Computation of eigenvalues of discontinuous dirac system using Hermite interpolation technique

Bhrawy, Tharwat

Description

We use the derivative sampling theorem (Hermite interpolations) to compute eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity numerically. We closely follow the analysis derived by Levitan and Sargsjan (1975) to establish the needed relations. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Numerical examples, illustrations and comparisons with the sinc methods are exhibited. Mathematical Subject Classification 2010: 34L16; 94A20; 65L15.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

Tharwat and BhrawyAdvances in Difference Equations2012,2012:59 http://www.advancesindifferenceequations.com/content/2012/1/59

R E S E A R C HOpen Access Computation of eigenvalues of discontinuous dirac system using Hermite interpolation technique 1,2* 1,2 Mohammed M Tharwatand Ali H Bhrawy

* Correspondence: [e-mail] 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Full list of author information is available at the end of the article

Abstract We use the derivative sampling theorem (Hermite interpolations) to compute eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity numerically. We closely follow the analysis derived by Levitan and Sargsjan (1975) to establish the needed relations. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Numerical examples, illustrations and comparisons with the sinc methods are exhibited. Mathematical Subject Classification 2010:34L16; 94A20; 65L15. Keywords:Dirac systems, Hermite interpolations, transmission conditions, discontinu ous boundary value problems, truncation and amplitude errors, sinc methods

1 Introduction 2 2 Lets> 0 and PW be the PaleyWiener space of allL(ℝ)entire functions of exponen 2 2 tial type types. Assume that f(t)∈PW⊂PW . Thenf(t) can be reconstructed via the sampling series ∞       nπ nπsin(σt−nπ) 2 f(t) =f S(t) +f S n (t) (1) n σ σσ =−∞ whereS n (t) is the sequences of sinc functions  sin(σt−nπ)nπ   ,t= (σt−nπ)σ S n (t) := (2) nπ   1,t= .

Series (1) converges absolutely and uniformly onℝ(cf. [14]). Sometimes, series (1) is called the derivative sampling theorem. Our task is to use formula (1) to compute eigenvalues of Dirac systems numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1) (cf. [5]). Both types of errors normally appear in numerical techniques that use interpolation procedures. In the following we summarize these estimates. The truncation error associated with (1) is defined to be

© 2012 Tharwat and Bhrawy; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tharwat and BhrawyAdvances in Difference Equations2012,2012:59 http://www.advancesindifferenceequations.com/content/2012/1/59

+ R N f t:=f t−f N t,N∈Z,t∈R (3) wheref N (t) is the truncated series       nπnπsin(σt−nπ) 2 f N (t) =f S(t) +f S n (t) n (4) σ σσ n≤N 2 It is proved in [5] that if f(t)∈PW andf(t) is sufficiently smooth in the sense that +k2 there existskÎℤ such thatt f(t)ÎL(ℝ), then, fortÎℝ, |t| <Nπ/s, we have 2 ξ k,σ E k |sinσt|1 1 |R N (f)(t)| ≤T N,k,σ (t+) := k3/2 3/2 3(N ( + 1) Nπ−σt) (Nπ+σt) (5)   2 ξ k,σ (σE k +k E k−1 )|sinσt|1 1 + +√ k σN+ 1 Nπ−σt N π+σt

where the constantsE k andξ k,s are given by

∞   k2 E k :=|t f(t)|dt, −∞

k+1/2 σ ξ k,σ := k+1−k π1−4

(6)

The amplitude error occurs when approximate samples are used instead of the exact ones, which we can not compute. It is defined to be    ∞  nπnπ ˜ 2 A (ε,f)(t) =f−f S(t) n σ σ n=−∞ (7)    nπnπsin(σt−nπ) ˜   +f−f S n (t) ,t∈R σ σσ   nπnπ ˜ ˜ nπ  nπ  where f and f− are approximate samples of f and f− , respectively. Let us assume that the differences      ˜ ˜ nπnπ   nπ  nπ −f,ε:= ε n :=f n f−f−,n∈Z, are bounded by a positive num  2 berε, i.e. |ε n |,|ε| ≤ε If f(t)∈PW satisfies the natural decay conditions    nπnπ     | ε n | ≤  f  ,ε n ≤  f  (8)

f  f(t)≤,t∈R− {0} (9) +1 t 0 <l≤1, then for 0< ε≤minπ/σ,σ/π, 1/e we have, [5], 1/4   4e || A (ε,f)|| ∞ ≤3e(1 +σ) + ((π/σ)A+M f )ρ(ε) + (σ+ 2 + log(2))M f εlog(1/ε) (10) σ + 1 where    3σ σ A:=|f(0)|+M f ,ρ(ε) :=γ+ 10 log(1/ε) (11) π π  n 1 ∼ and γ:= lim−logn= 0.57721 is the EulerMascheroni constant. n→∞ k=1 k

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