A Taylor expansion approach using Faà di Bruno's formula for solving nonlinear integral equations of the second and third kind

Zitoun, Feyed Ben; Cherruault, Y.

doi:10.1108/03684920910962687

Cited by 5 publications

(5 citation statements)

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“…Consider the following nonlinear Fredholm integral equation with exact solution y ( t )= e t : Equation 48 We apply the presented method equation (16) to solve this example and we applied our method with h=1/20 and using quintic polynomial the maximum absolute error in the solution at grid points x i =0,0.1, … ,1 are compared with the method given in Zitoun and Cherruault (2009) using polynomial of degree 7 the maximum absolute errors are tabulated in Table IV.…”

Section: Numerical Examplesmentioning

confidence: 99%

Collocation method for Fredholm and Volterra integral equations

Rashidinia

Mahmoodi

2013

Kybernetes

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PurposeThe purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.Design/methodology/approachThe solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.FindingsThe arising system of linear or nonlinear algebraic equations can solve the linear combination coefficients appearing in the representation of the solution in spline basic functions.Practical implicationsThe error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.Originality/valueThe paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.

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Section: Numerical Examplesmentioning

confidence: 99%

Collocation method for Fredholm and Volterra integral equations

Rashidinia

Mahmoodi

2013

Kybernetes

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“…To give a clear overview of the methodology, we have chosen to illustrate the method by the following examples. It is to be noted that Examples 4‐8 have also been treated and solved in our previous paper (Ben Zitoun and Cherruault, 2009a, b).…”

Section: Test Problemsmentioning

confidence: 83%

“…In our previous paper (Ben Zitoun and Cherruault, 2009a, b), we have introduced a technique for solving nonlinear boundary value problems. In the present work, our aim is to present a modified form of this technique.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this new paper is to present a method for solving nonlinear differential equations with constant and/or variable coefficients and with initial and/or boundary conditions. The proposed method rests in part on the ideas and the results of our previous paper (Ben Zitoun and Cherruault, 2009a, b). This method rests also on the following works: Adomian (1994), Kanwal and Liu (1989), Sezer (1994, 1996), Gülsu and Sezer (2006), Yalçinbaş (2002), Mahmoudi (2005), Maleknejad and Mahmoudi (2003), Wang (2006), Darania and Ebadian (2006) and Darania and Ivaz (2008).…”

Section: Introductionmentioning

confidence: 99%

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A method for solving nonlinear differential equations

Zitoun

Cherruault

2010

Kybernetes

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PurposeThe purpose of this paper is to present a method for solving nonlinear differential equations with constant and/or variable coefficients and with initial and/or boundary conditions.Design/methodology/approachThe method converts the nonlinear boundary value problem into a system of nonlinear algebraic equations. By solving this system, the solution is determined. Comparing the methodology with some known techniques shows that the present approach is simple, easy to use, and highly accurate.FindingsThe proposed technique allows us to obtain an approximate solution in a series form. Test problems are given to illustrate the pertinent features of the method. The accuracy of the numerical results indicates that the technique is efficient and well suited for solving nonlinear differential equations.Originality/valueThe present approach provides a reliable technique, which avoids the tedious work needed by classical techniques and existing numerical methods. The nonlinear problem is solved without linearizing or discretizing the nonlinear terms of the equation. The method does not require physically unrealistic assumptions, linearization, discretization, perturbation, or any transformation in order to find the solutions of the given problems.

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“…analytically) addressed by Taylor-method users, but the lack of a systematic formalization of the traditional approach could have limited their impact (see [93]- [103], for purely analytical approaches). However, if one considers studies with finite precision goal then, again, the traditional method has reached a higher level of development such as in the treatment of the differential algebraic case -a particular system of ODEs already well treated [23,32], [104]- [107]-or of integral and integro-differential equations (see, e.g., [96,108]). I will not discuss further the use of the DTM in such problems.…”

Section: Introductionmentioning

confidence: 99%

Status of the differential transformation method

Bervillier

2012

Applied Mathematics and Computation

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Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages -numerical routines-for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated. It is concluded that, for the sake of clarity, when the DTM is applied to ODEs, it should be mentioned that the DTM exactly coincides with the traditional Taylor method, contrary to what is currently written.

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A Taylor expansion approach using Faà di Bruno's formula for solving nonlinear integral equations of the second and third kind

Cited by 5 publications

References 39 publications

Collocation method for Fredholm and Volterra integral equations

Collocation method for Fredholm and Volterra integral equations

A method for solving nonlinear differential equations

Status of the differential transformation method

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