Numerical Solution to the System of Integro-Differential Equations Using Collocation Approximation

The system of integro-differential equations (IDEs) has been utilized to model a wide range of problems in finance, control systems, viscoelasticity, engineering, and the wide-ranging applications of these equations in various fields of science. In this paper, we present efficient solvers for solving a system of high-order linear Volterra–Fredholm integro-differential equations (VFIDEs) using numerical techniques. The system of integro-differential equations is reduced to equations into a system of linear algebraic equations, and matrix inversion is employed to solve the algebraic equations. Only a small number of S-polynomials are needed to obtain a satisfactory result. The method’s error analysis is presented. Several examples are provided to demonstrate the use of the collocation approach. The numerical simulation demonstrates the dependability and efficiency of the collocation method. The proposed method is highly effective, straightforward, and well-suited for solving systems of Volterra integro-differential equations.

S-polynomials, Integro-differential equations, Operational matrix, Collocation method.

1. K. Diethelm and N. J. Ford, (2002). “Analysis of Fractional Differential Equations,” J. Math. Anal. Appl., vol. 265, no. 2, pp. 229–248, Jan, doi: 10.1006/jmaa.2000.7194. [Google Scholar] [Crossref]

2. R. Gorenflo and F. (1997). Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics. [Google Scholar] [Crossref]

3. N. Bellomo, B. Firmani, and L. Guerri, (1999). “Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition,” Appl. Math. Lett., vol. 12, no. 2, pp. 39 – 44, doi: 10.1016/S0893-9659(98)00146-3. [Google Scholar] [Crossref]

4. M. S. Abou-Dina and M. A. Helal, (1998). “Reduction for the nonlinear problem of fluid waves to a system of integro-differential equations with an oceanographical application,” J. Comput. Appl. Math., vol. 95, no. 1–2, pp. 65 – 81, doi: 10.1016/S0377-0427(98)00072-7. [Google Scholar] [Crossref]

5. A.-M. Wazwaz, (2003). “The existence of noise terms for systems of inhomogeneous differential and integral equations,” Appl. Math. Comput., vol. 146, no. 1, pp. 81 – 92, doi: 10.1016/S0096-3003(02)00527-1. [Google Scholar] [Crossref]

6. S. G. Samko, A. A. Kilbas, and O. I. Marichev, (1993). “Fractional Integrals and Derivatives: Theory and Applications,”. [Online]. Available: https://api.semanticscholar.org/CorpusID:118631078 [Google Scholar] [Crossref]

7. F. Mainardi, (Feb. 2012). “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics. [Google Scholar] [Crossref]

8. S. Ullah, M. Khan, M. Farooq, Z. Hammouch, and D. Baleanu, (2019). “A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative,” Discrete and Continuous Dynamical Systems, doi: 10.3934/dcdss.2020057. [Google Scholar] [Crossref]

9. H. Sun, A. Chang, Y. Zhang, and W. Chen, (2018). “A review on variable-order fractional differential equations: Mathematical foundations, physical models, and its applications,” Fract. Calc. Appl. Anal., vol. 22, pp. 27–59, doi: 10.1515/fca-2019-0003. [Google Scholar] [Crossref]

10. Kochubei and Y. Luchko, (2019). Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory. doi: 10.1515/9783110571622. [Google Scholar] [Crossref]

11. O. Abdullaev and S. Kishin, (2016). “Non-local problems with integral gluing condition for loaded mixed type equations involving the Caputo fractional derivative,” Electronic Journal of Differential Equations, vol. [Google Scholar] [Crossref]

12. T. Sandev (Трифче Сандев) and Z. Tomovski, (2019). Fractional Equations and Models: Theory and Applications. doi: 10.1007/978-3-030-29614-8. [Google Scholar] [Crossref]

13. T. K. Yuldashev, (2018). “Nonlocal Boundary Value Problem for a Nonlinear Fredholm Integro-Differential Equation with Degenerate Kernel,” Differential Equations, vol. 54, no. 12, pp. 1646–1653, doi: 10.1134/S0012266118120108. [Google Scholar] [Crossref]

14. Y. Grigoriev, N. Ibragimov, V. Kovalev, and S. Meleshko, (2010). Symmetries of Integro-Differential Equations, vol. 806. doi: 10.1007/978-90-481-3797-8. [Google Scholar] [Crossref]

15. T. Atanackovic, S. Pilipovic, B. Stanković, and D. Zorica, (2014). Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. doi: 10.1002/9781118577530. [Google Scholar] [Crossref]

16. Podlubny, (1999). “Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,”. [Online]. Available: https://api.semanticscholar.org/CorpusID:124040597 [Google Scholar] [Crossref]

17. K. Maleknejad, B. Basirat, and E. Hashemizadeh, (2012). “A Bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations,” Math. Comput. Model., vol. 55, no. 3–4, pp. 1363–1372, doi: 10.1016/J.MCM.2011.10.015. [Google Scholar] [Crossref]

18. P. Rahimkhani, (Jan. 2025). “A numerical method for Ψ-fractional integro-differential equations by Bell polynomials,” Applied Numerical Mathematics, vol. 207, pp. 244–253, doi: 10.1016/J.APNUM.2024.09.011. [Google Scholar] [Crossref]

19. S. Mehranpour and J. Damirchi, (2025). “Numerical solution of fractional integro-differential equation by collocation Pell-Lucas polynomial method,” TWMS Journal of Applied and Engineering Mathematics. [Google Scholar] [Crossref]

20. S. Momani and S. Hadid, (Feb. 2003). “On the inequalities of integro-differential fractional equations,” IJAM. International Journal of Applied Mathematics, vol. 12. [Google Scholar] [Crossref]

21. J. T. Edwards, N. J. Ford, and C. A. Simpson, (2002). “The numerical solution of linear multi-term fractional differential equations: systems of equations,” J. Comput. Appl. Math., vol. 148, no. 2, pp. 401–418, Nov, doi: 10.1016/S0377-0427(02)00558-7. [Google Scholar] [Crossref]

22. S. Momani and M. Noor, (2006). “Numerical methods for fourth-order fractional integro-differential equations,” Appl. Math. Comput., vol. 182, pp. 754–760, doi: 10.1016/j.amc.2006.04.041. [Google Scholar] [Crossref]

23. Y. Nawaz, (Apr. 2011). “Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2330–2341, doi: 10.1016/j.camwa.2010.10.004. [Google Scholar] [Crossref]

24. L. Rong and C. Phang, (2016). “Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation,” J. Phys. Conf. Ser., vol. 693, p. 12002, doi: 10.1088/1742-6596/693/1/012002. [Google Scholar] [Crossref]

25. J. Wang, T.-Z. Xu, Y.-Q. Wei, and J.-Q. Xie, (2018). “Numerical simulation for coupled systems of nonlinear fractional order integro-differential equations via wavelets method,” Appl. Math. Comput., vol. 324, pp. 36–50, doi: 10.1016/j.amc.2017.12.010. [Google Scholar] [Crossref]

26. Yousefi, T. Rad, and S. Shafiei, (2015). “A Quadrature Tau Method for Solving Fractional Integro-differential Equations in The Caputo Sense,” Journal of Mathematics and Computer Science, vol. 15, pp. 97–107, doi: 10.22436/jmcs.015.02.01. [Google Scholar] [Crossref]

27. Akyüz-Daşcoğlu and M. Sezer, (2005). “Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations,” J. Franklin Inst., vol. 342, no. 6, pp. 688–701. [Google Scholar] [Crossref]

28. K. Maleknejad and M. T. Kajani, (2004). “Solving linear integro-differential equation system by Galerkin methods with hybrid functions,” Appl. Math. Comput., vol. 159, no. 3, pp. 603 – 612, doi: 10.1016/j.amc.2003.10.046. [Google Scholar] [Crossref]

29. C. T. H. Baker and A. Tang, (1997). “Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays,” Applied Numerical Mathematics, vol. 24, no. 2–3, pp. 153 – 173, doi: 10.1016/S0168-9274(97)00018-4. [Google Scholar] [Crossref]

30. M. Ganesh and I. H. Sloan, (1999). “Optimal order spline methods for nonlinear differential and integro-difrerential equations,” Applied Numerical Mathematics, vol. 29, no. 4, pp. 445 – 478, doi: 10.1016/S0168-9274(98)00067-1. [Google Scholar] [Crossref]

31. O. M. A. Al-Faour and R. K. Saeed, (2006). “Solution of a system of linear Volterra integral and integro-differential equations by spectral method,” Al-Nahrain Univ. J. Sci., vol. 6, no. 2, pp. 30 – 46, [Online]. Available: https://www.scopus.com/inward/record.uri?eid=2-s2.0-84855204718&partnerID=40&md5=4bf0d2f37368b28bb1b76e51c1871012 [Google Scholar] [Crossref]

32. T. Jangveladze, Z. Kiguradze, and B. Neta, (2009) “Finite difference approximation of a nonlinear integro-differential system,” Appl. Math. Comput., vol. 215, no. 2, pp. 615 – 628, doi: 10.1016/j.amc.2009.05.061. [Google Scholar] [Crossref]

33. G. Ajileye and S. A. Amoo, (2023). “Numerical solution to Volterra integro-differential equations using collocation approximation,” Mathematics and Computational Sciences, vol. 4, no. 1, pp. 1–8, doi: 10.30511/MCS.2023.1978083.1099. [Google Scholar] [Crossref]

34. R. C. Mittal and R. Nigam, (2008). “Solution of fractional integro-differential equations by Adomian decomposition method,” Int. J. of Appl. Math. and Mech., vol. 4. [Google Scholar] [Crossref]

35. G. Mehdiyeva, M. Imanova, and V. Ibrahimov, (2020). “On the Construction of the Multistep Methods to Solving the Initial-Value Problem for ODE and the Volterra Integro- Differential Equations,” pp. 978–979. [Google Scholar] [Crossref]

36. G. Mehdiyeva, M. Imanova, and V. Ibrahimov, (2015). “Solving volterra integro-differential equation by the second derivative methods,” Applied Mathematics and Information Sciences, vol. 9, no. 5, pp. 2521–2527, doi: 10.12785/amis/090536. [Google Scholar] [Crossref]

37. K. Issa and F. Salehi, (2017). “Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method,” Journal of Mathematics, vol. 2017, doi: 10.1155/2017/8213932. [Google Scholar] [Crossref]

38. Bhrawy, E. Tohidi, and F. Soleymani, (2013). “A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals,” Appl. Math. Comput. [Google Scholar] [Crossref]

39. N. Rohaninasab, K. Maleknejad, and R. Ezzati, (2018). “Numerical solution of high-order Volterra–Fredholm integro-differential equations by using Legendre collocation method,” Appl. Math. Comput., vol. 328, pp. 171–188, Jul, doi: 10.1016/j.amc.2018.01.032. [Google Scholar] [Crossref]

40. P. Rahimkhani, (Nov. 2023) “Numerical solution of nonlinear stochastic differential equations with fractional Brownian motion using fractional-order Genocchi deep neural networks,” Commun. Nonlinear Sci. Numer. Simul., vol. 126, p. 107466, doi: 10.1016/j.cnsns.2023.107466. [Google Scholar] [Crossref]

41. K. Maleknejad, E. Hashemizadeh, and R. Ezzati, (2011). “A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation,” Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 2, pp. 647 – 655, doi: 10.1016/j.cnsns.2010.05.006. [Google Scholar] [Crossref]

42. E. H. Doha, A. H. Bhrawy, and M. A. Saker, (2011). “Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations,” Appl. Math. Lett., vol. 24, no. 4, pp. 559 – 565, doi: 10.1016/j.aml.2010.11.013. [Google Scholar] [Crossref]

43. E. H. Doha, A. H. Bhrawy, and M. A. Saker, (2011). “On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations,” Boundary Value Problems, vol, no. 1, p. 829543, 2011, doi: 10.1155/2011/829543. [Google Scholar] [Crossref]

44. B. N. Mandal and S. Bhattacharya, (2007). “Numerical solution of some classes of integral equations using Bernstein polynomials,” Appl. Math. Comput., vol. 190, no. 2, pp. 1707–1716. [Google Scholar] [Crossref]

45. T. J. Rivlin, (1981). An introduction to the approximation of functions. Courier Corporation. [Google Scholar] [Crossref]

46. P. Rahimkhani and Y. Ordokhani, (2023). “Hahn wavelets collocation method combined with Laplace transform method for solving fractional integro-differential equations,” Mathematical Sciences, vol. 18, doi: 10.1007/s40096-023-00514-3. [Google Scholar] [Crossref]

47. R. M. Ganji, H. Jafari, M. Kgarose, and A. Mohammadi, (2021). “Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials,” Alexandria Engineering Journal, vol. 60, no. 5, pp. 4563–4571, Oct, doi: 10.1016/j.aej.2021.03.026. [Google Scholar] [Crossref]