A reproducing kernel method for solving nonlocal functional differential equations with delayed or advanced arguments

Document Type : Research Paper

Authors

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.

4 Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.

Abstract

This paper discusses an effective approach for solving non-local functional differential equations with delayed or advanced arguments. The reproducing kernel method is utilized to avoid the need for an orthogonalization process. The main objective of this technique is to successfully apply this method to solve singular multi-point boundary value problems with non-local conditions, resulting in an accurate approximate solution and a valid error analysis. This method greatly improves the accuracy of the solutions obtained.

Keywords

Main Subjects

References

  • [1] S. Abbasbandy, H. Sahihi, and T. Allahviranloo, Combining the reproducing kernel method with a practical technique to solve the system of nonlinear singularly perturbed boundary value problems, Comput. Methods Differ. Equ., 10(4) (2022), 942–953.
  • [2] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, Fractional order differential systems involving right Caputo and left Riemann–Liouville fractional derivatives with nonlocal coupled conditions, Bound. Value Probl., 109 (2019), 2019.
  • [3] T. Allahviranloo and H. Sahihi, Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions, Numer. Method. Partial Diff. Equ., 36 (2020), 1758–1772.
  • [4] T. Allahviranloo and H. Sahihi, Reproducing kernel method to solve fractional delay differential equations, Appl. Math. Comput., 400 (2021), 126095.
  • [5] M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, and I. K. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157.
  • [6] N. Aronszajn, Theory of reproducing kernel, Trans. Amer. Math. Soc., 68 (1950), 337–404.
  • [7] K. Atkinson and W. Han, Theoretical Numerical Analysis A Functional Analysis Framework, Third Edition. Springer Science, 2009.
  • [8] E. Babolian and D. Hamedzadeh, A splitting iterative method for solving second kind integral equations in reproducing kernel spaces, J. Comput. Appl. Math., 326 (2017), 204–216.
  • [9] E. Babolian, S. Javadi, and E. Moradi, Error analysis of reproducing kernel Hilbert space method for solving functional integral equations, J. Comput. Appl. Math., 300 (2016), 300–311.
  • [10] J. Caballero, L. Plociniczak, and K. Sadarangani, Existence and uniqueness of solutions in the Lipschitz space of a functional equation and its application to the behavior of the paradise fish, Appl. Math. Comput., 477 (2024), 128798.
  • [11] J. Cermak and L. Nechvatal, On stability of linear differential equations with commensurate delayed arguments, Appl. Math. Lett., 125 (2022), 107750.
  • [12] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, Hauppauge, New York, United States, 2009.
  • [13] J. Diblik, Novel criterion for the existence of solutions with positive coordinates to a system of linear delayed differential equations with multiple delays, Appl. Math. Lett., 152 (2024), 109032.
  • [14] S. S. Ezz-Eldien, On solving systems of multi-pantograph equations via spectral tau method, Appl. Math. Comput., 321 (2018), 63–73.
  • [15] A. Ghasemi and A. Saadatmandi, A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions, Comput. Methods Differ. Equ., 12 (2024), 329–337.
  • [16] C. S. Goodrich, Pointwise conditions in discrete boundary value problems with nonlocal boundary conditions, Appl. Math. Comput., 67 (2017), 7–15.
  • [17] S. Guo and S. Li, On the stability of reaction–diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 106197.
  • [18] R. Ketabchi, R. Mokhtari, and E. Babolian, Some error estimates for solving Volterra integral equations by using the reproducing kernel method, J. Comput. Appl. Math., 273 (2015), 245–250.
  • [19] X. Li, Y. Gao, and B. Wu, Mixed reproducing kernel-based iterative approach for nonlinear boundary value problems with nonlocal conditions, Comput. Methods Differ. Equ., 9 (2021), 649–658.
  • [20] Z. Y. Li, Y. L. Wang, F. G. Tan, X. H. Wan, H. Yu, and J. S. Duan, Solving a class of linear nonlocal boundary value problems using the reproducing kernel, Appl. Math. Comput., 265 (2015), 1098–1105.
  • [21] X. Y. Li and B. Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math., 243 (2013), 10–15.
  • [22] X. Y. Li and B. Y. Wu, A continuous method for nonlocal functional differential equations with delayed or advanced arguments, J. Math. Anal. Appl., 409 (2014), 485–493.
  • [23] Y. Muroya, E. Ishiwata, and H. Brunner, On the attainable order of collocation methods for pantograph integrodifferential equations, J. Comput. Appl. Math., 152 (2003), 347–366.
  • [24] M. Sezer, S. Yalc¸inba¸s, and M. Gu¨lsu, A Taylor polynomial approach for solving generalized pantograph equations with nonhomogeneous term, Int. J. Comput. Math., 85 (2008), 1055–1063.
  • [25] Y. Wang, T. Chaolu, and P. Jing, New algorithm for second-order boundary value problems of integro-differential equation, J. Comput. Appl. Math., 229 (2009), 1–6.
  • [26] Y. Wang, T. Chaolu, and Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math., 87 (2010), 367–380.
  • [27] Y. Wang, X. Cao, and X. Li, A new method for solving singular fourth-order boundary value problems with mixed boundary conditions, Appl. Math. Comput., 217 (2011), 7385–7390.
  • [28] Y. Wang, M. Du, F. Tan, Z. Li, and T. Nie, Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions, Appl. Math. Comput., 219 (2013), 5918–5925.
Computational Methods for Differential Equations
Volume 14, Issue 1
January 2026
Pages 1-13

Files

History

  • Receive Date: 28 September 2024
  • Revise Date: 26 February 2025
  • Accept Date: 27 February 2025

Share

How to cite

Statistics