Combining the reproducing kernel method with Taylor series expansion to solve systems of nonlinear fractional Volterra integro-differential equations

Document Type : Research Article

Authors

T. Amoozad ¹

S. Abbasbandy ²

H. Sahihi ¹

T. Allahviranloo ³

¹ Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

² Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.

³ Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey.

10.22067/ijnao.2025.90460.1542

Abstract

In this article, we present a novel approach for solving systems of nonlinear fractional Volterra integro-differential equations $(NFVI-DEs)$ by reproducing the Hilbert kernel method. Kernel methods are powerful tools for addressing both linear and nonlinear problems. The reproducing kernel method stands out for its wide-ranging applications in solving complex scientific challenges. Our method combines the reproducing kernel method with a truncated Taylor series expansion, resulting in a more precise solution. This transformation converts the original $NFVI-DEs$ into a system of nonlinear fractional differential equations. Our numerical results showcase this approach’s effectiveness and align with theorems about error analysis.

Keywords

Reproducing kernel method

Fractional Volterra integro-differential equations

Taylor series expansion

Error analysis

Subjects

• Computational Mathematics

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References

[1] Akbar M., Nawaz R., Ahsan S., Nisar K.S., Abdel-Aty A.H. and Eleuch H. New approach to approximate the solution for the system of fractional order Volterra integro-differential equations, Results Phys. 19 (2020) 103453.
[2] Alvandi A. and Paripour M. Reproducing kernel method with Taylor expansion for linear Volterra integro-differential equations, Commun. Numer. Anal. 1 (2017) 1–10.
[3] Alvandi A. and Paripour M. The combined reproducing kernel method and Taylor series for handling nonlinear Volterra integro-differential equations with derivative type kernel, Appl. Math. Comput. 355 (2019) 151–160.
[4] Amoozad T., Abbasbandy S., Sahihi H. and Allahviranloo T. A new application of the reproducing kernel method for solving linear systems of fractional order Volterra integro-differential equations, Phys. Scripta. 99 (2024) 075209.
[5] Amoozad T., Allahviranloo T., Abbasbandy S. and Rostamy Malkhal-ifeh M. Using a new implementation of reproducing kernel Hilbert space method to solve a system of second-order BVPs, Int. J. Dynam. Control. 12(6) (2024) 1694–1706.
[6] Amoozad T., Allahviranloo T., Abbasbandy S. and Rostamy Malkhal-ifeh M. Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients, Phys. Scripta 99 (2024) 025246.
[7] Babolian E., Javadi S. and Moradi E. Error analysis of reproducing kernel Hilbert space method for solving functional integral equations, Comput. Appl. Math. 300 (2016) 300–311.
[8] Bakodah H.O., Al-Mazmumy M. and Almuhalbedi S.O. Solving system of integro differential equations using discrete Adomian decomposition method, Taibah Univ. Sci. 13 (2019) 805–812.

[9] Cui M.G. and Lin Y. Nonlinear numerical analysis in the reproducing kernel space, Nova Science, Hauppauge, Inc., Hauppauge, 2009.
[10] Das P., Rana S. and Ramos H. A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis, Int. J. Comput. Math. 97 (2020) 1994–2014.
[11] Das P., Rana S. and Ramos H. On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis, Comp. Appl. Math. 404 (2022) 113116.
[12] Geng F.Z. and Cui M.G. Solving a nonlinear system of second order boundary value problems, Math. Anal. Appl. 327 (2007) 1167–1181.
[13] Ghanbari F., Mokhtary P. and Ghanbari K. Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz–Jacobi Tau method, Comp. Appl. Math. 362 (2019) 172–184.
[14] Hansen V.L. Functional analysis, entering Hilbert space, World Scientific Publishing Co. Pte. Ltd., 2006.
[15] Jiang W. and Chen Z. Solving a system of linear Volterra integral equations using the new reproducing kernel method, Appl. Math. Comput. 219 (2013) 10225–10230.
[16] Kumar S., Nieto J.J. and Ahmad B. Chebyshev spectral method for solving fuzzy fractional Fredholm–Volterra integro-differential equation, Math. Comput. Simul. 192 (2022) 501–513.
[17] Li H. and Ma J. On generalized multistep collocation methods for Volterra integro-differential equations, Partial Differ. Equ. Appl. Math. 226 (2024) 399–412.
[18] Mandal M. Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations, Math. Comput. Simul. 225 (2024) 889–913.

[19] Mei L. and Lin Y. Simplified reproducing kernel method and convergence order for linear Volterra integral equations with variable coefficients, Comput. Appl. Math. 346 (2019) 390–398.
[20] Podlubny I. Fractional differential equations, Academic Press, Academic Press, 1999.
[21] Sahihi H., Allahviranloo T. and Abbasbandy S. Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space, Appl. Num. Math. 151 (2020) 27–39.
[22] Shen L., Zhu S., Liu B., Zhang Z. and Cui Y. Numerical implementation of nonlinear system of fractional Volterra integral-differential equations by Legendre wavelet method and error estimation, Numer. Methods Par-tial Differ. Equ. 37 (2021) 1344–1360.
[23] Wang J., Xu T.Z., Wei Y.Q. and Xie J.Q. Numerical simulation for cou-pled systems of nonlinear fractional order integro-differential equations via wavelets method, Appl. Math. Comput. 324 (2018) 36–50.
[24] Wang Y., Chaolu T. and Chen Z. Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Comput. Appl. Math. 87 (2010) 367–380.
[25] Xie J. and Yi M. Numerical research of nonlinear system of fractional Volterra-Fredholm integral-differential equations via Block-Pulse functions and error analysis, Comput. Appl. Math. 345 (2019) 159–167.
[26] Yang, L.H. Shen, J.H. and Wang, Y. The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients, Comput. Appl. Math. 236 (2012) 2398–2405.
[27] Zhou, F. and Xu, X. Numerical solution of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions via Chebyshev wavelet method, Int. J. Comput. Math. 96 (2019) 436–456.