The main objective of this work is to present a high-order numerical method to solve a class of nonlinear Fredholm integro-differential equations. By multiplying appropriate efficient factors and constructing an appropriate approximate function, as well as employing a numerical integration method of order $\gamma$, the above-mentioned problem can be simplified to a nonlinear system of algebraic equations. Furthermore, we discuss the convergence analysis of the presented method in detail and demonstrate that it converges with an order $\mathcal{O}(h^{3.5})$ in the $L^2$-norm. Some test examples are provided to demonstrate that the claimed order of convergence is obtained.
[1] R. Amin, I. Mahariq, K. Shah, M. Awais, and F. Elsayed, Numerical solution of the second order linear and nonlinear integro-differential equations using Haar wavelet method, Arab Journal of Basic and Applied Sciences, 28(1) (2021), 11–19.
[2] S. Amiri, Effective numerical methods for nonlinear singular two-point boundary value Fredholm integro-differential equations, Iranian Journal of Numerical Analysis and Optimization, 13(3) (2023), 444–459.
[3] S. H. Behiry and H. Hashish, Wavelet methods for the numerical solution of Fredholm integro-differential equations, Int. J. Appl. Math., 11(1) (2002), 27–35.
[4] S. H. Behiry and S. I. Mohamed, Solving high-order nonlinear Volterra–Fredholm integro-differential equations by differential trasform method, Nat. Sci., 4(8) (2012), 581–587.
[5] A. H. Bhrawy, E. Tohidi, and F. Soleymani, A new Bernoulii matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals, Appl. Math. Comput., 219 (2012), 482– 497.
[6] J. Chen, M. F. He, and Y. Huang, A fast multiscale Galerkin method for solving second order linear fredholm integro-differential equation with Dirichlet boundary conditions, J. Comput. Appl. Math., 364 (2020), 112352.
[7] J. Chen, M. He, and T. Zeng, A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation II: Effcient algorithm for the discrete linear system, J. Vis. Commun. Image Represent., 58 (2019), 112–118.
[8] J. Chen, Y. Huang, H. Rong, T. Wu, and T. Zeng, A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation, J. Comput. Appl. Math., 290 (2015), 633–640.
[9] M. Dehghan and A. Saadatmandi, Chebyshev finite difference method for Fredholm integro-differential equation, Int. J. Comput. Math. 85 (2008), 123–130.
[10] D. S. Dzhumabaev, On one approach to solve the linear boundary value problems for Fredholm integro-differential equations, J. Comput. Appl. Math., 294 (2016), 342–357.
[11] O. A. Gegele, O. P. Evans, and D. Akoh, Numerical solution of higher order linear Fredholm integro-differential equations, Applied Journal of Engineering, 8 (2014), 243–247.
[12] S. Islam, I. Aziz, and M. Fayyaz, A new approach for numerical solution of integro-differential equations via Harr wavelets, Int. J. Comput. Math., 90(9) (2013), 1971–1989.
[13] R. Jalilian and T. Tahernezhad, Exponential spline method for approximation solution of Fredholm integrodifferential equation, Int. J. Comput. Math., 97(4) (2020), 791–801.
[14] P. Kanwal, Linear integral equations theory and technique, London, Academic Press, 1971.
[15] G. Long, G. Nelakanti, and X. Zhang, Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels, Appl. Numer. Math., 62 (2012), 201–211.
[16] F. Mirzaee, Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials, Comput. Methods Differ. Equ., 5(2) (2017), 88–102.
[17] A. Molabahrami, Direct computation method for solving a general nonlinear Fredholm integro-differential equation under the mixed conditions: Degenerate and non-degenerate kernels, J. Comput. Appl. Math., 282 (2015), 34–43.
[18] Y. Ordokhani, An application of Walsh functions for Fredholm–Hammerstein integro-differential equations, Int. J. Contemp. Math. Sci., 5(22) (2010), 1055–1063.
[19] P. K. Pandey, Non-standard difference method for numerical solution of linear Fredholm integro-differential type two-point boundary value problems, Open Access Lib. J., 2 (2015), 1–10.
[20] M. A. L. Ramadan, K. M. Raslan, and M. A. E. G. Nassear, A rational Chebyshev functions approach for Fredholm-Volterra integro-differential equations, Comput. Methods Differ. Equ., 3(4) (2015), 284-297.
[21] M. Razzaghi and S. Yousefi, Legendre wavelets method for the nonlinear Volterra Fredholm integral equations, Math. Comput. Simul., 70 (2005), 1–8.
[22] A. Saadatmandi and M. Dehghan, Numerical solution of high-order linear Fredholm integro-differential–difference equation with variable coefficients, Comput. Math. Appl., 59 (2010), 2996–3004.
[23] T. L. Saaty, Modern nonlinear equations, New York, Dover publications., 1981.
[24] A. Shidfar, A. Molabahrami, A. Babaei, and A. Yazdanian, A series solution of the nonlinear Volterra and Fredholm integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 205–215.
[25] H. R. Thiem, A model for spatio spead of an epidemic, J. Math. Biol., 4 (1977), 337–351.
[26] Q. Xue, J. Niu, D. Yu, and C. Ran, An improved reproducing kernel method for fredholm integro-differential type two-point boundary value problems, Int. J. Comput. Math., 95 (2017), 1015–1023.
[27] S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002), 195–206.
[28] S. Yalcinbas and M. Sezer, The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput., 112 (2000), 291–308.
[29] S. Yang, X. Luo, F. Li, and G. Long, A fast multiscale Galerkin method for the first kind ill-posed integral equations via iterated regularization, Appl. Math. Comput., 219 (2013), 10527–10537.
[30] S. Yeganeh, Y. Ordokhani, and A. Saadatmandi, A Sinc-collocation method for second-order boundary value problems of nonlinear integro-differential equation, J. Inf. Comput. Sci., 7(2) (2012), 151–160.
[31] W. Yulan, T. Chaolu, and P. Jing, New algorithm for second-order boundary value problems of integro-differential equation, J. Comput. Appl. Math., 229 (2009), 1–6.
[32] F. Zhang, Matrix Theory, Springer, New York, 2011.
Amiri, S. and Eshaghnezhad, M. (2025). High-order numerical solution for a class of nonlinear Fredholm integro-differential equations. Computational Methods for Differential Equations, 13(3), 1059-1073. doi: 10.22034/cmde.2024.58818.2489
MLA
Amiri, S. , and Eshaghnezhad, M. . "High-order numerical solution for a class of nonlinear Fredholm integro-differential equations", Computational Methods for Differential Equations, 13, 3, 2025, 1059-1073. doi: 10.22034/cmde.2024.58818.2489
HARVARD
Amiri, S., Eshaghnezhad, M. (2025). 'High-order numerical solution for a class of nonlinear Fredholm integro-differential equations', Computational Methods for Differential Equations, 13(3), pp. 1059-1073. doi: 10.22034/cmde.2024.58818.2489
CHICAGO
S. Amiri and M. Eshaghnezhad, "High-order numerical solution for a class of nonlinear Fredholm integro-differential equations," Computational Methods for Differential Equations, 13 3 (2025): 1059-1073, doi: 10.22034/cmde.2024.58818.2489
VANCOUVER
Amiri, S., Eshaghnezhad, M. High-order numerical solution for a class of nonlinear Fredholm integro-differential equations. Computational Methods for Differential Equations, 2025; 13(3): 1059-1073. doi: 10.22034/cmde.2024.58818.2489
)