Numerical Algorithms for Computing Eigenvalues of Discontinuous Dirac System Using Sinc-Gaussian Method

and Applied Analysis 3 function of exponential type σ > 0, then for h ∈ 0, π/σ , ω : π − hσ /2, N ∈ N, | λ| < N, we have ∣f λ − ( Gh,Nf ) λ ∣∣ ≤ 2 ∣∣sin ( h−1πλ ∣∣ ∥f ∥∥ ∞ e−ωN √ πωN βN ( h−1 λ ) , λ ∈ C, 1.7


Introduction
The mathematical modeling of many practical problems in mechanics and other areas of mathematical physics requires solutions of boundary value problems see, e.g., 1-7 . Boundary value problems with discontinuity conditions arise in different branches of mathematics, mechanics, radio, electronics, geophysics, and other fields of natural science and technology. For example, discontinuous conditions inside an interval are connected with discontinuous or nonsmooth properties of media see [8][9][10][11] and references there in .
Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. A fundamental result in information theory is the Whittaker-Kotel'nikov-Shannon WKS sampling theorem 12-14 . It states that any f ∈ PW 2 σ , where PW 2 σ is the space of all entire functions of exponential type σ > 0 which lie in L 2 R when restricted to R, can be reconstructed from its sampled values {f nπ/σ : n ∈ Z} by the formula The speed of convergence of the series in 1.3 is determined by the decay of |Φ λ |. But the decay of an entire function of exponential type cannot be as fast as e −c|x| as |x| → ∞, for some positive c, 28 . In 29 , Qian has introduced the following regularized sampling formula. For h ∈ 0, π/σ , N ∈ N and r > 0, where G t : exp −t 2 , which is called the Gaussian function, S n h −1 πx : sinc h −1 πx − nπ , Z N x : {n ∈ Z : | h −1 x − n| ≤ N} and x denotes the integer part of x ∈ R, see also 30, 31 . Qian also derived the following error bound. If f ∈ PW 2 σ , h ∈ 0, π/σ and a : min{r π − hσ , N − 2 /r} ≥ 1, then In 28 , Schmeisser and Stenger extended the operator 1.4 to the complex domain C. For σ > 0, h ∈ 0, π/σ and ω : π − hσ /2, they defined the operator 28 , where Z N λ : {n ∈ Z : | h −1 λ 1/2 − n| ≤ N} and N ∈ N. Note that the summation limits in 1.6 depend on the real part of λ. Schmeisser and Stenger,28 , proved that if f is an entire Abstract and Applied Analysis 3 function of exponential type σ > 0, then for h ∈ 0, π/σ , ω : π − hσ /2, N ∈ N, | λ| < N, we have The amplitude error arises when the exact valuesf nh of 1.6 are replaced by the approximations f nh . We assume that f nh are close to f nh , that is, there is ε > 0, sufficiently small such that sup n∈Z n λ f nh − f nh < ε. 1.9 Let h ∈ 0, π/σ , ω : π − hσ /2 and N ∈ N be fixed numbers. The authors in 9 proved that if 1.9 is held, then for | λ| < N, we have We are concerned with the computation of eigenvalues the Dirac system where λ ∈ C; the real valued function p 1 · and p 2 · are continuous in −1, 0 and 0, 1 , and have finite limits In this paper we will use the sinc-Gaussian sampling formula 1.6 to compute eigenvalues of the Dirac system 1.12 -1. 16 . As expected, the new method reduced the error bounds remarkably, see examples at the end of this paper. Special attention is given to the comparison of the numerical results obtained by the new method with those found by classical sinc-method. We would like to mention that works in direction of computing eigenvalues with the sinc-Gaussian, are few, see for example, 9 . Also papers in computing of eigenvalues with discontinuous are few, see 10, 32 . However, the computing of eigenvalues by sinc-Gaussian technique which has discontinuity conditions, do not exist as for as we know.
The paper is organized as follows: Section 2 contains some preliminary results and the approximated values of the eigenvalues of the Dirac system with discontinuous. The method with error estimates are contained in Section 3. The last section involves some illustrative examples for showing the high accuracy of the proposed technique.

The Approximated Eigenvalues of Dirac System
In this section we derive approximate values of the eigenvalues of problem 1.12 -1.16 . Recall that problem 1.12 -1.16 has denumerable set of real and simple eigenvalues, compare with 33, 34 . Let be the solution of 1.12 satisfying the following initial conditions: Since φ ·, λ satisfies 1.13 , then the eigenvalues of the problem 1.12 -1.16 are the zeros of the function Δ λ −δ 2 sin βφ 12 1, λ cos βφ 22 1, λ .

2.8
For convenience, we define the constants

2.13
Using the inequalities | sin z| ≤ e | z| and | cos z| ≤ e | z| for z ∈ C, leads for λ ∈ C to

2.14
The above inequality can be reduced to Similarly, we can prove that In a similar manner, we will prove the following lemma for f 0,1 ·, λ and f 0,2 ·, λ .

2.18
Then from 2.4 and 2.5 and Lemma 2.1, we get

The Method and Error Analysis
In this section we derive the method of computing the eigenvalues of problem 1.12 -1.16 numerically. We aim to approximate Δ λ and hence its zeros, that is, the eigenvalues. The idea is to split Δ λ into two parts, one is known and the other is unknown, but is an entire function of exponential type. Then we approximate the unknown part using 1.6 to get the approximate Δ λ and then compute the approximate zeros. Now, let us split Δ λ into where U λ is the unknown part involving integral operators Abstract and Applied Analysis and K λ is the known part Then, from Lemmas 2.1 and 2.2, we have the following result.
Lemma 3.1. The function U λ is entire in λ and the following estimate holds:
Then U λ is an entire function of exponential type 2. In the following we let λ ∈ R since all eigenvalues are real. Now we approximate the function U λ using the operator 1.6 where h ∈ 0, π/2 and ω : π − 2h /2 and then we obtain The samples U nh Δ nh − K nh , n ∈ Z N λ cannot be computed explicitly in the general case. We approximate these samples numerically by solving the initial-value problems defined by 1.12 and 2.2 to obtain the approximate values U nh , n ∈ Z N λ , that is, U nh Δ nh − K nh . Accordingly we have the explicit expansion Therefore we get, compare with 1.10 ,

3.10
Abstract and Applied Analysis 9 Now let Δ N λ : K λ G h,N U λ . From 3.7 and 3.10 we obtain Let λ * be an eigenvalue and λ N be its desired approximation, that is, Δ λ * 0 and Δ N λ N 0. From 3.11 we have | Δ N λ * | ≤ T h,N λ * A ε,N 0 . Now we define an enclosure interval I ε,N for λ * . Define the curves The curves a λ , a − λ trap the curve of Δ λ for suitably large N. Hence the closure interval is determined by solving a ± λ 0, gives an interval I ε,N : a − , a . Next we estimate the error |λ * − λ N | for the eigenvalue λ * .

Theorem 3.2.
Let λ * be an eigenvalue of 1.12 -1.16 and λ N be its approximation. Then, for λ ∈ R, one has the following estimate: where the interval I ε,N is defined above.

Numerical Examples and Comparisons
This section includes two examples illustrating the sinc-Gaussian method. In the following examples, we consider λ k,N being the kth root of K λ G h,N U 0. Also, it is observed that the approximation λ k,N and the exact solution λ k are all inside the interval a − , a . We indicate in these two examples the effect of the amplitude error in the proposed method by determining enclosure intervals for different values of ε. All examples are computed in 32 by using the classical sinc method. We see that the sinc-Gaussian method gives remarkably better results.

4.2
Here therefore the eigenvalues are λ k 6kπ − 5 /12, k ∈ Z. Let E and E G denote the absolute errors associated with the results of classical sinc method and sinc-Gaussian method, respectively. In Table 1, we give comparison between the absolute error of sinc-Gaussian and the classical sinc-method.
In Table 2, we observe that the approximation λ k,N and the exact solution λ k are all inside the interval a − , a for different values of ε.
Example 4.2. In this example we consider the system  therefore the eigenvalues are λ k 6k 1 π − 5 /12, k ∈ Z. Table 3 gives comparison between the absolute error of sinc-Gaussian and the classical sinc-method. Moreover, in Table 4 the approximation λ k,N and the exact solution λ k are all inside the interval a − , a for different values of ε.