Tau Method Approximation Research Articles

On a fractional generalization of the free electron laser equation

The following fractional generalization of the free electron laser equation is investigated: D τ αh(τ)=λ∫ 0 τt δh(τ−t)Φ(b;δ+1; iνt) dt+μτ γΦ(β,γ+1; iντ), 0⩽τ⩽1, where β, γ, λ∈ C; ν, b, β∈ R, α>0, γ>−1 and δ>−1. A closed form solution is derived in terms of Kummer’s function Φ( α, β; z) by the application of Riemann–Liouville fractional integration operators. Tau method approximation is used in the evaluation of the results in a suitable form for numerical computation. The results derived are of general character and provide extension of the work reported by Boyadjiev et al. and Al-Shammery et al. [Math. Comput. Model. 25 (1997) 1; Integral Transform. Spec. Funct. 9 (2000) 81; J. Fract. Cal. Appl. Anal. 2 (1999) 501].

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

Tau method approximation of the Hubbell rectangular source integral

The Tau method is applied to obtain expansions, in terms of Chebyshev polynomials, which approximate the Hubbell rectangular source integral: I a, b =∫ b 0 1 1+x 2 arctan a 1+x 2 dx This integral corresponds to the response of an omni-directional radiation detector situated over a corner of a plane isotropic rectangular source. A discussion of the error in the Tau method approximation follows.

On a generalized fractional intero-differential equation of volterra-type

The following fractional generalization of the free electron laser (FEL) equation is studied: . An analytical treatment of this fractional integral equation leads to a closed of form solution in terms of Kummer functions φ(a,b;x). The solution is also expressed in terms of incomplete gamma functions . Finally, Tau method approximation is used in the numerical evaluation of the results.

Tau method approximation of a generalized Epstein-Hubbell elliptic-type integral

We consider the evaluation of a recent generalization of the Ep- stein-Hubbell elliptic-type integral using the tau method approximation with a Chebyshev polynomial basis. This also leads to an approximation of Lauricella’s hypergeometric function of three variables. Numerical results are given for polynomial approximations of degree 6.

Chebyshev series approximations for the Bessel function Yn( z) of complex argument

We employ the truncated Chebyshev series to approximate the Bessel function of the second kind Y n ( z) for | z| ≤ 8. Detailed manipulations and discussions for Y 0( z) and Y 1( z) are given. Results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the Chebyshev series approximations. Advantages and disadvantages of the Chebyshev series approximation compared with other polynomial approximation methods, e.g., the tau-method approximations, are discussed.

Analytical and numerical treatment of a fractional integro-differential equation of volterra-type

An analytical treatment of a fractional generalisation of the free electron laser (FEL) equation is given, whereby a closed form solution is obtained in terms of Kummer functions Φ( a, c; z). Subsequently, Tau method approximation of these functions facilitates the numerical evaluation of the solution of the FEL equation.

Tau method approximation of some integrals related to radiation field problems

A more general and unified form of the Hubbell rectangular source integral is defined as H a,b,p,⋋,μ,v δ,β,γ = ∞ b D ϰ ⋋(ϰ 2 + p) ν 1 − ϰ 2 b 2 μ F δ,beta;,γ;− a 2 ϰ 2 + p dϰ subject to γ > β > 0; a, b, p > 0; λ, μ > −1. Special cases of this integral occur in some radiation field problems. Under certain restrictions, H may be expressed in terms of Appell's double hypergeometric functions F 2 and F 4. Tau method approximations of the Gauss hypergeometric function 2 F 1 are employed in the evaluation of this integral, and upper bound estimates on the Tau error are given.

The uniform convergence of the Tau Method for singularly perturbed problems

We prove the stability and uniform convergence of Tau Method approximations to the analytic solution of a model singularly perturbed boundary value problem, independently of the values taken by the singularity parameter ϵ.

A note on the tau-method approximations for the Bessel functions Y0(z) and Y1(z)

This paper is to complete and improve the work reported in [1,2], using the Lanczos τ-method (in Coleman's version) to approximate the Bessel functions Y 0 ( z ) and Y 1 ( z ). We introduce symbolic representations of the scaled Faber polynomials on any fan-shaped section of the complex plane. These Faber polynomials are used as the perturbation terms in the τ-method. Numerical comparison among the power series, the Chebyshev series and the τ-method are conducted to show the accuracy improvement achieved by this new version of the τ-method. Some concluding remarks and suggestions on future research are given.

Ultra-high precision computations

We describe a machine independent Fortran subroutine which performs the four basic arithmetic operations with a degree of accuracy prescribed by the user. Tables of Chebyshev expansions of orders 48 and 50 for some basic mathematical functions are obtained as a result of applying this subroutine in conjunction with the recursive formulation of the Tau Method. A recently devised technique for the sharp determination of upper and lower error bounds for Tau Method approximations (see [1]) enables us to find the degree n required to achieve a prescribed accuracy ϵ over a given interval [ a, b]. A number of practical illustrations are given.

Error analysis of the Tau Method: Dependence of the error on the degree and on the length of the interval of approximation

In this paper, we discuss the dependence of Tau Method approximations on (i) the degree of approximant n and (ii) the length of the interval of approximation h. We shall show that the Tau Method parameters τ i (i) decay exponentially in terms of n and (ii) for a fixed n they decay in terms of h as ( h/2) n .

Error analysis of the Tau method: dependence of the approximation error on the choice of perturbation term

We consider a system of ordinary differential equations with constant coefficients and deduce asymptotic estimates for the Tau Method approximation error vector per step for different choices of the perturbation term H n ( x). The cases considered are Legendre polynomials, Chebyshev polynomials, powers of x and polynomials of the form ( x 2 − r 2) n, −r ⩽ x ⩽ r . The first two are standard choices for the Tau Method, for Chebyshev and Legendre series expansion techniques and also for collocation; the third one realizes the classical power series expansion techniques in the framework of the Tau Method and the last is related to the trial functions used in weighted residuals methods; we shall refer to it as the weighted residuals choice. We show that the resulting Tau Method implementations can be arranged into the following scale of increasing error estimates at the end point x = r : Legendre < Chebyshev ⪡ Power series < Weighted residuals. For the interesting case of Legendre Tau approximations, we offer upper and lower error bounds for the end point of the interval of approximation. In particular, this last estimates solve a conjecture on increased accuracy at the end point of the interval of approximation formulated by Lanczos in 1956. Such conjecture has equivalent forms for other polynomial methods for the numerical solution of differential equations. Although formulated in the convenient framework of the recursive Tau Method (see Ortiz [1]), the results given here apply, without essential modifications, to Chebyshev or Legendre series expansion techniques for differential equations, collocation and spectral methods. We give numerical examples which confirm the sharpness of the lemmas and theorems given in this paper. Finally, we discuss in an example the application of our results to the analysis of singularly perturbed differential equations.

Upper and lower error estimation for the Tau Method and related polynomial techniques

A new method is discussed by which estimates of upper and lower bounds of the maximum Tau Method approximation error are obtained. It improves on the upper bound estimate of Lanczos and other more recently proposed estimates. We give an example of a non-linear Ode where our technique is applied and show how it is implemented for Chebyshev series expansions, collocation and spectral methods.

Upper and lower error estimation for the Tau method: Some remarks on a problem of rational approximation

We show that an error analysis of Y.L. Luke for a rational Tau Method approximation of the solution of the first order differential equation with constant coefficients, a technique due to G. Meinardus and a new relation between Legendre and Bernstein polynomials we give in this paper, can be combined to give upper and lower error bounds for the error vector of a Tau Method rational approximation of a system of ordinary differential equation with constant coefficients.