Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases

Abstract

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

Investigation approach for a nonlinear singular Fredholm integro-differential equation

“…Several articles have been published in the approximation of integro-differential equations, such as using Haar wavelet bases [3], with RH wavelet method [4,7], or using cubic B-spline finite element method [6].…”

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

“…Fredholm integro-dierential equation has been solved by some other methods, such as weighted mean value theorem [9]. The approximate solution for solving the nonlinear Fredholm integro-dierential equation of the second kind in the complex plane by using the properties of rationalized Haar wavelet has been obtained in [14]. A two-dimensional nonlinear Volterra-Fredholm integro-dierential equation by using some iterative methods is presented [13].…”

Solving nonlinear Fredholm integral equations with PQWs in complex plane

“…A numerical solution of the Integral equations is the subject of recent research. Some recent relevant contributions in this area include the Haar wavelet method (Aziz and Siraj-ul-Islam, 2013;Siraj-ul-Islam et al, 2013;Erfanian, 2018aErfanian, , 2018bErfanian and Zeidabadi, 2019a;Erfanian et al, 2015Erfanian et al, , 2017, the block boundary-value method (Chen and Zhang, 2012), the Adomian decomposition method (Hashim, 2006), with cubic Bspline finite element method (Erfanian and Zeidabadi, 2019b), the Hybrid Legendre polynomials and Block-Pulse functions approach (Maleknejad et al, 2011), the variational iteration method (VIM) (Sweilam, 2007), the homotopy perturbation method (HPM) (Yildirim, 2008), the Sinc collocation method (Zarebnia, 2010), the meshless method (Dehghan and Salehi, 2012), and the sequential approach (Berenguer et al, 2012). An overview of the existing numerical methods establishes the fact that each of the existing methods, whether semi-analytical or fully numerical has limited in the real space.…”