ABC fractional, Existence, fractional Gronwall inequality

Supporting Institution

Agencia Estatal de Investigacion (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER), project and by Xunta de Galicia

Project Number

PID2020-113275GB-I00; ED431C 2019/02.

References

• [1] O.A. Arqub and B. Maayah, Numerical solutions of integrodi?erential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator, Chaos Solitions Fractals. 117, (2018), 117-124.
• [2] A. Atagana and D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernal: Theory and applications to heat transfer model, Therm sci. 20(2), 2016, 763−769.
• [3] A. Atangana, J.F. Gomez-Aguilar, Numerical approximation of Riemann-Liouville de?nition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differ Equ. 2018;34(5):1502−23 .
• [4] D. Baleanu, A.M. Lopes, Handbook of Fractional Calculus with Applications, Applications in Engineering, Life and Social Sciences, De Gruyter., Vol. 7:2019.
• [5] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ Appl. 1(2015), 73−85.
• [6] Ding X.L., Daniel C.L., Nieto J.J., A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations Stochastic Analysis and Applications, 019, 37(6), pp. 1042−1056.
• [7] Djida J., Atangana A. and Area I., Numerical computation of a Fractional Derivative with nonlocal and non-singular kernal, Math. Model. Nat. Phenom. 12(3), 2017, 4−13.
• [8] A. Fernandez and D. Baleanu, The mean value theorem and Taylors theorem for fractional derivatives with Mittag-Le?er kernel, Advances in Difference Equations, 2018: 86.
• [9] R. Hilfer, Applications of fractional calculusin physics, World Scienti?c. 2000.
• [10] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary diferential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitions Fractals. 117(2018), 16−20.
• [11] U. Katugampola, New approach to generalized fractional derivatives, Bull. Math. Anal. Appl. (2014), 6(4), 16−20.
• [12] I. Koca and A. Atangana, Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order, Chaos Solitons Fractals. (2016),89:447−54.
• [13] K.D. Kucche and S.T. Sutar, Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative, Chaos, Solitons Fractals, Volume 143, 2021, 110556.
• [14] A. Kumar and D. Pandey, Existence of mild solutions of Atangana-Baleanu fractional differential equations with noninstaneous impulses and with non-local conditions, Chaos Solitions Fractals, 132 (2020), 109551
• [15] A. Kilbas, H. Srivastava and J. Trujillio, Theory and application of fractional differential equations, North Holland Math- ematics Studies, 2006.
• [16] M. MalikkaArjunan, A. Hamiaz and V. Kavitha, Existence results for Atangana-Baleanu fractional neutral integro- differential systems with infinite delay through sectorial operators, Chaos, Solitons Fractals. 149, 2021, 111042.
• [17] S. Mekki, J.J. Nieto, A. Ouahab, Stochastic version of Henry type Gronwalls inequality, Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2021, 24(02), 2150013.
• [18] K.M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional, derivative to model chaotic problems. Chaos Solitions Fractals. 2019;29(2):1−12.
• [19] K.M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional, derivative, Eur. Phys. J. Plus. 2018;133(1):1−13.
• [20] S.K. Panda, T. Abdeljawad and C. Ravichandran, Novel fixed point approach to Atangana-Baleanu fractional and Lp- Fredholm integral equations, Alexandria Engineering Journal. (2020) 59,1959−1970.
• [21] I. Petras, Handbook of Fractional Calculus with Applications, Applications in Control, De Gruyter.. Vol. 6:2019.
• [22] C. Ravichandran, K. Logeswari and F. Jarad, New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-di?erential equations, Chaos Solitions Fractals. 125(2019), 194−200.
• [23] M. Syam and M. Al-Refai, Fractional differential equations with Atangana-Baleanu fractional drivative: Analysis and Applications, Chaos Solitions Fractals:X. (2019), 100013
• [24] S. Ucar, E. Ucar, N. Ozdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals, (2019), 118:300−6 .
• [25] Vanterler da J., C. Sousa and E.C. De Oliveira, A Gronwall inequality and the Cauchy type problem by means of ψ-Hilfer Operator, Differ. Equ. Appl., 11(1), (2019), 87−106.
• [26] H. Vu, B. Ghanbari and N.V. Hoa, Fuzzy fractional differential equations with the generalized Atangana-Baleanu fractional derivative, Fuzzy Sets and Systems , 2020
• [27] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to fractional differential equation, J. Math. Anal. Appl., 328(2), (2007), 1075−1081.