Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods

The paper presents the modifications of the variational iteration method (MVIM), Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-differential equation of the second kind. In these methods a series is created, the summation of which gives the solution of the discussed equation. Conditions ensuring convergence of this series are presented in the paper. An example illustrating the usage of the investigated methods is presented as well and the results reveal that the proposed methods are very effective, able, and simple. comparison between our proposed methods with the exact solution and some traditional methods is presented during a numerical example. The results reveal that (MHPM) and (MLADM) lead to an exact solution and (MVIM) leads to a limited solution. The uniqueness of the solutions and the convergence of the proposed methods are also proved.

Keywords:

Nonlinear Fredholm integro-differential equation, modification of the variational iteration method (MVIM) modification of homotopy perturbation method(MHPM), modification of Laplace Adomian decomposition method (MLADM),

PDF

___

[1] S. Abbasbandy and E. Shivanian, A new analytical technique to solve Fredholm's integral equations, Numer. algorithms, 2011, 56(1), 27-43.
[2] Q.M. Al-Mdallal, Monotone iterative sequences for nonlinear integro-differential equations of second order, Nonlinear Analysis: Real World Applications, 2011, 12(6), 3665-3673.
[3] M.D. Aloko, O.J. Fenuga, and S.A. Okunuga, Modified variational iteration method for the numerical solutions of some non-linear Fredholm integro-di?erential equations of the second kind, J. Appl. Computat. Math. 2017, 6(4), 1-4.
[4] F. Al-Saar, K. Ghadle, and P. Pathade, The approximate solutions of Fredholm integral equations by Adomian decompo- sition method and its modification, Int. J. Math. Appl. 2018, 6, 327-336.
[5] F. Al-Saar and K. Ghadle, An approximate solution for solving the system of Fredholm integral equations of the second kind, Bull. Pure Appl. Sci. Math. 2019, 1, 208-215.
[6] F. Al-Saar and K. Ghadle, The numerical solutions of linear and non-linear Volterra integral equations of the second kind using variational iteration method, Acta Univ. M. Belii Ser. Math. 2019, 27, 3-13.
[7] F. Al-Saar and K. Ghadle, Combined Laplace transform with analytical methods for solving Volterra integral equations with a convolution kernel, KSIAM, 2018, 22(2), 125-136.
[8] F. Al-Saar, A. Hamoud, and K. Ghadle, Some numerical methods to solve a system of Volterra integral equations, Int. J. Open Problems Compt. Math. 2019, 12(4), 22-35.
[9] A. Alturk, Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem, SpringerPlus, 2016, 5(1), 1-15.
[10] M. Asif, I. Khan, N. Haider and Q. Al-Mdallal, Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations, Alexandria Engineering Journal, 2020, 59(6), 5099-5109.
[11] Z. P. Atabakan, A. K. Nasab, and A. Kiliçman, On solution of Fredholm integro-differential equations using composite Chebyshev finite difference method, Abstr. Appl. Anal. 2013, 2013, Hindawi.
[12] Z.P. Atabakan, A.K. Nasab, A. Kiliçman, and Z.K. Eshkuvatov, Numerical solution of nonlinear Fredholm integrodifferential equations using spectral homotopy analysis method, Math. Probl. Eng. 2013, 2013, 1-9.
[13] S.S. Behzadi, The use of iterative methods to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equations, Communications in Numerical Analysis, 2012, 2012, 1-20.
[14] M. Erfanian and H. Zeidabadi, Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases, Asian Eur. J. Math. 2019, 12(04), 1-15.
[15] S.M. Mirzaei, Homotopy perturbation method and variational iteration method for Voltra integral equations, Journal of Applied Mathematics and Bioinformatics, 2011, 1(1), 105-113.
[16] M.O. Omotosho, and T.O. Adebayo, Modified homotopy perturbation method for solving high-order integro-differential equation, Mathematical Theory and Modeling, 2020, 10(1), 15-29.
[17] P.K. Pandey, Numerical solution of linear Fredholm integro-differential equations by non-standard finite difference method, Int. J. Math. Model Comput. 2015, 5, 259-266.
[18] M. Safavi and A.A. Khajehnasiri, Numerical solution of nonlinear mixed Volterra-Fredholm integro-differential equations by two-dimensional block-pulse functions, Cogent math. stat. 2018, 5(1), 1-12.
[19] M.I. Syam, Q.M. Al-Mdallal and M.N. Anwar, An eficient numerical algorithm for solving fractional higher-order nonlinear integro-differential equations, In Abstract and Applied Analysis 2015, (2015), Hindawi.