Mathematics and Computational Sciences

A comparative analysis of Daftardar-Gejji and Jafari method and Chebyshev least squares method in solving Volterra integro-differential equations

Document Type : Original Article

Authors

Maria Rahman
Dipta Roy
Faria Tasnim
Tania Sharmin Khaleque

Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh

https://doi.org/10.30511/mcs.2026.2073962.1491
Abstract
This study presents a comparative analysis of two efficient numerical approaches for solving linear
and nonlinear Volterra integro-differential equations: the Daftardar–Gejji and Jafari Method (DJM)
and the Chebyshev Least Squares Method (CLSM). The Daftardar–Gejji and Jafari Method (DJM)
provides robust results without linearization or perturbation assumptions, whereas CLSM employs
shifted Chebyshev polynomial approximations together with least-squares minimization to obtain
highly accurate spectral solutions. Convergence and stability properties of both methods are briefly
discussed within fixed-point and weighted residual frameworks. Several benchmark first- through
fourth-order VIDEs are solved to evaluate the performance of the proposed approaches. The ob-
tained numerical results show excellent agreement with exact solutions and previously published
methods. Comparative analyses based on absolute error, L∞-norm, residual behavior, and CPU
time are presented to examine the convergence, computational efficiency, and numerical stability
of the proposed methods.

Keywords

Volterra Integro-Differential equation
Daftardar-Gejji and Jafari method
Chebyshev Least Squares method
Shifted Chebyshev polynomial
Computational stability analysis
Subjects
[1] Abed, A. M., Younis, M. F., & Hamoud, A. A. (2022). Numerical solutions of nonlinear VolterraFredholm integro-differential equations by using MADM and VIM. Nonlinear Functional Analysis and Applications, 189201. 
[2] Adebisi, F., Ojurongbe, T. A., Okunlola, K. A., & Peter, O. J. (2021). Application of Chebyshev polynomial basis function on the solution of Volterra integro-differential equations using Galerkin method. Mathematics and Computational Sciences, 2(4), 4151. 
[3] Adebisi, A. F., & Okunola, K. A. (2025). Solution of VolterraFredholm integro-differential equations using Chebyshev least square method. Science World Journal, 20(1), 271275.
[4] Ahmed, A. (2025). Solving nonlinear fractional autonomous dynamical systems using a LaplaceDaftardarJafari method. Mathematics and Computational Sciences, 6(3), 143152.
[5] Akgun, F. A., & Rasulov, Z. (2025). Numerical method for solution of fourth-order Volterra integro-differential equations by Greens function. Kragujevac Journal of Mathematics, 49(3), 485494.
[6] Akgun, F. A., & Rasulov, Z. (2021). Generalized iteration method for the solution of fourth order BVP via Greens function. European Journal of Pure and Applied Mathematics, 14(3). 
[7] Akram, G., & Rehman, H. (2014). Homotopy perturbation method with reproducing kernel method for third order nonlinear boundary value problems. Journal of Basic and Applied Science Research, 4(1), 6067. 
[8] Al-Hayani, W. (2017). DaftardarJafari method for fractional heat-like and wave-like equations with variable coefficients. Applied Mathematics, 8(2), 215228.
[9] Al-Mdallal, Q. M. (2011). Monotone iterative sequences for nonlinear integro-differential equations of second order. Nonlinear Analysis: Real World Applications, 12(6), 36653673. 
[10] Alao, S., Akinboro, F. S., Akinpelu, F. O., & Oderinu, R. (2014). Numerical solution of integro-differential equation using Adomian decomposition and variational iteration methods. IOSR Journal of Mathematics, 10(4), 1822. 
[11] Ali, K. K., Cattani, C., Gómez-Aguilar, J. F., Baleanu, D., & Osman, M. S. (2020). Analytical and numerical study of the DNA dynamics arising in oscillator-chain of PeyrardBishop model. Chaos, Solitons & Fractals, 139, 110089.
[12] Ali, K. K., Abd El Salam, M. A., Mohamed, E. M., Samet, B., Kumar, S., & Osman, M. S. (2020). Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Advances in Difference Equations, 2020, 123. 
[13] Ali, F., Ali, J., & Uddin, I. (2021). A novel approach for the solution of BVPs via Greens function and fixed point iterative method. Journal of Applied Mathematics and Computation, 66(1), 167181.
[14] Ansari, A., Ahmad, N., & Ali, A. H. (2024). Numerical study of the series solution method to analysis of Volterra integro-differential equations. Journal of Applied Mathematics and Informatics, 42(4), 899913. 
[15] Avudainayagam, A., & Vani, C. (2000). WaveletGalerkin method for integro-differential equations. Applied Numerical Mathematics, 32(3), 247254. 
[16] Brunner, H. (1989). The numerical treatment of Volterra integro-differential equations with unbounded delay.Journal of Computational and Applied Mathematics, 28, 523.
[17] Boyd, J. P. (2001). Chebyshev and Fourier spectral methods. Dover Publications. 
[18] Cakir, M., Gunes, B., & Duru, H. (2021). A novel computational method for solving nonlinear Volterra integro-differential equation. Kuwait Journal of Science, 48(1). 
[19] Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (2006). Spectral methods (Vol. 285). Springer. 
[20] El-Sayed, S. M., & Abdel-Aziz, M. R. (2003). A comparison of Adomians decomposition method and wavelet Galerkin method for solving integro-differential equations. Applied Mathematics and Computation, 136(1), 151159. 
[21] Hamoud, A. A., & Ghadle, K. P. (2018). Existence and uniqueness of the solution for VolterraFredholm integro differential equations. Journal of the Siberian Federal University. Mathematics and Physics, 11(6), 692701.
[22] Hosseini, S. M., & Shahmorad, S. (2003). Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Applied Mathematical Modelling, 27(2), 145154. 
[23] Jeffery, G. B. (1914). On the form of the Volterra equations. Proceedings of the Royal Society A, 90(619), 204215. 
[24] Jiang, Y., & Ma, J. (2013). Spectral collocation methods for Volterra integro-differential equations with noncompact kernels. Journal of Computational and Applied Mathematics, 244, 115124. 
[25] Kot, M. (2001). Elements of mathematical ecology. Cambridge University Press. 
[26] Kopeikin, I. D., & Shishkin, V. P. (1984). Integral form of the general solution of equations of steady-state thermoelasticity. Journal of Applied Mathematics and Mechanics, 48(1), 117119. 
[27] Kreyszig, E. (1991). Introductory functional analysis with applications. John Wiley & Sons. 
[28] Liu, Z., Sun, J., & Szántó, I. (2013). Monotone iterative technique for RiemannLiouville fractional integro differential equations with advanced arguments. Results in Mathematics, 63(3), 12771287. 
[29] Lotka, A. J. (1939). On an integral equation in population analysis. The Annals of Mathematical Statistics, 10(2),144161. 
[30] Maleknejad, K., & Mahmoudi, Y. (2003). Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations. Applied Mathematics and Computation, 145(23), 641653.
[31] Mishra, A. K., Verma, L., Nikan, O., & Molavi-Arabshahi, M. (2025). Numerical pricing of European options under time-fractional BlackScholes equation in financial markets. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(8). 
[32] Ndidiamaka, D. E., Oyedepo, T., & Adenipekun, A. E. (2024). Fourth-kind Chebyshev computational approach for integro-differential equations. Mathematics and Computational Sciences, 5(1), 5564. 
[33] Nikan, O., Rashidinia, J., & Jafari, H. (2025). An improved local radial basis function method for pricing options under the time-fractional BlackScholes model. Journal of Computational Science, 89, 102610. 
[34] Pougaza, D. B. (2007). The Lotka integral equation as a stable population model. AIMS, 130. 
[35] Rashid, S., Kubra, K. T., Jafari, H., & Lehre, S. U. (2022). A semi-analytical approach for fractional order Boussinesq equation in gradient unconfined aquifers. Mathematical Methods in the Applied Sciences, 45(2), 10331062.
[36] Rashed, M. T. (2004). Lagrange interpolation to compute numerical solutions of differential, integral and integro differential equations. Applied Mathematics and Computation, 151(3), 869878. 
[37] Sahin, N., Yüzba, ., & Sezer, M. (2011). A Bessel polynomial approach for solving general linear Fredholm integro differentialdifference equations. International Journal of Computer Mathematics, 88(14), 30933111.
[38] Sunthrayuth, P., Ullah, R., Khan, A., Shah, R., Kafle, J., Mahariq, I., & Jarad, F. (2021). Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations. Journal of Function Spaces, 2021, Article ID 1537958. 
[39] Tahernezhad, T., & Jalilian, R. (2020). Exponential spline for the numerical solutions of linear Fredholm integro-differential equations. Advances in Difference Equations, 2020, 115.
[40] Tisdell, C. C. (2019). On Picards iteration method to solve differential equations and a pedagogical space for otherness. International Journal of Mathematical Education in Science and Technology, 50(5), 788799. 
[41] Trefethen, L. N. (2000). Spectral methods in MATLAB. SIAM.
[42] Verma, L., Meher, R., Nikan, O., & Al-Saedi, A. A. (2025). Numerical study on fractional order nonlinear SIR-SI model for dengue fever epidemics. Scientific Reports, 15(1), 30677. 
[43] Volterra, V. (1910). Sulle equazioni integro-differenziali. Tipografia della R. Accademia dei Lincei. 
[44] Volterra, V. (1930). Theory of functionals and of integral and integro-differential equations. Blackie & Son Limited.
[45] Zaboli, M., & Tajadodi, H. (2025). A spectral collocation method for solving stochastic fractional integro-differential equation. Mathematics and Computational Sciences, 6(2), 131. 
[46] Zeidler, E. (2013). Nonlinear functional analysis and its applications II/B: nonlinear monotone operators. Springer.
[47] Zhao, J., & Corless, R. M. (2006). Compact finite difference method for integro-differential equations. Applied Mathematics and Computation, 177(1), 271288.
Mathematics and Computational Sciences
Volume 7, Issue 3
Summer 2026
Pages 142-159

  • Receive Date 09 October 2025
  • Revise Date 13 June 2026
  • Accept Date 17 June 2026

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