10.1007/s40096-022-00492-y

A neural computational method for solving renewal delay integro-differential equations constrained by the half-line

  • Ömür Kıvanç Kürkçü*1,
  1. Department of Engineering Basic Sciences, Konya Technical University, Konya, 42250, TR

Published in Issue 2022-10-03

How to Cite

Kürkçü, Ömür K. (2022). A neural computational method for solving renewal delay integro-differential equations constrained by the half-line. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00492-y
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Abstract

Abstract This study aims to solve the renewal delay integro-differential equations constrained by the half-line, introducing a computational method composed of the matrix relations of the Stieltjes–Wigert polynomials at the collocation points. In order to mathematically interpret their robust integral part, the method is also fed neurally by the Stieltjes–Wigert polynomials and a hybrid polynomial dependent upon the alteration of the Taylor and exponential polynomial bases. Thus, the method easily gathers the matrix relations into a matrix equation and immediately produces a desired solution. An error bound analysis is established to discuss the accuracy of the method by employing the collaboration of the mentioned polynomials. A population model with time-lags (delays), the detection of the displaced atoms versus kinetic energy, an integral delay equation, and a delayed problem with functional kernel are firstly treated by the method. Consequently, it is evident that the method presents a novel consistent approach and is directly programmable on a mathematical software thanks to its neural structure.

Keywords

  • Delay forces,
  • Error bound,
  • Infinite boundary,
  • Neural computation,
  • Renewal equation

References

  1. Wangersky and Cunningham (1957) Time lag in population models 2(2) (pp. 329-338) https://doi.org/10.1101/SQB.1957.022.01.031
  2. Bellman and Cooke (1963) Academic Press
  3. Dads and Ezzinbi (2000) Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems (pp. 1-13) https://doi.org/10.1016/S0362-546X(98)00219-3
  4. Fein (1958) Influence of a variable ejection probability on the displacement of atoms (pp. 1076-1083) https://doi.org/10.1103/PhysRev.109.1076
  5. Leitman and Mizel (1970) On linear hereditary laws (pp. 46-58) https://doi.org/10.1007/BF00251540
  6. Miller and Michel (1980) On the response of nonlinear multivariable interconnected feedback systems to periodic input signals (pp. 1088-1097) https://doi.org/10.1109/TCS.1980.1084746
  7. Annunziato et al. (2012) Asymptotic stability of solutions to Volterra-renewal integral equations with space maps (pp. 766-775) https://doi.org/10.1016/j.jmaa.2012.05.080
  8. Zhao and Huang (2020) Exponential fitting collocation methods for a class of Volterra integral equations
  9. Chamayou (1973) Volterra’s functional integral equations of the statistical theory of damage (pp. 70-93) https://doi.org/10.1016/0021-9991(73)90126-5
  10. Brunner et al. (1997) Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution (pp. 381-402) https://doi.org/10.1016/S0168-9274(96)00075-X
  11. Cardone et al. (2010) Exponential fitting direct quadrature methods for Volterra integral equations (pp. 467-480) https://doi.org/10.1007/s11075-010-9365-1
  12. Minh and Chau (2015) Asymptotic equilibrium of integro-differential equations with infinite delay (pp. 189-192) https://doi.org/10.1007/s40096-015-0166-5
  13. Franco and O’Regan (2008) Solutions of Volterra integral equations with infinite delay (pp. 325-336) https://doi.org/10.1002/mana.200510605
  14. Maleknejad et al. (2009) A wavelet Petrov–Galerkin method for solving integro-differential equations 86(9) (pp. 1572-1590) https://doi.org/10.1080/00207160801923056
  15. Maleknejad et al. (2006) Using the Petrov-Galerkin elements for solving Hammerstein integral equations 172(2) (pp. 831-845)
  16. Deep A., Deepmala, Rabbani M.: A numerical method for solvability of some non-linear functional integral equations. Appl. Math. Comput.
  17. 402
  18. , 125637 (2021)
  19. Stieltjes (1894) Recherches sur les fractions continues (pp. 1-122) https://doi.org/10.5802/afst.108
  20. Boelen and Assche (2015) Variations of Stieltjes-Wigert and qdocumentclass[12pt]{minimal}
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  28. setlength{oddsidemargin}{-69pt}
  29. begin{document}$$q$$end{document}-Laguerre polynomials and their recurrence coefficients (pp. 56-73) https://doi.org/10.1016/j.jat.2014.06.012
  30. Chihara (1970) A characterization and a class of distribution functions for the Stieltjes-Wigert polynomials (pp. 529-532) https://doi.org/10.4153/CMB-1970-098-7
  31. Wigert (1923) Sur les polynômes orthogonaux et l’approximation des fonctions continues
  32. Szegö (1975) American Mathematical Society
  33. Koekoek and Swarttouw (1998) Delft University of Technology
  34. Wang and Wong (2006) Uniform asymptotics of the Stieltjes–Wigert polynomials via the Riemann–Hilbert approach (pp. 698-718) https://doi.org/10.1016/j.matpur.2005.11.005
  35. Christiansen (2003) The moment problem associated with the Stieltjes–Wigert polynomials (pp. 218-245) https://doi.org/10.1016/S0022-247X(02)00534-6
  36. Kürkçü et al. (2019) An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function https://doi.org/10.1007/s40314-019-0771-2