SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 5

Morgan-Voyce polynomial approach for ordinary linear delay integro-differential equations with variable delays and variable bounds

Mustafa Özel(DOKUZ EYLUL UNIVERSITY)0000-0003-3771-8625
Mehmet Tarakçı(DOKUZ EYLUL UNIVERSITY)0000-0002-8643-7232
Mehmet Sezer(MANISA CELAL BAYAR UNIVERSITY)0000-0002-7744-2574
Pages
—
Publication Date
2021-10-15
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Also, some examples are given to show the validity and the residual error analysis is investigated.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:An effective matrix method to solve the ordinary linear integro-differential equations with variable coefficients and variable delays under initial conditions is offered in this article.
📊
Key Findings & Data:Our method consists of determining the approximate solution of the matrix form of Morgan-Voyce and Taylor polynomials and their derivatives in the collocation points.
💡
Conclusion & Impact:Also, some examples are given to show the validity and the residual error analysis is investigated.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Morgan-Voyce polynomials · integro differential equations with variable delays · matrix-collocation method

Abstract

An effective matrix method to solve the ordinary linear integro-differential equations with variable coefficients and variable delays under initial conditions is offered in this article. Our method consists of determining the approximate solution of the matrix form of Morgan-Voyce and Taylor polynomials and their derivatives in the collocation points. Then, we reconstruct the problem as a system of equations and solve this linear system. Also, some examples are given to show the validity and the residual error analysis is investigated.

🔖

Cite This Article

Özel, M., Tarakçı, M., Sezer, M. (2021). Morgan-Voyce polynomial approach for ordinary linear delay integro-differential equations with variable delays and variable bounds. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 5. https://doi.org/10.15672/hujms.569245

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (42)

  1. H.A. Ali, Expansion Method For Solving Linear Delay Integro-Differential Equation Using B-Spline Functions, Engineering and Technology Journal, 27 (10), 1651–1661, 2009.
  2. C. Angelamaria, I.D. Prete and C. Nitsch, Gaussian direct quadrature methods for double delay Volterra integral equations, Electron. Trans. Numer. Anal. 35, 201–216, 2009.
  3. A. Ardjouni and D. Ahcene, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal. Theory, Methods Appl. 74 (6), 2062– 2070, 2011.
  4. Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc. 23 (2), 424-428, 2015.
  5. M.A. Balci and M. Sezer, Hybrid Euler-Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations, Appl. Math. Comput. 273, 33–41, 2016.
  6. A. Bellour and M. Bousselsal, Numerical solution of delay integrodifferential equations by using Taylor collocation method, Math. Methods Appl. Sci. 37 (10), 1491–1506, 2014.
  7. A.H. Bhrawy and S.S. Ezz-Eldien, A new Legendre operational technique for delay fractional optimal control problems, Calcolo, 53 (4), 521–543, 2016.
  8. A.H. Bhrawy et al., A Legendre-Gauss collocation method for neutral functional- differential equations with proportional delays, Adv. Differ. Equ. 2013 (1), 63, 2013.
  9. J. Biazar et al., Numerical solution of functional integral equations by the variational iteration method, J. Comput. Appl. Math. 235 (8), 2581–2585, 2011.
  10. A.M. Bica and M. Sorin, Smooth dependence by LAG of the solution of a delay integro- differential equation from Biomathematics, Commun. Math. Anal. 1 (1), 64–74, 2006.
  11. H. Brunner and H. Qiya, Optimal superconvergence results for delay integro- differential equations of pantograph type, SIAM J. Numer. Anal. 45 (3), 986–1004, 2007.
  12. H. Brunner, Recent advances in the numerical analysis of Volterra functional dif- ferential equations with variable delays, J. Comput. Appl. Math. 228 (2), 524–537, 2009.
  13. B. Cahlon and D. Schmidt, Stability criteria for certain delay integral equations of Volterra type, J. Comput. Appl. Math.84 2, 161–188, 1997.
  14. A. Canada and A. Zertiti, Positive solutions of nonlinear delay integral equations modelling epidemics and population growth, Extracta Math. 8 (2-3), 153–157, 1993.
  15. A. Canada and A. Zertiti, Systems of nonlinear delay integral equations modelling population growth in a periodic environment, Comment. Math. Univ. Carolinae, 35 (4), 633–644, 1994.
  16. E.A. Dads and K. Ezzinbi, Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems, Non- linear Anal. Theory, Methods Appl. 41 (1), 1–13, 2000.
  17. H.S. Dink, T-J. Xiao and J. Liang, Existence of positive almost automorphic solutions to nonlinear delay integral equations, Nonlinear Anal. 70, 2216–2231, 2009.
  18. E. Eleonora, E. Russo and A. Vecchio, Comparing analytical and numerical solution of a nonlinear two-delay integral equations, Math. Comput. Simul. 81 (5), 1017–1026, 2011.
  19. S.S. Ezz-Eldien, On solving systems of multi-pantograph equations via spectral tau method, Appl. Math. Comput. 321, 63–73, 2018.
  20. S.S. Ezz-Eldien and E.H. Doha, Fast and precise spectral method for solving panto- graph type Volterra integro-differential equations, Numer. Algorithms, 81 (1), 57–77, 2019.
  21. A.M. Fink and J.A. Gatica, Positive almost periodic solutions of some delay integral equations, J. Differ. Equ. 83 (1), 166–178, 1990.
  22. L. Fortuna and M. Frasca, Generating passive systems from recursively defined poly- nomials, Int. J. Circuits, Syst. Signal Process. 6, 179–188, 2012.
  23. E. Gokmen, G. Yuksel and M. Sezer, A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays, J. Comput. Appl. Math. 311, 354–363, 2017.
  24. M. Gulsu and M. Sezer, Approximations to the solution of linear Fredholm integrod- ifferentialdifference equation of high order, J. Franklin Inst. 343 (7), 720–737, 2006.
  25. H.G. He and L.P. Wen, Dissipativity of -methods and one-leg methods for nonlinear neutral delay integro-differential equations, WSEAS Trans. Math. textbf12, 405–415, 2013.
  26. Ö.K. Kürkçü, E. Aslan and M. Sezer, A numerical approach with error estimation to solve general integro-differentialdifference equations using Dickson polynomials, Appl. Math. Comput. 276, 324–339, 2016.
  27. Ö.K. Kürkçü, E. Aslan and M. Sezer, A novel collocation method based on residual er- ror analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials, Sains Malays. 46, 335–347, 2017.
  28. Ö.K. Kürkçü, E. Aslan and M. Sezer, A numerical method for solving some model problems arising in science and convergence analysis based on residual function, Appl. Numer. Math. 121, 134–148, 2017.
  29. T. Luzyanina, D. Roose and K. Engelborghs, Numerical stability analysis of steady state solutions of integral equations with distributed delays, Appl. Numer. Math.50 (1), 75–92, 2004.
  30. M. Özel, Ö.K. Kürkçü and M. Sezer, Morgan-Voyce matrix method for generalized functional integro-differential equations of Volterra type, Journal of Science and Arts, 47 (2), 295–310, 2019.
  31. M. Özel, M. Tarakçı and M. Sezer, A numerical approach for a nonhomogeneous differential equations with variable delays, Mathematical Sciences, 12, 145–155, 2018.
  32. I. Özgül and N. Şahin, On Morgan-Voyce polynomials approximation for linear dif- ferential equations, Turk. J. Math. Comput. Sci. 2 (1), 1–10, 2014.
  33. M.T. Rashed, Numerical solution of functional differential, integral and integro- differential equations, Appl. Math. Comp. 156 (2), 485–492, 2004.
  34. S.Y. Reutskiy, The backward substitution method for multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type, J. Comput. Appl. Math. 296, 724–738, 2016.
  35. M. Sezer and A.A. Daşcıoğlu, Taylor polynomial solutions of general linear differential–difference equations with variable coefficients, Appl. Math. Comput. 174 (2), 1526–1538, 2006.
  36. M. Sezer and A.A. Daşcıoğlu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math. 200 (1), 217–225, 2007.
  37. T. Stoll and R.F. Tichy, Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials, Math. Slovaca, 58 (1), 11–18, 2008.
  38. M.N.S. Swamy, Further Properties of Morgan-Voyce Polynomials, Fibonacci Quart. 6.2, 167–175, 1968.
  39. N. Şahin, Ş. Yüzbaşı and M. Sezer, A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations, Int. J. Comput. Math. 88 (14), 3093–3111, 2011.
  40. K. Toshiyuki, Stability of RungeKutta methods for delay integro-differential equations, J. Comput. Appl. Math. 145 (2), 483–492, 2002.
  41. K. Toshiyuki, Stability of -methods for delay integro-differential equations, J. Com- put. Appl. Math. 161 (2), 393–404, 2003.
  42. M.N.S. Swamy, Further Properties of Morgan-Voyce Polynomials, Fibonacci Quart. 6.2, 167–175, 1968.