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Keywords:
second order integro-differential equation; trapezoidal quadrature formula with corrections; iterative method; high accuracy numerical method
second order integro-differential equation; trapezoidal quadrature formula with corrections; iterative method; high accuracy numerical method
Summary:
We develop high-order numerical methods for solving second-order nonlinear Fredholm Integro-Differential Equations (FrIDEs). First, we establish the existence and uniqueness of solutions under easily verified conditions and propose a continuous iterative method. Subsequently, we construct a discrete iterative scheme by discretizing this continuous method using the trapezoidal quadrature formula augmented with correction terms. We rigorously prove that the proposed numerical method achieves eighth-order convergence. Extensive experiments on both linear and nonlinear equations demonstrate that our method attains eighth-order accuracy, surpassing existing methods whose accuracy does not exceed fifth order.
References:
[1] Amirali, I., Durmaz, M. E., Amiraliyev, G. M.: A monotone second-order numerical method for Fredholm intergro-differential equation. Mediterr. J. Math. 21 (2024), Article ID 198, 15 pages. DOI 10.1007/s00009-024-02746-6 | MR 4811238 | Zbl 1550.65288
[2] Amiri, S., Hajipour, M.: A compact discretization of the boundary value problems of the nonlinear Fredholm integro-differential equations. J. Math. Model. 12 (2024), 233-246. DOI 10.22124/jmm.2023.24380.2184 | MR 4777338 | Zbl 1563.65500
[3] Chen, J., He, M., Zeng, T.: A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation. II: Efficient algorithm for the discrete linear system. J. Visual Commun. Image Represent. 58 (2019), 112-118. DOI 10.1016/j.jvcir.2018.11.027
[4] Dang, Q. A, Dang, Q. L.: Simple numerical methods of second- and third-order convergence for solving a fully third-order nonlinear boundary value problem. Numer. Algorithms 87 (2021), 1479-1499. DOI 10.1007/s11075-020-01016-2 | MR 4287899 | Zbl 1483.34026
[5] Dang, Q. A, Dang, Q. L., Ngo, T. K. Q.: Numerical methods of fourth, sixth and eighth orders convergence for solving third order nonlinear ODEs. Math. Comput. Simul. 221 (2024), 397-414. DOI 10.1016/j.matcom.2024.03.018 | MR 4719240 | Zbl 1570.65219
[6] Dang, Q. A, Nguyen, T. H., Vu, V. Q.: Construction of high order numerical methods for solving fourth order nonlinear boundary value problems. Numer. Algorithms 99 (2025), 323-354. DOI 10.1007/s11075-024-01879-9 | MR 4892873 | Zbl 1564.65152
[7] Dang, Q. A, Nguyen, T. H., Vu, V. Q.: Eighth order numerical method for solving second order nonlinear BVPs and applications. J. Appl. Math. Comput. 71 (2025), 3577-3600. DOI 10.1007/s12190-025-02368-5 | MR 4910863 | Zbl 1570.65222
[8] Dang, Q. L., Dang, Q. A: Existence results and numerical method for solving a fourth-order nonlinear integro-differential equation. Numer. Algorithms 90 (2022), 563-576. DOI 10.1007/s11075-021-01198-3 | MR 4418363 | Zbl 1491.65168
[9] Hildebrand, F. B.: Introduction to Numerical Analysis. International Series in Pure and Applied Mathematics. McGraw Hill, New York (1974). MR 0347033 | Zbl 0279.65001
[10] Li, J.: General explicit difference formulas for numerical differentiation. J. Comput. Appl. Math. 183 (2005), 29-52 \99999DOI99999 10.1016/j.cam.2004.12.026 . DOI 10.1016/j.cam.2004.12.026 | MR 2156098 | Zbl 1077.65021
[11] Sidi, A., Pennline, J. A.: Improving the accuracy of quadrature method solutions of Fredholm integral equations that arise from nonlinear two-point boundary value problems. J. Integral Equations Appl. 11 (1999), 103-139. DOI 10.1216/jiea/1181074263 | MR 1685181 | Zbl 0974.65123
[12] Tahernezhad, T., Jalilian, R.: Exponential spline for the numerical solutions of linear Fredholm integro-differential equations. Adv. Difference Equ. 2020 (2020), Article ID 141, 15 pages. DOI 10.1186/s13662-020-02591-3 | MR 4085947 | Zbl 1482.65243
[13] Zhao, J., Corless, R. M.: Compact finite difference method for integro-differential equations. Appl. Math. Comput. 177 (2006), 271-288. DOI 10.1016/j.amc.2005.11.007 | MR 2234518 | Zbl 1102.65144
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