On Taylor-series expansion methods for the second kind integral equations

Abstract

a b s t r a c tIn this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.

The effect of magnetic field on the tunneling yield of ammonia molecules

“…where , , ′ ( ) = ∑ ( Eq. ( 13) can be computed by using the Taylor-series expansion methods for solving integral equations (HUABSOMBOON et al, 2010). As a result the folowing equation can be obtained:…”

Collocation method for Fredholm and Volterra integral equations

“…In Mandal and Chakrabarti (2011), Frankel (1995) and Chakrabarti and Hamsapriye (1999) was described a method of solution an integro‐differential equation arising in the study of a problem concerning heat conduction and radiation, in elastic contact problems, etc. For solution of the integral equations, several numerical approaches have been proposed such as, the Homotopy‐perturbation method (Ganji et al , 2007; Ghasemi et al , 2007; Ghorbani and Saberi‐Nadjafi, 2008; Yusufoglu, 2008), the wavelet basis (Yousefi and Razzaghi, 2005; Rabbani et al , 2012, 2009), the collocation method basis (Oja, 2002; Taiwo et al , 2010; Maleknejad et al , 2007; Nizami and Elnur, 2009), the converting to optimization problem (Nazemi et al , 2007), the decompositional method of Adomian (Cherruault and Seng, 1997; Babolian and Davari, 2005; Javadi et al , 2007; Ngarasta, 2009; Ngarasta et al , 2009; Bougoffa and Al‐Haqbani, 2012), the semi‐orthogonal B‐spline (Lakestani et al , 2005), the inversion of linear system produced by quadrature (Saberi et al , 2010), the iterated collocation method and their discretizations (Brunner, 1984), the iterations of the quasilinear technique (Maleknejad and Najafi, 2011), the Taylor‐series expansion methods (Huabsomboona et al , 2010; Yalçinbaş, 2002), the Newton‐Kantorovich‐quadrature method (Saberi‐Nadjafi and Heidari, 2010), the Tau approximation (Ghoreishi and Hadizadeh, 2009), the infinite delay (Islam, 1991). In this paper we will develop a collocation method based on quintic B‐spline to approximate the unknown functions in equations (1) and (2) then the Gauss‐Kronrod‐Legendre quadrature formula in the case n =2 is used to approximate the linear and nonlinear Fredholm and Volterra integral equations of second kind.…”

Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind

“…Attestation to the widespread and continuing interest in the numerical solution of FIEs is reflected in the development and/or application of a diverse range of approximation techniques including (in chronological order of appearance) degenerate-kernel [5], multigrids [6], approximation theory [7], discrete product integration [8], Chebyshev polynomials [9,, Adomian decomposition [10], Taylor series [11,12], multi-FIE systems [13], Haar wavelets [14], fast matrix-vector algorithms [15] and piecewise-linear basis functions [16].…”