Some novel analysis of two different Caputo-type fractional-order boundary value problems

Year 2022, , 299 - 311, 30.09.2022

Zouaoui Bekrı , Vedat Suat Ertürk , Pushpendra Kumar , Venkatesan Govindaraj

https://doi.org/10.53006/rna.1114063

Cited By: 5

Abstract

Nowadays, a number of classical order results are being analyzed in
the sense of fractional derivatives. In this research work, we
discuss two different boundary value problems. In the first half of
the paper, we generalize an integer-order boundary value problem
into fractional-order and then we demonstrate the existence and
uniqueness of the solution subject to the Caputo fractional
derivative. First, we recall some results and then justify our main
results with the proofs of the given theorems. We conclude our
results by presenting an illustrative example. In the other half of
the paper, we extend the Banach's contraction theorem to prove the
existence and uniqueness of the solution to a sequential Caputo
fractional-order boundary value problem.

Keywords

Caputo fractional derivative, Existence and uniqueness, Boundary value problem

References

• [1] W.G. Kelley and A.C. Peterson, theory of differential equations, Springer, 2010.
• [2] P.B. Bailey, L.F.Shampine and P.E. Waltman, Nonlinear two-point boundaryvalue problem,Academic Press, 1968.
• [3] R.P. Agarwal and Donal O’Regan, An Introduction to Ordinary Differential Equations, Springer-Verlag, 2008.
• [4] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations Elsevier, 2006.
• [5] C.F.Li, X.N.Luo and Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers and Mathematics with Applications, 59(3),1363–1375, 2010.
• [6] S. Zhang, Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Analysis, 71, 2087–2093, 2009.
• [7] T. Trif, Existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis, Fractional Calculus and Applied Analysis 16 (3), 595-612, 2013.
• [8] Y. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Applied Mathematics Letters, 51, 48-54, 2016.
• [9] Z. Bekri, V.S. Erturk, & P. Kumar, On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order. International Journal of Modeling, Simulation, and Scientific Computing, 13(01), 2250011, (2022).
• [10] V.S. Erturk, A. Ali, K. Shah, P. Kumar, & T. Abdeljawad, Existence and stability results for nonlocal boundary value problems of fractional order. Boundary Value Problems, 2022(1), 1–15.
• [11] P. Kumar, V. Govindaraj, Z.A. Khan, Some novel mathematical results on the existence and uniqueness of generalized Caputo-type initial value problems with delay. AIMS Mathematics, 7(6), 10483–10494.
• [12] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [14] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
• [15] V.S. Erturk, A. Ahmadkhanlu, P. Kumar, & V. Govindaraj, Some novel mathematical analysis on a corneal shape model by using Caputo fractional derivative. Optik, 261, 169086,(2022).
• [16] V.S. Erturk, A.K. Alomari, P. Kumar, & M. Murillo-Arcila, Analytic Solution for the Strongly Nonlinear Multi-Order Fractional Version of a BVP Occurring in Chemical Reactor Theory. Discrete Dynamics in Nature and Society, 2022.
• [17] M. Klimek, Sequential fractional differential equations with Hadamard derivative, Commun. Nonl Sci. Numer. Simul. 16 (2011) 4689- 4697.
• [18] D. Baleanu, O.G. Mustafa, R.P. Agarwal, On Lp-solutions for a class of sequential fractional differential equations, Appl.Math.Comput. 218 (2011) 2074–2081.
• [19] C. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative, J. Math. Anal. Appl. 384 (2011), 211–231.
• [20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
• [21] A.S. Vatsala, B. Sambandham, Sequential Caputo versus Nonsequential Caputo Fractional Initial and Boundary Value Problems, Int J of Diff Equ, V 15, Number 2, pp. 531-546 (2020).
• [22] B. Ahmed, J.J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput and Math with Appl 64 (2012) 3046-3052.
• [23] N. Phuangthong, S.K. Ntouyas, J. Tariboon and K. Nonlaopon, Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions, Mathematics 2021, 9, 615.
• [24] J. Tariboon, A. Cuntavepanit, S.K. Ntouyas and W. Nithiarayaphaks, Separated Boundary Value Problems of Sequential Caputo and Hadamard Fractional Differential Equations, Hindawi J of Funct Spac, V 2018, Art ID 6974046, 8 p.
• [25] A. Tudorache and R. Luca, Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives, Symmetry 2021, 13, 1489.
• [26] Z. Baitiche, K. Guerbati, M. Benchohra, J. Henderson, Boundary Value Problems for Hybrid Caputo Sequential Fractional Differential Equations, Communi on Appl Nonl Analy, Vol 27(2020), N 4, 1–16.
• [27] R.A.C. Ferreira, Note on a uniqueness result for a two-point fractional boundary value problem, Applied Mathematics Letters 90 (2019) 75-78.
• [28] B. Ahmad, Sharp estimates for the unique solution of two-point fractional-order boundary value problems, Appl. Math. Lett. 65(2017) 77–82.
• [29] Y. Zhou, Yong, Basic theory of fractional differential equations, World Scientific, Publishing Co: Pte. Ltd, 2014.
• [30] K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, 2004, Springer, Berlin, 2010.