Solving a class of nonlinear delay Fredholm integro-differential equations with convergence analysis

2022;
: pp. 375–384
https://doi.org/10.23939/mmc2022.02.375
Received: November 01, 2021
Accepted: February 10, 2022

Mathematical Modeling and Computing, Vol. 9, No. 2, pp. 375–384 (2022)

1
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
2
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
3
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

The main idea proposed in this article is an efficient shifted Legendre pseudospectral method for solving a class of nonlinear delay Fredholm integro-differential equations.  In this method, first we transform the problem into an equivalent continuous-time optimization problem and then utilize a shifted pseudospectral method to discrete the problem. By this method, we obtained a nonlinear programming problem.  Having solved the last problem, we can obtain an approximate solution for the original delay Fredholm integro-differential equation.  Here, the convergence of the method is presented under some mild conditions.  Illustrative examples are included to demonstrate the efficiency and applicability of the presented technique.

  1. Dehghan M., Saadatmandi A.  Chebyshev finite difference method for Fredholm integro-differential equation.  International Journal of Computer Mathematics. 85 (1), 123–130 (2008).
  2. Lakestani M., Razzaghi M., Dehghan M.  Semiorthogonal wavelets approximation for Fredholm integro-differential equations.  Mathematical Problems in Engineering. 2006, Article ID 096184 (2006).
  3. Jackiewicz Z., Rahman M., Welfert B. D.  Numerical solution of a Fredholm integro-differential equation modelling neural networks.  Applied Numerical Mathematics. 56 (3–4), 423–432 (2006).
  4. Wazwaz A. M.  A First Course in Integral Equations.  World Scientific, River Edge (1997).
  5. Mahmoudi M., Ghovatmand M., Noori Skandari M. H.  A novel numerical method and its convergence for nonlinear delay Voltrra integro-differential equations.  Mathematical Methods in the Applied Sciences. 43 (5), 2357–2368 (2020).
  6. Mahmoudi M., Ghovatmand M., Noori Skandari M. H.  A New Convergent Pseudospectral Method for Delay Differential Equations.  Iranian Journal of Science and Technology, Transactions A: Science. 44, 203–211 (2020).
  7. Belloura A., Bousselsal M.  Numerical solution of delay integro-differential equations by using Taylor collocation method.  Mathematical Methods in the Applied Sciences. 37 (10), 1491–1506 (2013).
  8. Wu S., Gan S.  Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations.  Computers & Mathematics with Applications. 55 (11), 2426–2443 (2008).
  9. Yüzbasi S.  Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations.  Journal of Taibah University for Science. 11 (2), 344–352 (2017).
  10. Saadatmandi A., Dehghan M.  Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients. Computers and Mathematics with Applications. 59 (8), 2996–3004 (2010).
  11. Kucche K. D., Shikhare P. U.  Ulam stabilities for nonlinear Volterra–Fredholm delay integro-differential equations.  International Journal of Nonlinear Analysis and Applications. 9 (2), 145–159 (2018).
  12. Gülsu M., Sezer M.  Approximations to the solution of linear Fredholm integro-differential-difference equation of high order.  Journal of the Franklin Institute. 343 (7), 720–737 (2006).
  13. Issa K., Biazar J., Yisa B. M.  Shifted Chebyshev Approach for the Solution of Delay  Fredholm and Volterra Integro-Differential Equations via Perturbed Galerkin Method.  Iranian Journal of Optimization. 11 (2), 149–159 (2019).
  14. Boichuk A. A., Medved M., Zhuravliov V. P.  Fredholm boundary-value problems for linear delay systems defined by pairwise permutable matrices.  Electronic Journal of Qualitative Theory of Differential Equations. 23 (1), 1–9 (2015).
  15. Şahin N., Yüzbaşi S., Sezer M.  A Bessel polynomial approach for solving general linear Fredholm integro-differential-difference equations.  International Journal of Computer Mathematics. 88 (14), 3093–3111 (2011).
  16. Sezer M., Gülsu M.  Polynomial solution of the most general linear    Fredholm–Volterra integro-differential-difference equations by means of Taylor collocation method.  Applied Mathematics and Computation. 185 (1), 646–657 (2007).
  17. Ordokhani Y., Mohtashami M. J.  Approximate solution of nonlinear Fredholm integro-differential equations with time delay by using Taylor method.  J. Sci. Tarbiat Moallem University. 9 (1), 73–84 (2010).
  18. Shen J., Tang T., Wang L.-L.  Spectral Methods: Algorithms, Analysis and Applications.  Springer, Berlin (2011).
  19. Canuto C., Hussaini M. Y., Quarteroni A., Zang T. A.  Spectral method in Fluid Dynamics. Springer, New York (1988).
  20. Freud G.  Orthogonal Polynomials.  Pergamom Press, Elmsford (1971).