A compact discretization of the boundary value problems of the nonlinear Fredholm integro-differential equations

Document Type : Research Article

Authors

1 Department of Basic Sciences, Shahid Sattari Aeronautical University of Science and Technology, P.O. Box: 13846-63113, Tehran, Iran

2 Department of Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran

Abstract

In this paper, we propose a  fourth-order compact discretization method   for solving a second-order boundary value problem governed by the nonlinear Fredholm integro-differential equations.  Using an efficient approximate polynomial,  a  compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations.  It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some  numerical examples are presented to demonstrate its  approximation accuracy and computational efficiency,   as well as to compare the derived results with those  obtained in the literature.

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[1] R. Amin, I. Mahariq, K. Shah, M. Awais, F. Elsayed, Numerical solution of the second order linear
and nonlinear integro-differential equations using Haar wavelet method, Arab J. Basic Appl. Sci.
28 (2021) 11–19.
[2] S.H. Behiry, H. Hashish, Wavelet methods for the numerical solution of Fredholm integro-
differential equations, Int. J. Appl. Math. 11 (2002) 27–35.
[3] S.H. Behiry, S.I. Mohamed, Solving high-order nonlinear VolterraFredholm integro-differential
equations by differential trasform method, Nat. Sci. 4 (2012) 581–587.
[4] A.H. Bhrawy, E. Tohidi, 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.
[5] S.H. Behiry, Solution of nonlinear Fredholm integro-differential equations using a hybrid of block
pulse functions and normalized Bernstein polynomials, J. Comput. Appl. Math. 260 (2014) 258–
265.
[6] J. Chen, M.F. He, 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, Y. Huang, H. Rong, T. Wu, 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.
[8] M. Dehghan, A. Saadatmandi, Chebyshev finite difference method for Fredholm integro-differential
equation, Int. J. Comput. Math. 85 (2008) 123–130.
[9] 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.
[10] R. Jalilian, T. Tahernezhad, Exponential spline method for approximation solution of Fredholm
integro-differential equation, Int. J. Comput. Math. 97 (2020) 791–801.
[11] S. Islam, I. Aziz, M. Fayyaz, A new approach for numerical solution of integro-differential equa-
tions via Harr wavelets, Int. J. Comput. Math. 90 (2013) 1971–1989.
[12] F. Mirzaee, Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polyno-
mials, Comput. Methods Differ. Equ. 5 (2017) 88–102.
[13] A. Molabahrami, Direct computation method for solving a general nonlinear Fredholm integro-
differential equation under the mixed conditions: Degenerate and non-degenerate kernels, J. Com-
put. Appl. Math. 282 (2015) 34–43.
[14] Y. Ordokhani, An application of Walsh functions for FredholmHammerstein integro-differential
equations, Int. J. Contemp. Math. Sci. 5 (2010) 1055–1063.
[15] 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.
[16] A. Saadatmandi, M. Dehghan, Numerical solution of high-order linear Fredholm integro-
differentialdifference equation with variable coefficients, Comput. Math. Appl. 59 (2010) 2996–
3004.
[17] A. Shidfar, A. Molabahrami, A. Babaei, A. Yazdanian, A series solution of the nonlinear Volterra
and Fredholm integro-differential equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010)
205–215.
[18] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,
Oxford university press, 1985.
[19] H.R. Thiem, A model for spatio spead of an epidemic, J. Math. Biol. 4 (1977) 337–351.
[20] A.J. Weir, Lebesgue Integration and Measure, Cambridge: Cambridge University Press, 1973.
[21] Q. Xue , J. Niu, D. Yu, 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.
[22] S. Yeganeh, Y. Ordokhani, A. Saadatmandi, A Sinc-collocation method for second-order boundary
value problems of nonlinear integro-differential equation, J. Inf. Comput. Sci. 7 (2012) 151–160.
[23] W. Yulan, T. Chaolu, P. Jing, New algorithm for second-order boundary value problems of integro-
differential equation, J. Comput. Appl. Math. 229 (2009) 1–6.