Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

Abdon Atangana; Ilknur Koca

doi:10.11121/ijocta.1639

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

Authors

Abdon Atangana Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, 9301, South Africa https://orcid.org/0000-0002-1886-3125
Ilknur Koca Department of Economics and Finance, Fethiye Business Faculty, Mugla Sitki Kocman University, Türkiye https://orcid.org/0000-0003-4393-1588

DOI:

https://doi.org/10.11121/ijocta.1639

Keywords:

Fractal calculus, Witte's conditions, Uniqueness, Nonlocal operators

Abstract

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

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Author Biographies

Abdon Atangana, Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, 9301, South Africa

Abdon Atangana is a professor at the University of the Free State, Bloemfontein, South Africa.

Ilknur Koca, Department of Economics and Finance, Fethiye Business Faculty, Mugla Sitki Kocman University, Türkiye

Ilknur Koca received the B.Sc. and M.Sc. degrees in 2007 and 2009 from the Department of Mathematics from Ankara University, Turkey respectively. She received her Ph.D. degree from the same university in 2013. She is working as an Professor at Department of Economics and Finance, Fethiye Business Faculty, Mugla Sitki Kocman University, Mugla, Türkiye. Her research interests are methods and applications of partial and ordinary differential equations,fractional differential equations, iterative methods.

References

Zill, D.G. (2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 978-1-285-40110-2.

Boyce, W. E., DiPrima, R. C. & Meade, D. B. (2017). Elementary Differential Equations. John Wiley & Sons.

Zaitsev, V. F. & Polyanin, A. D. (2002). Handbook of Exact Solutions for Ordinary Differential Equations. Chapman and Hall/CRC. https://doi.org/10.1201/9781420035339

Sher, M., Khan, A., Shah, K. & Abdeljawad, T. (2023). Existence and stability theory of pantograph conformable fractional differential problem. Thermal Science, 27(Spec. issue 1), 237-244. https://doi.org/10.2298/TSCI23S1237S

Shah, K., Abdeljawad, T. & Abdalla, B. (2023). On a coupled system under coupled integral boundary conditions involving non-singular differential operator. AIMS Mathematics, 8(4), 9890-9910. https://doi.org/10.3934/math.2023500

Koca, I. & Atangana, A. (2023). Theoretical and numerical analysis of a chaotic model with nonlocal and stochastic differential operators. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 13(2), 181– 192. https://doi.org/10.11121/ijocta.2023.1398

Sene, N. & Ndiaye, A. . (2024). Existence and uniqueness study for partial neutral functional fractional differential equation under Caputo derivative. An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 14(3), 208–219. https://doi.org/10.11121/ijocta.1464

Teschl, G. (2024). Ordinary Differential Equations and Dynamical Systems (Vol. 140). American Mathematical Society.

Agarwal, R. P., Agarwal, R. P., & Lakshmikantham, V. (1993). Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. (Vol. 6). World Scientific. https://doi.org/10.1142/1988

Atangana, A. (2017). Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons & Fractals, 102, 396-406. https://doi.org/10.1016/j.chaos.2017.04.027

Chen, W. (2006). Time-space fabric underlying anomalous diffusion. Chaos, Solitons & Fractals, 28(4), 923-929. https://doi.org/10.1016/j.chaos.2005.08.199

Metzler, R. & Klafter, J. (2000). The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1), 1- 77. https://doi.org/10.1016/S0370-1573(00)00070-3

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Published

2024-10-09

CITATION

DOI: 10.11121/ijocta.1639

Published: 2024-10-09

How to Cite

Atangana, A., & Koca, I. (2024). Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(4), 322–335. https://doi.org/10.11121/ijocta.1639

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Issue

Vol. 14 No. 4 (2024): IJOCTA

Section

Research Articles

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