On Existence and Uniqueness of Some Fractional Order Integro-Differential Equation

Year 2023, Volume: 16 Issue: 2, 297 - 310, 31.08.2023

Lale Cona Esmahan Bal

https://doi.org/10.18185/erzifbed.1220243

Abstract

In this study, a sufficient condition for the existence and uniqueness of some fractional order Integral-Differential equations is obtained. Therefore, the fixed point method is used to solve the differential equation problem involving nonlinear degree integrals. In addition, the results found is supported by examples.

Keywords

Initial value problem, Fractional order Integro-Differential equation, Riemann-Liouville derivative, Riemann-Liouville integral, Fixed point.

References

• [1] Petras, I. (2011). Fractional-order nonlinear systems modeling, Analysis and Simulation, Springer. New York.
• [2] Ashyralyev A. (2009). A note on fractional derivatives and fractional powers of operators. JMAA. 357; 232-236.
• [3] Ashyralyev A. (2013). Well-posedness of fractional parabolic equations, Boundary Value Problems. 31; 1-18.
• [4] Ashyralyev A. and Cakir, Z. (2013). FDM for fractional parabolic equations with the Neumann condition. Advances in Difference Equations, 2013(120), doi:10.1186/1687-1847-2013-120.
• [5] Ashyralyev A., Emirov N. and Cakir Z. (2014). Well-posedness of fractional parabolic differential and difference equations with Dirchlet-Neumann condition, Electronic Journal of Differential Equations. 2014 (97); 1-17.
• [6] Podlubny, I. (1999). Fractional differential equations. Mathematics in science and engineering, vol 198. Academic Press. San Diego.
• [7] Samko S.G., Kilbas A.A., Marichev O.I. (1993). Fractional Integrals and Derivatives. Gordon & Breach. Yverdon.
• [8] Baleanu D, Garra, R, Petras, I. (2013). A fractional variational approach to the fractional Basset-Type equation, Reports on Mathematical Physics. 72 (1); 57-64.
• [9] Diethelm, K. and Ford, J.N. (2002). Analysis of fractional differential equations. JMAA. 265; 229-248.
• [10] Cona, L. (2017). Fixed point approach to Basset problem. New Trends in Mathematical Sciences. 5(3); 175-181.
• [11] Cona, L. (2017). Fixed point approach to Bagley Torvik problem. Communication in Mathematical Modeling and Applications. 2(3), 50--57.
• [12] Bai, Z., Sun, S. and Chen, Y.Q. (2014).The existence and uniqueness of a class of fractional differential equations. Abstract and Applied Analysis, Hindawi Publishing Corporation.2014;1-6.
• [13] Ashyralyev A. (2013). Computational mathematics textbook, Shymkent.
• [14] Kimeu, Joseph M. (2009). Fractional calculus: Definitions and applications. Masters Theses & Specialist Projects. Paper 115. (https://digitalcommons.wku.edu/theses/115)
• [15] Kreyszig, E. (1978). Introductory functional analysis with applications, John Willey & Sons, New York.
• [16] Caputo, M., Fabrizo, M. (2015). A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2), 73-85.
• [17] Atangana, A. Baleanu, D. (2016). New fractional derivative with non-local and non-singular kernel, Therm. Sci., 20(2), 757-763.
• [18] Yang, X.J., Sirivasta, H.M., Machado, J.A.T. (2016). A new fractional derivatives without singular kernel: application to the modelling of the steady heat flow, Therm. Sci., 20, 753-756.
• [19] Vanterler da Sausa, J., Capelas de Oliveira C., E. (2017). A new fractional derivative of variable order with non-singular order and fractional differential equations, arXiv: 1712.06506v1.
• [20] Vanterler da Sausa, J., Capelas de Oliveira C., E. (2020). On the Ψ-Hilfer fractional derivative, Commun Nonlinear Sci. Numer, Simul. 60, 72-91.