Computing eigenvalues and Hermite interpolation for Dirac systems with eigenparameter in boundary conditions

Tharwat Boundary Value Problems 2013, 2013:36 http://www.boundaryvalueproblems.com/content/2013/1/36 RESEARCH Open Access Computing eigenvalues and Hermite interpolation for Dirac systems with eigenparameter in boundary conditions Mohammed M Tharwat* * Correspondence: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Abstract Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper we use the derivative sampling theorem ‘Hermite interpolations’ to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Using computable error bounds, we obtain eigenvalue enclosures. Examples with tables and illustrative figures are given. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc-method in Annaby and Tharwat (BIT Numer. Math. 47:699-713, 2007) and explain that the Hermite interpolations method gives remarkably better results. MSC: 34L16; 94A20; 65L15 Keywords: Dirac systems; eigenvalue problems with eigenparameter in the boundary conditions; Hermite interpolations; truncation error; amplitude error; sinc methods 1 Introduction Let σ >  and PW σ be the Paley-Wiener space of all L (R)-entire functions of exponential type σ . Assume that f (t) ∈ PW σ ⊂ PW σ . Then f (t) can be reconstructed via the Hermitetype sampling series f (t) =     ∞    nπ  nπ sin(σ t – nπ) f Sn (t) , Sn (t) + f  σ σ σ n=–∞ (.) where Sn (t) is the sequences of sinc functions ⎧ ⎨ sin(σ t–nπ ) , t = nπ , (σ t–nπ ) σ Sn (t) := ⎩, t = nπ . (.) σ Series (.) converges absolutely and uniformly on R, cf. [–]. Sometimes, series (.) is called the derivative sampling theorem. Our task is to use formula (.) 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 (.), © 2013 Tharwat; 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 Boundary Value Problems 2013, 2013:36 http://www.boundaryvalueproblems.com/content/2013/1/36 Page 2 of 21 cf. []. 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 (.) is defined to be N ∈ Z+ , t ∈ R, RN (f )(t) := f (t) – fN (t), (.) where fN (t) is the truncated series fN (t) =       nπ  nπ sin(σ t – nπ) f Sn (t) . Sn (t) + f  σ σ σ |n|≤N (.) It is proved in [] that if f (t) ∈ PW σ and f (t) is sufficiently smooth in the sense that there exists k ∈ Z+ such that t k f (t) ∈ L (R), then, for t ∈ R, |t| < Nπ/σ , we have RN (f )(t) ≤ TN,k,σ (t)   ξk,σ Ek | sin σ t|   := √ + (Nπ – σ t)/ (Nπ + σ t)/ (N + )k   ξk,σ (σ Ek + kEk– )| sin σ t|   + +√ , √ σ (N + )k Nπ – σ t Nπ + σ t (.) where the constants Ek and ξk,σ are given by ∞  t k f (t) dt, Ek := ξk,σ := –∞ σ k+/ . √ π k+  – –k (.) The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be A(ε, f )(t) = ∞       nπ nπ – f Sn (t) σ σ n=–∞      sin(σ t – nπ) nπ  nπ   Sn (t) , –f + f σ σ σ f t ∈ R, (.) ) and  f  ( nπ ) are approximate samples of f ( nπ ) and f  ( nπ ), respectively. Let us where  f ( nπ σ σ σ σ nπ nπ nπ   assume that the differences εn := f ( σ ) – f ( σ ), εn := f ( σ ) –  f  ( nπ ), n ∈ Z, are bounded by σ   a positive number ε, i.e., |εn |, |εn | ≤ ε. If f (t) ∈ PW σ satisfies the natural decay conditions  nπ |εn | ≤ f σ f (t) ≤  Mf , |t|α+ , εn ≤ f   nπ σ  , t ∈ R – {}, (.) (.) √  < ω ≤ , then for  < ε ≤ min{π/σ , σ /π, / e}, we have, [],   A(ε, f ) ≤ ∞   e/ √ e( + σ ) + (π/σ )A + Mf ρ(ε) σ (ω + )    + σ +  + log() Mf ε log(/ε), (.) Tharwat Boundary Value Problems 2013, 2013:36 http://www.boundaryvalueproblems.com/content/2013/1/36 Page 3 of 21 where σ A := π   ω  σ f () + Mf , π ρ(ε) := γ +  log(/ε), (.)  and γ := limn→∞ [ nk= k – log n] ∼ = . is the Euler-Mascheroni constant. The classical [] sampling theorem of Whittaker, Kotel’nikov and Shannon (WKS) for f ∈ PW σ is the series representation f (t) =   ∞  nπ Sn (t), f σ n=–∞ t ∈ R, (.) where the convergence is absolute and uniform on R and it is uniform on compact sets of C, cf. [–]. Series (.), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, e.g., [–]. The use of (.) in numerical analysis is known as the sinc-method established by Stenger, cf. [–]. In [, ], the authors applied (.) and the regularized sinc-method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [, ]. In [] the Dirac system has an eigenparameter appearing in the boundary conditions. The aim of this paper is to investigate the possibilities of using Hermite interpolations rather than Lagrange interpolations, to compute the eigenvalues numerically. Notice that, due to Paley-Wiener’s theorem [], f ∈ PW σ if and only if there is g(·) ∈ L (–σ , σ ) such that  f (t) = √ π σ g(x)eixt dx. (.) –σ Therefore f  (t) ∈ PW σ , i.e., f  (t) also has an expansion of the form (.). However, f  (t) can be also obtained by the term-by-term differentiation formula of (.) f  (t) =   ∞  nπ  Sn (t), f σ n=–∞ (.) see [, p.] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples f ( nπ ) will be used to compute both σ f (t) and f  (t) according to (.) and (.), respectively. Consider the Dirac system which consists of the system of differential equations u (x) – r (x)u (x) = λu (x), u (x) + r (x)u (x) = –λu (x), x ∈ [, ] (.) and the boundary conditions   α u () – α u () = –λ α u () – α u () ,   β u () – β u () = –λ β u () – β u () , (.) (.) where r (·), r (·) ∈ L (, ) and αi , βi , αi , βi ∈ R, i = , , satisfying      α , α = (, ) or α α – α α >  and      β , β = (, ) or β β – β β >  . (.) Tharwat Boundary Value Problems 2013, 2013:36 http://www.boundaryvalueproblems.com/content/2013/1/36 Pag (...truncated)