A class of nonlinear systems with new boundary conditions: existence of solutions, stability and travelling waves
Abstract
In this work, we begin by introducing a new notion of coupled closed fractional boundary conditions to study a class of nonlinear sequential systems of Caputo fractional differential equations. The existence and uniqueness of solutions for the class of systems is proved by applying Banach contraction principle. The existence of at least one solution is then accomplished by applying Schauder fixed point theorem. The Ulam Hyers stability, with a limiting-case example, is also discussed. In a second part of our work, we use the tanh method to obtain a new travelling wave solution for the coupled system of Burgers using time and space Khalil derivatives. By bridging these two aspects, we aim to present an understanding of the system’s behaviour.
Keyword : Caputo sequential derivative, coupled system of FDEs, existence and uniqueness, Ulam-Hyers stability, fixed point theorem, travelling wave, Khalil derivative
How to Cite
Lamamri, A., Gouari, Y., Dahmani, Z., Rakah, M., & Sarıkaya, M. Z. (2025). A class of nonlinear systems with new boundary conditions: existence of solutions, stability and travelling waves. Mathematical Modelling and Analysis, 30(2), 233–253. https://doi.org/10.3846/mma.2025.20920
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Z. Dahmani, M.A. Abdellaoui and M. Houas. Coupled systems of fractional integro-differential equations involving several functions. Theory and Applications of Mathematics & Computer Science, 5(1):53–61, 2015.
Z. Dahmani, A. Anber and I. Jebril. Solving conformable evolution equations by an extended numerical method. Jordan Journal of Mathematics and Statistics (JJMS), 15(2):363–380, 2022. https://doi.org/10.47013/15.2.14
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R. Khalil, M. Al Horani, A. Yousef and M. Sababheh. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264:65– 70, 2014. https://doi.org/10.1016/j.cam.2014.01.002
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J. Sugie. Interval criteria for oscillation of second-order self-adjoint impulsive differential equations. Proceedings of the American Mathematical Society, 148:1095–1108, 2020. https://doi.org/10.1090/proc/14797
S.T.M. Thabet, T. Abdeljawad, I. Kedim and M. Iadh Ayari. A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law. Boundary Value Problems, 2023(100):1–16, 2023. https://doi.org/10.1186/s13661-023-01790-7
S.T.M. Thabet, S. Al-Sadi, I. Kedim, A.S. Rafeeq and S. Rezapour. Analysis study on multi-order ϱ-Hilfer fractional pantograph implicit differential equation on unbounded domains. AIMS Mathematics, 8(8):18455–18473, 2023. https://doi.org/10.3934/math.2023938
S.T.M. Thabet, M. Vivas-Cortez and I. Kedim. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 8(10):23635–23654, 2023. https://doi.org/10.3934/math.20231202
S.T.M. Thabet, M. Vivas-Cortez, I. Kedim, M.E. Samei and M.I. Ayari. Solvability of a ρ Hilfer fractional snap dynamic system on unbounded domains. Fractal and Fractional, 7(8):607, 2023. https://doi.org/10.3390/fractalfract7080607
A. Tudorache and R. Luca. On a system of sequential Caputo fractional differential equations with nonlocal boundary conditions. Fractal and Fractional, 7(2):1–23, 2023. https://doi.org/10.3390/fractalfract7020181
H.K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education Limited, Harlow, UK, Harlow, UK, 2nd edition, 2007.
A.-M. Wazwaz. The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the k(n,n)-Burger equations. Physica D: Nonlinear Phenomena, 213(2):147–151, 2006. https://doi.org/10.1016/j.physd.2005.09.018
S.-W. Yao, T. Rasool, R. Hussain, H. Rezazadeh and M. Inc. Exact soliton solutions of conformable fractional coupled Burger’s equation using hyperbolic funtion approach. Results in Physics, 30:104776, 2021. https://doi.org/10.1016/j.rinp.2021.104776
M. Younis, A. Zafar, K. Ul-Haq and M. Rahman. Travelling wave solutions of fractional order coupled Burgers’ equation by g/g′-expansion method. American Journal of Computational and Applied Mathematics, 3(2):81–85, 2013. https://doi.org/10.5923/j.ajcam.20130302.04
S.B. Yuste, E. Abad and K. Lindenberg. Reactions in subdiffusive media and associated fractional equations in fractional dynamics. Fractional Dynamics, pp. 77–106, 2011. https://doi.org/10.1142/9789814340595_0004
B. Ahmad, M. Alnahdi and S.K. Ntouyas. Existence results for a differential equation involving the right Caputo fractional derivative and mixed nonlinearities with nonlocal closed boundary conditions. Fractal and Fractional, 7(2):129, 2023. https://doi.org/10.3390/fractalfract7020129
B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas. Second-order ordinary differential equations and inclusions with a new kind of integral and multistrip boundary conditions. Differential Equations & Applications, 11(2):183–202, 2019. https://doi.org/10.7153/dea-2019-11-07
A. Alsaedi, M. Alnahdi, B. Ahmad and S. K. Ntouyas. On a nonlinear coupled Caputo-type fractional differential system with coupled closed boundary conditions. AIMS Mathematics, 8(8):17981–17995, 2023. https://doi.org/10.3934/math.2023914
A. Anber, I. Jebril, Z. Dahmani, N. Bedjaoui and A. Lamamri. The tanh method and the (g′/g)-expansion method for solving the space-time conformable FZK and FZZ evolution equations. International Journal of Innovative Computing, Information and Control, 20(2):557–573, 2024. https://doi.org/10.24507/ijicic.20.02.557
M.I. Ayari and S.T.M. Thabet. Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator. Arab Journal of Mathematical Sciences, 30(2):197–217, 2024. https://doi.org/10.1108/AJMS-06-2022-0147
Z. Dahmani, M.A. Abdellaoui and M. Houas. Coupled systems of fractional integro-differential equations involving several functions. Theory and Applications of Mathematics & Computer Science, 5(1):53–61, 2015.
Z. Dahmani, A. Anber and I. Jebril. Solving conformable evolution equations by an extended numerical method. Jordan Journal of Mathematics and Statistics (JJMS), 15(2):363–380, 2022. https://doi.org/10.47013/15.2.14
J.M. Fogac¸a and E.L. Cardoso. A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 46(4):197, 2024. https://doi.org/10.1007/s40430-024-04756-7
V. Gafiychuk, B. Datsko and V. Meleshko. Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics, 220(1-2):215–225, 2008. https://doi.org/10.1016/j.cam.2007.08.011
R. Khalil, M. Al Horani, A. Yousef and M. Sababheh. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264:65– 70, 2014. https://doi.org/10.1016/j.cam.2014.01.002
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Amsterdam, Amsterdam, 2006.
A. Lamamri, I. Jebril, Z. Dahmani, A. Anber, M. Rakah and S. Alkhazaleh. Fractional calculus in beam deflection: Analyzing nonlinear systems with Caputo and conformable derivatives. AIMS Mathematics, 9(8):21609–21627, 2024. https://doi.org/10.3934/math.20241050
W. Malfliet and W. Hareman. The tanh method: I. exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54(6):563–568, 1996. https://doi.org/10.1088/0031-8949/54/6/003
M. Mohammadimehr, S.V. Okhravi and S.M.A. Alavi. Free vibration analysis of magneto-electro-elastic cylindrical composite panel reinforced by various distributions of CNTs with considering open and closed circuits boundary conditions based on FSDT. Journal of Vibration and Control, 24(8):1551–1569, 2018. https://doi.org/10.1177/1077546316664022
W. Rudin. Functional Analysis. McGraw-Hill, Springer-Verlag, New York, 1973.
J. Sabatier, P. Lanusse, P. Melchior and A. Oustaloup. Fractional order differentiation and robust control design, volume 77 of Intelligent Systems, Control and Automation: Science and Engineering. Springer, 2015. https://doi.org/10.1007/978-94-017-9807-5
J. Sugie. Interval criteria for oscillation of second-order self-adjoint impulsive differential equations. Proceedings of the American Mathematical Society, 148:1095–1108, 2020. https://doi.org/10.1090/proc/14797
S.T.M. Thabet, T. Abdeljawad, I. Kedim and M. Iadh Ayari. A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law. Boundary Value Problems, 2023(100):1–16, 2023. https://doi.org/10.1186/s13661-023-01790-7
S.T.M. Thabet, S. Al-Sadi, I. Kedim, A.S. Rafeeq and S. Rezapour. Analysis study on multi-order ϱ-Hilfer fractional pantograph implicit differential equation on unbounded domains. AIMS Mathematics, 8(8):18455–18473, 2023. https://doi.org/10.3934/math.2023938
S.T.M. Thabet, M. Vivas-Cortez and I. Kedim. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 8(10):23635–23654, 2023. https://doi.org/10.3934/math.20231202
S.T.M. Thabet, M. Vivas-Cortez, I. Kedim, M.E. Samei and M.I. Ayari. Solvability of a ρ Hilfer fractional snap dynamic system on unbounded domains. Fractal and Fractional, 7(8):607, 2023. https://doi.org/10.3390/fractalfract7080607
A. Tudorache and R. Luca. On a system of sequential Caputo fractional differential equations with nonlocal boundary conditions. Fractal and Fractional, 7(2):1–23, 2023. https://doi.org/10.3390/fractalfract7020181
H.K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education Limited, Harlow, UK, Harlow, UK, 2nd edition, 2007.
A.-M. Wazwaz. The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the k(n,n)-Burger equations. Physica D: Nonlinear Phenomena, 213(2):147–151, 2006. https://doi.org/10.1016/j.physd.2005.09.018
S.-W. Yao, T. Rasool, R. Hussain, H. Rezazadeh and M. Inc. Exact soliton solutions of conformable fractional coupled Burger’s equation using hyperbolic funtion approach. Results in Physics, 30:104776, 2021. https://doi.org/10.1016/j.rinp.2021.104776
M. Younis, A. Zafar, K. Ul-Haq and M. Rahman. Travelling wave solutions of fractional order coupled Burgers’ equation by g/g′-expansion method. American Journal of Computational and Applied Mathematics, 3(2):81–85, 2013. https://doi.org/10.5923/j.ajcam.20130302.04
S.B. Yuste, E. Abad and K. Lindenberg. Reactions in subdiffusive media and associated fractional equations in fractional dynamics. Fractional Dynamics, pp. 77–106, 2011. https://doi.org/10.1142/9789814340595_0004