Tau method approximation of the Hubbell rectangular source integral

Numerical Methods Partial

Shakeri

2009

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second-order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included.

Section: Legenrde Tau Methodsmentioning

confidence: 99%

The numerical solution of the second Painlevé equation

Numerical Methods Partial

Shakeri

2009

“…The number of the unknown coefficients a ij , b i and c i is equal to (n + 1)(m + 1) + 2(n + 1) and can be obtained from Equations (33)-(37). Consequently, u n,m (x, t) given in Equation (21), p n (t) given in Equation (19) and q n (t) given in Equation (20) can be calculated.…”

Section: A Saadatmandi and M Dehghanmentioning

confidence: 99%

“…Recently, there have been several published works in the literature on the applications of the tau method, for instance see [19,22,25,26]. The method consists of expanding the required approximate solution as the elements of a complete set of orthogonal polynomials.…”

mentioning

confidence: 99%

Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

Saadatmandi

International Journal of Computer Mathematics

2010

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

“…The method consists of expanding the required approximate solution as the elements of a complete set of orthogonal polynomials [23,24]. Recently there have been several published works in the literature on the applications of the tau method [25][26][27][28]. For more details of Legendre polynomials see [29,30] and also some technique for solving integrodifferential equations can be found in [31,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

Saadatmandi

Computers & Mathematics with Applications

2010

a b s t r a c tThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.