Finite difference method for solving fractional cordial Volterra integro-differential equations with Caputo derivative

Articles in Press

Document Type : Research Paper

Author
Zahra Barikbin
Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, 34149-16818, Iran
10.22075/ijnaa.2025.37393.5447
Abstract
This study introduces, for the first time, a fractional formulation of cordial Volterra integro-differential equations and proposes an efficient numerical algorithm for their solution. The method combines a finite difference scheme of order $\mathscr{O}(\tau^{3-\alpha})$ for discretizing the Caputo fractional derivative with a composite trapezoidal rule $\mathscr{O}(\tau^{2})$ to approximate the integral term. Theoretical analysis establishes that the unified scheme achieves an overall convergence order of $\mathscr{O}(\tau^{2})$, a result rigorously confirmed through numerical experiments. The presented examples not only validate the algorithm's accuracy but also demonstrate its effectiveness in handling fractional-order systems, addressing a gap in the numerical analysis of such equations. This work provides a foundational framework for future research on fractional cordial integro-differential problems.
Keywords
Fractional cordial Volterra integro-differential equations
Finite difference
Caputo fractional derivative
Composite trapezoidal rule
[1] Y.E. Aghdam, H. Mesgarani, G.M. Moremedi, and M. Khoshkhahitnat, High‑accuracy numerical scheme for solving the space‑time fractional advection‑diffusion equation with convergence analysis, Alexandria Engin. J. 61 (2022), no. 1, 217-225.
[2] A. Azin and F. Mohammadi, Approximate solution of Volterra‑Fredholm integral equations using generalized barycentric rational interpolant, Appl. Math.‑J. Chinese Univ. 39 (2024), no. 2, 220-238.
[3] D. Baleanu, S. Etemad, and Sh. Rezapour, On a fractional hybrid integro‑differential equation with mixed hybrid integral boundary value conditions by using three operators, Alexandria Engin. J. 59 (2020), no. 5, 3019-3027.
[4] L. Banjai and M. Ferrari, Generalized convolution quadrature based on the trapezoidal rule, arXiv preprint arXiv:2305.11134, 2023.
[5] Y. Cao, O. Nikan, and Z. Avazzadeh, A localized meshless technique for solving 2D nonlinear integro‑differential equation with multi‑term kernels, Appl. Numerical Math. 183 (2023), 140-156.
[6] M. Dehghan and A. Saadatmandi, Chebyshev finite difference method for Fredholm integro‑differential equation, Int. J. Comput. Math. 85 (2008), no. 1, 123-130.
[7] T. Diogo and G. Vainikko, Applicability of spline collocation to cordial Volterra equations, Math. Modell. Anal. 18 (2013), no. 1, 1-21.
[8] F. Fakhar‑Izadi and M. Dehghan, The spectral methods for parabolic Volterra integro‑differential equations, J. Comput. Appl. Math. 235 (2011), no. 14, 4032-4046.
[9] U. Kangro, Cordial Volterra integral equations and singular fractional integro‑differential equations in spaces of analytic functions, Math. Modell. Anal. 22 (2017), no. 4, 548-567.
[10] K. Karthikeyan, P. Karthikeyan, H.M. Baskonus, K. Venkatachalam, and Y.-M. Chu, Almost sectorial operators on ψψ‑Hiffer derivative fractional impulsive integro‑differential equations, Math. Meth. Appl. Sci. 45 (2022), no. 13, 8045-8059.
[11] M. Kudu, I. Amirali, and G.M. Amiraliyev, A finite‑difference method for a singularly perturbed delay integro‑differential equation, Journal of Computational and Applied Mathematics 308 (2016), 379-390.
[12] K. Latt and A. Pedas, Numerical schemes for a class of singular fractional integro‑differential equations, Appl. Numer. Math. 2023 (2023).
[13] C. Li and F. Zeng, Finite difference methods for fractional differential equations, Int. J. Bifurc. Chaos 22 (2012), no. 4, 1230014.
[14] C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim. 34 (2013), no. 2, 149-179.
[15] H. Majidian, Efficient quadrature rules for a class of cordial Volterra integral equations: A comparative study, Bull. Iran. Math. Soc. 43 (2017), no. 5, 1245-1258.
[16] J. Pande and P. Nasikkar, A maximum power point tracking technique for a wind power system based on the trapezoidal rule, Energies 16 (2023), no. 6, 2799.
[17] N. Pandiangan, D. Johar, and S. Purwani, Fractional integral approximation and Caputo derivatives with modification of trapezoidal rule, World Sci. News 153 (2021), no. 2, 169-180.
[18] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[19] C. Ravichandran, K. Logeswari, and F. Jarad, New results on existence in the framework of Atangana‑Baleanu derivative for fractional integro‑differential equations, Chaos Solitons Fractals 125 (2019), 194-200.
[20] H. Song, Z. Yang, and T. Diogo, Collocation methods for cordial Volterra integro‑differential equations, J. Comput. Appl. Mathe. 393 (2021), 113321.
[21] M. Toranj‑Simin and M. Hadizadeh, A priori mesh grading in collocation solution of noncompact Volterra integral equations with diagonal singularity, Int. J. Comput. Math. 100 (2023), no. 5, 1078-1093.
[22] C. Tunc and O. Tunc, New qualitative criteria for solutions of Volterra integro‑differential equations, Arab J. Basic Appl. Sci. 25 (2018), no. 3, 158-165.
[23] C. Tunc and O. Tunc, A note on the qualitative analysis of Volterra integro‑differential equations, J. Taibah Univ. Sci. 13 (2019), no. 1, 490-496.
[24] G. Vainikko, Spline collocation for cordial Volterra integral equations, Numer. Funct. Anal. Optim. 31 (2010), no. 3, 313-338.
[25] J. Vanualailai and S. Nakagiri, Stability of a system of Volterra integro‑differential equations, J. Math. Analy. Appl. 281 (2003), no. 2, 602-619.
[26] T. Wang, N. Li, and G. Gao, The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities, Int. J. Comput. Math. 92 (2015), no. 3, 579-590.
[27] H. Wang, Z. Yang, and M. Khumalo, Cordial Volterra integral equations with vanishing delays, J. Appl. Math. Phys. 5 (2017), no. 2, 294.
[28] A.-M. Wazwaz, The combined Laplace transform‑Adomian decomposition method for handling nonlinear Volterra integro‑differential equations, Appl. Math. Comput. 216 (2010), no. 4, 1304-1309.
[29] J. Zhao and R.M. Corless, Compact finite difference method for integro‑differential equations, Appl. Math. Comput. 177 (2006), no. 1, 271-288.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 27 July 2026

  • Receive Date 19 April 2025
  • Revise Date 30 July 2025
  • Accept Date 12 August 2025