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investigating nonlinear fractional systems: reproducing kernel Hilbert space method

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Abstract

The reproducing kernel Hilbert space method (RK-HS method) is used in this research for solving some important nonlinear systems of fractional ordinary differential equations, such as the fractional Susceptible-Infected-Recovered (SIR) model. Nonlinear systems are widely used across various disciplines, including medicine, biology, technology, and numerous other fields. To evaluate the RK-HS method’s accuracy and applicability, we compare its numerical solutions with those obtained via Hermite interpolation, the Adomian decomposition method, and the residual power series method. To further support the reliability of the RK-HS method, the convergence analysis is discussed.

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References

  • Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)

    Article  MathSciNet  Google Scholar 

  • Abu Arqub, O., Osman, M.S., Park, C., Lee, J.R., Alsulam, H., Alhodaly, M.: Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation. Alex. Eng. J. 61(12), 10539–10550 (2022)

    Article  Google Scholar 

  • Akgül, A.: A novel method for a fractional derivative with non-local and non-singular kernel. Chaos, Solitons Fractals 114, 478–482 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  • Allahviranloo, T., Sahihi, H.: Reproducing kernel method to solve fractional delay differential equations. Appl. Math. Comput. 400, 126095 (2021)

    MathSciNet  Google Scholar 

  • Al-Smadi, M., Gumah, G.: On the homotopy analysis method for fractional SEIR epidemic model. Res. J. Appl. Sci. Eng. Technol. 7(18), 3809–3820 (2014)

    Article  Google Scholar 

  • Attia, N., Akgül, A., Seba, D., Nour, A.: An efficient numerical technique for a biological population model of fractional order. Chaos, Solitons Fractals 141, 110349 (2020)

    Article  MathSciNet  Google Scholar 

  • Babolian, E., Javadi, S., Moradi, E.: Error analysis of reproducing kernel Hilbert space method for solving functional integral equations. J. Comput. Appl. Math. 300, 300–311 (2016)

    Article  MathSciNet  Google Scholar 

  • Chen, S.-B., Soradi-Zeid, S., Dutta, H., Mesrizadeh, M., Jahanshahi, H., Chu, Y.-M.: Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay. Chaos, Solitons Fractals 144, 110674 (2021)

    Article  MathSciNet  Google Scholar 

  • Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers Inc, New York (2009)

    Google Scholar 

  • Dubey, V.P., Kumar, D., Dubey, S.: A Modified Computational Scheme and Convergence Analysis for Fractional Order Hepatitis E Virus Model. In: Advanced Numerical Methods for Differential Equations, CRC Press, Boca Raton, pp. 279–312 (2021)

  • Dubey, V.P., Dubey, S., Kumar, D., Singh, J.: A computational study of fractional model of atmospheric dynamics of carbon dioxide gas. Chaos, Solitons Fractals 142, 110375 (2021)

    Article  MathSciNet  Google Scholar 

  • Dubey, V.P., Singh, J., Alshehri, A.M., Dubey, S., Kumar, D.: Forecasting the behavior of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique. Chaos, Solitons Fractals 164, 112691 (2022)

    Article  MathSciNet  Google Scholar 

  • Dubey, S., Dubey, V.P., Singh, J., Alshehri, A.M., Kumar, D.: Computational study of a local fractional Tricomi equation occurring in fractal transonic flow. J. Comput. Nonlinear Dyn. 17(8), 081006 (2022)

    Article  Google Scholar 

  • Dubey, V.P., Singh, J., Dubey, S., Kumar, D.: Some integral transform results for Hilfer–Prabhakar fractional derivative and analysis of free-electron laser equation. Iran. J. Sci. 47, 1333–1342 (2023)

    Article  MathSciNet  Google Scholar 

  • Fardi, M.: A kernel-based pseudo-spectral method for multi-term and distributed order time-fractional diffusion equations. Numer. Methods Partial Differ. Equ. 39(3), 2630–2651 (2023)

    Article  MathSciNet  Google Scholar 

  • Fardi, M.: A kernel-based method for solving the time-fractional diffusion equation. Numer. Methods Partial Differ. Equ. 39(3), 2719–2733 (2023)

    Article  MathSciNet  Google Scholar 

  • Fardi, M., Al-Omari, S.K.Q., Araci, S.: A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation. Adv. Contin. Discret. Models 2022, 54 (2022)

    Article  MathSciNet  Google Scholar 

  • Fardi, M., Zaky, M.A., Hendy, A.S.: Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian. Math. Comput. Simul. 206, 614–635 (2023)

    Article  MathSciNet  Google Scholar 

  • Fernandez, A., Baleanu, D., Fokas, A.S.: Solving PDEs of fractional order using the unified transform method. Appl. Math. Comput. 339, 738–749 (2018)

    MathSciNet  Google Scholar 

  • Freihat, A., Handam, A.: Solution of the SIR models of epidemics using MSGDTM. Appl. Appl. Math. 9(2), 622–636 (2014)

    Google Scholar 

  • Hasan, S., Al-Zoubi, A., Freihet, A., Al-Smadi, M., Momani, S.: Solution of fractional SIR epidemic model using residual power series method. Appl. Math. Inf. Sci. 13(2), 153–161 (2019)

    Article  MathSciNet  Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)

    Google Scholar 

  • Mohammadi, S., Ghasemi, M., Fardi, M.: A fast Fourier spectral exponential time-differencing method for solving the time-fractional mobile-immobile advection-dispersion equation. Comput. Appl. Math. 41, 264 (2022)

    Article  MathSciNet  Google Scholar 

  • Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  • Singh, H., Srivastava, H.M., Baleanu, D.: Methods of Mathematical Modeling: Infectious Diseases, Elsevier Science: Amsterdam (2022) (ISBN: 9780323998888)

  • Singh, H., Srivastava, H.M., Nieto, J.J.: Handbook of Fractional Calculus for Engineering and Science, CRC Press, Taylor & Francis Group, Boca Raton (2022)

  • Singh, H.: Analysis for fractional dynamics of Ebola virus model. Chaos, Solitons Fractals 138, 109992 (2020)

    Article  MathSciNet  PubMed  Google Scholar 

  • Singh, H.: Analysis of drug treatment of the fractional HIV infection model of CD4+ T-cells. Chaos, Solitons Fractals 146, 110868 (2021)

    Article  MathSciNet  Google Scholar 

  • Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  ADS  Google Scholar 

  • Yildirim, E.N., Akgül, A., Inc, M.: Reproducing kernel method for the solutions of non-linear partial differential equations. Arab J. Basic Appl. Sci. 28(1), 80–86 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. (221412044).

Funding

This research was funded by Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia grant number 221412044. Data Availability Statement: Data are included within this research.

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Authors and Affiliations

  1. National High School for Marine Sciences and Coastal (ENSSMAL), Dely Ibrahim University Campus, Bois des Cars, B.P. 19, 16320, Algiers, Algeria

    Nourhane Attia

  2. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Ali Akgül

  3. Department of Mathematics, Art and Science Faculty, Siirt University, 56100, Siirt, Turkey

    Ali Akgül

  4. Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, PC: 99138, Nicosia/Mersin 10, Turkey

    Ali Akgül

  5. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University(IMSIU), Riyadh, Saudi Arabia

    Rubayyi T. Alqahtani

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  1. Nourhane Attia
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  2. Ali Akgül
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  3. Rubayyi T. Alqahtani
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Contributions

Conceptualization, RTA; Methodology, AA and NA; Software, AA; Formal analysis, NA; Investigation, RTA; Resources, NA; Data curation, AA; Writing-original draft, NA; Writing-review & editing, NA; Visualization, AA; Supervision, RTA; Project administration, AA, RTA; Funding acquisition, RTA. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Nourhane Attia.

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Attia, N., Akgül, A. & Alqahtani, R.T. investigating nonlinear fractional systems: reproducing kernel Hilbert space method. Opt Quant Electron 56, 8 (2024). https://doi.org/10.1007/s11082-023-05591-1

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