A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

• Autorzy: Aljazzazi Mazin , Maayah Banan , Djeddi Nadir , Al-Smadi Mohammed , Momani Shaher

Identyfikatory

DOI

10.1515/dema-2022-0237

• Języki publikacji: EN

Abstrakty

EN

In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ3[a,b]. From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

Słowa kluczowe

EN

fractional Begley-Torvik equations; integral boundary condition; reproducing kernel Hilbert space method; numerical approximation

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20220237
• Opis fizyczny: Bibliogr. 45 poz., rys., tab.

Twórcy

autor

Aljazzazi Mazin

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan

autor

Maayah Banan

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan

autor

Djeddi Nadir

• Department of Mathematics and Computer Science, Larbi Tebessi University, Tebessa 12002, Algeria

autor

Al-Smadi Mohammed

• College of Commerce and Business, Lusail University, Lusail, Qatar; Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
• Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun, 26816, Jordan

autor

Momani Shaher

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan
• Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

Bibliografia

• [1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
• [2] P. Veeresha, D. G. Prakasha, and S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Methods Appl. Sci. (2020), 1–15, DOI: https://doi.org/10.1002/mma.6335.
• [3] S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alex. Eng. J. 52 (2013), no. 4, 813–819.
• [4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [5] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010, DOI: https://doi.org/10.1007/978-3-642-14574-2.
• [6] S. Abbas, M. Benchohra, J. E. Lazreg, J. J. Nieto, and Y. Zhou, Fractional Differential Equations and Inclusions, Classical and Advanced Topics, vol. 10, World Scientific, Hackensack, USA, 2023, p. 328, DOI: https://doi.org/10.1142/12993.
• [7] Y. Yan and S. Luo, Local polynomial smoother for solving Bagley-Torvik fractional differential equations, Preprints, (2016), 2016080231. DOI: https://doi.org/10.20944/preprints201608.0231.v1.
• [8] Y. Çenesiz, Y. Keskin, and A. Kurnaz, The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, J. Franklin Inst. 347 (2010), 452–466.
• [9] S. Yuzbasi, Numerical solution of the Bagley-Torvik equation by the Bessel collocation method, Math. Methods Appl. Sci. 36 (2013), 300–312.
• [10] S. Kumar, A. Kumar, B. Samet, J. F. Gomez-Aguilar, and M. S. Osman, A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment, Chaos Solitons Fractals 141 (2021), 110321, DOI: https://doi.org/10.1016/j.chaos.2020.110321.
• [11] A. Arikoglu and A. L. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals 34 (2017), 1473–1481.
• [12] H. Mohammadi, S. Kumar, S. Rezapour, and S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fractals 144 (2021), 110668, DOI: https://doi.org/10.1016/j.chaos.2021.110668.
• [13] F. Mohammadi and S. T. Mohyud-Din, A fractional-order Legendre collocation method for solving the Bagley-Torvik equations, Adv. Differential Equations 2016 (2016), 269, DOI: https://doi.org/10.1186/s13662-016-0989-x
• [14] S. Hasan, N. Djeddi, M. Al-Smadi, S. Al-Omari, S. Momani, and A. Fulga, Numerical solvability of generalized Bagley-Torvik fractional models under Caputo-Fabrizio derivative, Adv. Differential Equations 2021 (2021), 469, DOI: https://doi.org/10.1186/s13662-021-03628-x.
• [15] O. Abu Arqub and B. Maayah, Solutions of Bagley-Torvik and Painleve equations of fractional order using iterative reproducing kernel algorithm with error estimates, Neural Comput. Appl. 29 (2016), 1465–1479.
• [16] X. Zhong, X. Liu, and S. Liao, On a generalized Bagley-Torvik equation with a fractional integral boundary condition, Int. J. Appl. Comput. Math. 3 (2017), 727–746.
• [17] A. V. Bicadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk 185 (1969), 739–740.
• [18] J. Andres, A four-point boundary value problem for the second-order ordinary differential equations, Arch. Math. 53 (1989), 384–389.
• [19] P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions, Appl. Math. Lett. 18 (2005), 521–527.
• [20] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. Lond. Math. Soc. 74 (2006), 673–693.
• [21] H. Khalil, M. Al-Smadi, K. Moaddy, R. A. Khan, and I. Hashim, Toward the approximate solution for fractional order nonlinear mixed derivative and nonlocal boundary value problems, Discrete Dyn. Nat. Soc. 2016 (2016), 5601821, DOI: https://doi.org/10.1155/2016/5601821.
• [22] G. Gumah, M. F. M. Naser, M. Al-Smadi, S. K. Q. Al-Omari, and D. Baleanu, Numerical solutions of hybrid fuzzy differential equations in a Hilbert space, Appl. Numer. Math. 151 (2020), 402–412, DOI: https://doi.org/10.1016/j.apnum.2020.01.008.
• [23] M. Al-Smadi, Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation, Ain Shams Eng. J. 9 (2018), no. 4, 2517–2525, DOI: https://doi.org/10.1016/j.asej.2017.04.006.
• [24] M. Al-Smadi, O. Abu Arqub, and S. Momani, Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense, Phys. Scr. 95 (2020), no. 7, 075218, DOI: https://doi.org/10.1088/1402-4896/ab96e0.
• [25] N. Djeddi, S. Hasan, M. Al-Smadi, and S. Momani, Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative, Alexandr. Eng. J. 59 (2020), no. 6, 5111–5122, DOI: https://doi.org/10.1016/j.aej.2020.09.041.
• [26] Q. Ding and P. J. Y. Wong, A higher order numerical scheme for solving fractional Bagley-Torvik equation, Math. Methods Appl. Sci. 45 (2022), 1241–1258.
• [27] L. Shi, S. Tayebi, O. Abu Arqub, M. S. Osman, P. Agarwal, W. Mahamoud, et al., The novel cubic B-spline method for fractional Painlevé and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense. Alexandria Eng. J. 65 (2023), 413–426.
• [28] H. Fazli, H. Sun, S. Aghchi, and J. J. Nieto, On a class of nonlinear nonlocal fractional differential equations, Carpathian J. Math. 37 (2021), 441–448.
• [29] C. L. Li and M. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl. Math. Comput. 143 (2003), 393–399.
• [30] G. N. Gumah, M. F. M. Naser, M. Al-Smadi, and S. K. Al-Omari, Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations, Adv. Differential Equations 2018 (2018), 475.
• [31] M. Al-Smadi, H. Dutta, S. Hasan, and S. Momani, On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space, Math. Model. Nat. Phenom. 16 (2021), 41, DOI: https://doi.org/10.1051/mmnp/2021030.
• [32] C. Li and M. Cui, The exact solution for solving aclass nonlinear operator equations in the reproducing kernel space, Appl. Math. Comput. 143 (2003), 393–399.
• [33] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh, and S. Momani, Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method, Appl. Math. Comput. 291 (2016), 137–148.
• [34] M. Al-Smadi, S. Momani, N. Djeddi, A. El-Ajou, and Z. Al-Zhour, Adaptation of reproducing kernel method in solving Atangana-Baleanu fractional Bratu model, Int. J. Dyn. Control. 11 (2023), 136–148. DOI: https://doi.org/10.1007/s40435-022-00961-1.
• [35] X. Li and B. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math. 243 (2013), 10–15.
• [36] M. Al-Smadi, Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces, Filomat 33 (2019), no. 2, 583–597, DOI: https://doi.org/10.2298/FIL1902583A.
• [37] M. Al-Smadi, N, Djeddi, S. Momani, S. Al-Omari, and S. Araci, An attractive numerical algorithm for solving nonlinear Caputo-Fabrizio fractional Abel differential equation in a Hilbert space, Adv. Differential Equations 2021 (2021), 271, DOI: https://doi.org/10.1186/s13662-021-03428-3.
• [38] S. Momani, N. Djeddi, M. Al-ŘSmadi, and S. Al-Omari, Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method, Appl. Numer. Math. 170 (2021), 418–434.
• [39] S. Hasan, M. Al-Smadi, H. Dutta, S. Momani, and S. Hadid, Multi-step reproducing kernel algorithm for solving Caputo-Fabrizio fractional stiff models arising in electric circuits, Soft Computing 26 (2022), no. 2, 3713–3727, DOI: https://doi.org/10.1007/s00500-022-06885-4.
• [40] M. Al-Smadi and O. Abu Arqub, Computational algorithm for solving Fredholm time-fractional partial integro differential equations of Dirichlet functions type with error estimates, Appl. Math. Comput. 342 (2019), 280–294.
• [41] H. Xu, L. Zhang, and G. Wang, Some new inequalities and extremal solutions of a Caputo-Fabrizio fractional Bagley-Torvik differential equation, Fractal Fract. 6 (2022), 488.
• [42] M. Al-Smadi, S. Al-Omari, Y. Karaca, and S. Momani, Effective analytical computational technique for conformable time-fractional nonlinear Gardner equation and Cahn-Hilliard equations of fourth and sixth order emerging in dispersive media, J. Funct. Spaces, 2022 (2022), 4422186, DOI: http://dx.doi.org/10.1155/2022/4422186.
• [43] G. Gumah, K. Moaddy, M. AL-Smadi, and I. Hashim, Solutions to uncertain Volterra integral equations by fitted reproducing kernel Hilbert space method, J. Funct. Spaces 2016 (2016), 2920463, DOI: http://dx.doi.org/10.1155/2016/2920463.
• [44] M. Al-Smadi, S. Al-Omari, S. Alhazmi, Y. Karaca, and S. Momani, Novel travelling-wave solutions of spatial-temporal fractional model of dynamical Benjamin-Bona-Mahony system, Fractals 31 (2023), no. 10, 2340189, DOI: https://doi.org/10.1142/S0218348X23401898.
• [45] M. Alabedalhadi, S. Al-Omari, M. Al-Smadi, S. Momani, and D. L. Suthar, New chirp soliton solutions for the space-time fractional perturbed Gerdjikov-Ivanov equation with conformable derivative, Appl. Math. Sci. Eng. 32 (2024), 2292175, DOI: https://doi.org/10.1080/27690911.2023.2292175.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).