Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative

• Autorzy: Mansouri Ikram , Bekkouche Mohammed Moumen , Ahmed Abdelaziz Azeb

Identyfikatory

DOI

10.17512/jamcm.2023.1.04

• Języki publikacji: EN

Abstrakty

EN

Within this work, we discuss the existence of solutions for a coupled system of linear fractional differential equations involving Caputo-Fabrizio fractional orders. We prove the existence and uniqueness of the solution by using the Picard-Lindel ̈of method and fixed point theory. Also, to compute an approximate solution of problem, we utilize the Adomian decomposition method (ADM), as this method provides the solution in the form of a series such that the infinite series converge to the exact solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Słowa kluczowe

EN

Caputo-Fabrizio fractional derivative; fractional integral; coupled system; fractional differential equation; fixed point; Adomian decomposition method

PL

pochodna ułamkowa Caputo-Fabrizio; całkowanie ułamkowe; system sprzężony; ułamkowe równanie różniczkowe; punkt stały; metoda dekompozycji Adomiana

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2023
• Tom: Vol. 22, nr 1
• Strony: 46--56
• Opis fizyczny: Bibliogr. 30 poz., rys.

Twórcy

autor

Mansouri Ikram

• Faculty of Exact Sciences – Department of Mathematics El oued University – El oued 39000, Algeria
• Laboratory of Operator Theory and PDE: Foundations and Applications

autor

Bekkouche Mohammed Moumen

• Faculty of Exact Sciences – Department of Mathematics El oued University – El oued 39000, Algeria

autor

Ahmed Abdelaziz Azeb

• Faculty of Exact Sciences – Department of Mathematics El oued University – El oued 39000, Algeria

Bibliografia

• [1] Sabatier, J., Lanusse, P., Melchior, P., & Oustaloup, A. (2015). Fractional order differentiation and robust control design. Intelligent Systems, Control and Automation: Science and Engineering 77, Springer, 13-18.
• [2] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and applications of fractional differential equations, ser. North-Holland Mathematics Studies. Amsterdam: Elsevier, 204.
• [3] Yuste, S.B., Abad, E., & Lindenberg, K. (2011). Reactions in Subdiffusive Media and Associated Fractional Equations in Fractional Dynamics. Recent Advances. J. Klafter, SC Lim, and R. Metzler.
• [4] Yang, X.J., Gao, F., & Yang, J. (2020). General Fractional Derivatives with Applications in Viscoelasticity. Academic Press.
• [5] Tarasova, V.V., & Tarasov, V. E. (2017). Logistic map with memory from economic model. Chaos, Solitons & Fractals, 95, 84-91.
• [6] Hilfer, R. (2000). Fractional diffusion based on Riemann-Liouville fractional derivatives.The Journal of Physical Chemistry B, 104(16), 3914-3917.
• [7] Iskenderoglu, G., & Kaya, D. (2020). Symmetry analysis of initial and boundary value problems for fractional differential equations in Caputo sense. Chaos, Solitons & Fractals, 134, 109684.
• [8] Mehandiratta, V., Mehra, M., & Leugering, G. (2019). Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph. Journal of Mathematical Analysis and Applications, 477(2), 1243-1264. DOI: 10.1016/j.jmaa.2019.05.011.
• [9] Verma, P., & Kumar, M. (2022). Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative. Engineering with Computers, 1-18. DOI: 10.1007/s00366-020-01067-4.
• [10] Mahmudov, N.I., & Al-Khateeb, A. (2019). Stability, existence and uniqueness of boundary value problems for a coupled system of fractional differential equations. Mathematics, 7(4), 354.
• [11] Zhang, H., & Gao, W. (2014). Existence and uniqueness results for a coupled system of nonlinear fractional differential equations with antiperiodic boundary conditions. In Abstract and Applied Analysis (Vol. 2014). Hindawi. DOI: 10.1155/2014/463517.
• [12] Ahmad, B., Alghanmi, M., Alsaedi, A., & Nieto, J.J. (2021). Existence and uniqueness results for a nonlinear coupled system involving Caputo fractional derivatives with a new kind of coupled boundary conditions. Applied Mathematics Letters, 116, 107018. DOI: 10.1016/j.aml.2021.107018.
• [13] Nisar, K.S., Rahman, M.U., Laouini, G., Shutaywi, M., & Arfan, M. (2022). On nonlinear fractional-order mathematical model of food-chain. Fractals, 30(01), 2240014. DOI: 10.1142/S0218348X2240014X.
• [14] Odibat, Z., & Momani, S. (2006). Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), 27-34.
• [15] Li, B., Liang, H., & He, Q. (2021). Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model. Chaos, Solitons & Fractals, 146, 110856. DOI: 10.1016/j.chaos.2021.110856.
• [16] Sousa, E., & Li, C. (2015). A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative. Applied Numerical Mathematics, 90, 22-37.
• [17] Adomian, G. (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(2), 501-544.
• [18] Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation & Applications, 1(2), 73-85.
• [19] Gul, R., Sarwar, M., Shah, K., Abdeljawad, T., & Jarad, F. (2020). Qualitative analysis of implicite Dirichlet boundary value problem for Caputo-Fabrizio fractional differentia equations. Journal of Function Spaces, 2020, 1-9.
• [20] Bekkouche, M.M., & Guebbai, H. (2020). Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral. Journal of Applied Mathematics and Computational Mechanics, 19(2), 31-42.
• [21] Marasi, H.R., Joujehi, A.S., & Aydi, H. (2021). An Extension of the Picard theorem to fractional differential equations with a Caputo-Fabrizio derivative. Journal of Function Spaces, 2021, 1-6. DOI: 10.1155/2021/6624861.
• [22] Eskandari, Z., Avazzadeh, Z., Khoshsiar Ghaziani, R., & Li, B. (2022). Dynamics and bifurcations of a discrete-time Lotka-Volterra model using nonstandard finite difference discretization method. Mathematical Methods in the Applied Sciences. DOI: 10.1002/mma.8859.
• [23] Kongson, J., Thaiprayoon, C., Neamvonk, A., Alzabut, J., & Sudsutad, W. (2022). Investigation of fractal-fractional HIV infection by evaluating the drug therapy effect in the Atangana-Baleanu sense. Mathematical Biosciences and Engineering, 19, 10762-10808. DOI: 10.3934/mbe.2022504.
• [24] Li, B., Liang, H., Shi, L., & He, Q. (2022). Complex dynamics of Kopel model with non-symmetric response between oligopolists. Chaos, Solitons & Fractals, 156, 111860. DOI:10.1016/j.chaos.2022.111860.
• [25] Baleanu, D., Jajarmi, A., Mohammadi, H., & Rezapour, S. (2020). A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos, Solitons & Fractals, 134, 109705. DOI: 10.1016/j.chaos.2020.109705.
• [26] Boulaaras, S., Jan, R., Khan, A., & Ahsan, M. (2022). Dynamical analysis of the transmission of dengue fever via Caputo-Fabrizio fractional derivative. Chaos, Solitons & Fractals: X, 8, 100072. DOI: 10.1016/j.csfx.2022.100072.
• [27] Abbas, S., Benchohra, M., & Henderson, J. (2020). Coupled Caputo-Fabrizio fractional differential systems in generalized Banach spaces. Malaya Journal of Matematik (MJM), 9(1, 2021), 20-25. DOI: 10.26637/MJM0901/0003.
• [28] Alkahtani, B.S.T. (2023). Analytical study of the complexities in a three species food web model with modified Caputo-Fabrizio operator. Fractal and Fractional, 7(2), 105. DOI: 10.3390/fractalfract7020105.
• [29] Chasreechai, S., Sitthiwirattham, T., El-Shorbagy, M.A., Sohail, M., Ullah, U., & ur Rahman, M. (2022). Qualitative theory and approximate solution to a dynamical system under modified type Caputo-Fabrizio derivative. AIMS Mathematics, 7(8), 14376-14393. DOI: 10.3934/math.2022792.
• [30] Granas, A., & Dugundji, J. (2003). Fixed Point Theory (Vol. 14, pp. 15-16). New York: Springer.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).