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

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.

Let and be the Paley-Wiener space of all -entire functions of exponential type σ. Assume that . Then can be reconstructed via the Hermite-type sampling series

where is the sequences of sinc functions

Series (1.1) converges absolutely and uniformly on ℝ, cf.[1-4]. Sometimes, series (1.1) is called the derivative sampling theorem. Our task is to use formula (1.1) 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 (1.1), 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 (1.1) is defined to be

where is the truncated series

It is proved in [5] that if and is sufficiently smooth in the sense that there exists such that , then, for , , we have

where the constants and are given by

The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be

where and are approximate samples of and , respectively. Let us assume that the differences , , , are bounded by a positive number ε, i.e., . If satisfies the natural decay conditions

, then for , we have, [5],

where

and is the Euler-Mascheroni constant.

The classical [6] sampling theorem of Whittaker, Kotel’nikov and Shannon (WKS) for is the series representation

where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ, cf.[6-8]. Series (1.12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, e.g., [9-15]. The use of (1.12) in numerical analysis is known as the sinc-method established by Stenger, cf. [16-18]. In [10,12], the authors applied (1.12) and the regularized sinc-method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [19,20]. In [12] 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 [21], if and only if there is such that

Therefore , i.e., also has an expansion of the form (1.12). However, can be also obtained by the term-by-term differentiation formula of (1.12)

see [[6], p.52] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples will be used to compute both and according to (1.12) and (1.14), respectively.

Consider the Dirac system which consists of the system of differential equations

and the boundary conditions

where and , , satisfying

The eigenvalue problem (1.15)-(1.17) will be denoted by when . It is a Dirac system when the eigenparameter λ appears linearly in both boundary conditions. The classical problem when , which we denote by , is studied in the monographs of Levitan and Sargsjan [22,23]. Annaby and Tharwat [24] used Hermite-type sampling series (1.1) to compute the eigenvalues of problem numerically. In [25], Kerimov proved that has a denumerable set of real and simple eigenvalues with ±∞ as the limit points. Similar results are established in [26] for the problem when the eigenparameter appears in one condition, i.e., when , or equivalently when and , where also sampling theorems have been established. These problems will be denoted by and , respectively. The aim of the present work is to compute the eigenvalues of and numerically by the Hermite interpolations with an error analysis. This method is based on sampling theorem, Hermite interpolations, but applied to regularized functions hence avoiding any (multiple) integration and keeping the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see [24]. In Sections 2 and 3, we derive the Hermite interpolation technique to compute the eigenvalues of Dirac systems with error estimates. We briefly derive some necessary asymptotics for Dirac systems’ spectral quantities. The last section contains three worked examples with comparisons accompanied by figures and numerics with the Lagrange interpolation method.

In this section we derive approximate values of the eigenvalues of . Recall that has a denumerable set of real and simple eigenvalues, cf.[25]. Let be a solution of (1.15) satisfying the following initial:

Here denotes the transpose of a matrix A. Since satisfies (1.16), then the eigenvalues of the problem are the zeros of the function

Similarly to [[22], p.220], and satisfy the system of integral equations

where and , , are the Volterra operators defined by

For convenience, we define the constants

Define and to be

As in [12] we split into two parts via

where is the known part

and is the unknown one

Then the function is entire in λ for each for which, cf.[12],

The analyticity of as well as estimate (2.11) are not adequate to prove that lies in a Paley-Wiener space. To solve this problem, we will multiply by a regularization factor. Let and , , be fixed. Let be the function

We choose θ sufficiently small for which . More specifications on m, θ will be given latter on. Then , see [12], is an entire function of λ which satisfies the estimate

Moreover, and

where

What we have just proved is that belongs to the Paley-Wiener space with . Since , then we can reconstruct the functions via the following sampling formula:

Let , , and approximate by its truncated series , where

Since all eigenvalues are real, then from now on we restrict ourselves to . Since , the truncation error, cf. (1.5), is given for by

where

The samples and , in general, are not known explicitly. So, we approximate them by solving numerically initial value problems at the nodes . Let and be the approximations of the samples of and , respectively. Now we define , which approximates

Using standard methods for solving initial problems, we may assume that for ,

for a sufficiently small ε. From (2.13) we can see that satisfies the condition (1.9) when and therefore whenever , we have

where there is a positive constant for which, cf. (1.10),

Here

In the following, we use the technique of [27], where only the truncation error analysis is considered, to determine enclosure intervals for the eigenvalues; see also [24,28]. Let be an eigenvalue with , that is,

Then it follows that

and so

Since is given and has computable upper bound, we can define an enclosure for by solving the following system of inequalities:

Its solution is an interval containing , and over which the graph

is squeezed between the graphs

and

Using the fact that

uniformly over any compact set, and since is a simple root, we obtain, for large N and sufficiently small ε,

in a neighborhood of . Hence the graph of intersects the graphs and at two points with abscissae and the solution of the system of inequalities (2.23) is the interval

and in particular . Summarizing the above discussion, we arrive at the following lemma which is similar to that of [27] for Sturm-Liouville problems.

Lemma 2.1For any eigenvalue, we can findand sufficiently smallεsuch thatfor. Moreover,

Proof Since all eigenvalues of are simple, then for large N and sufficiently small ε, we have in a neighborhood of . Choose such that

has two distinct solutions which we denote by . The decay of as and as will ensure the existence of the solutions and as and . For the second point, we recall that as and as . Hence, by taking the limit, we obtain

that is, . This leads us to conclude that since is a simple root.

Let . Then (2.17) and (2.21) imply

Therefore θ, m must be chosen so that for

Let be an eigenvalue and be its approximation. Thus and . From (2.27) we have . Now we estimate the error for an eigenvalue . □

Theorem 2.2Letbe an eigenvalue of. For sufficient largeN, we have the following estimate:

Moreover, whenand.

Proof Since , then from (2.27) and after replacing λ by , we obtain

Using the mean value theorem yields that for some ,

Since the eigenvalues are simple, then for sufficiently large N and we get (2.28). The rest of the proof follows from the fact that converges uniformly to in ℝ and when . □

This section includes briefly a treatment similar to that of the previous section for the eigenvalue problem introduced in Section 1 above. Notice that the condition (1.18) implies that the analysis of problem is not included in that of . Let be a solution of (1.15) satisfying the following initial:

Therefore, the eigenvalues of the problem in question are the zeros of the function

Similarly to [[22], p.220], satisfies the system of integral equations

where and , , are the Volterra operators defined in (2.5) above. Define and to be

As in [12] we split into

where is the known part

and is the unknown one

Then is entire in λ for each for which, see [12],

Define to be

where θ is sufficiently small, for which and m are as in the previous section, but . Hence

and with

where

Thus, belongs to the Paley-Wiener space with . Since , then we can reconstruct the functions via the following sampling formula:

Let , , and approximate by its truncated series , where

Since all eigenvalues are real, then from now on we restrict ourselves to . Since , the truncation error, cf. (1.5), is given for by

where

The samples and , in general, are not known explicitly. So, we approximate them by solving numerically initial value problems at the nodes . Let and be the approximations of the samples of and , respectively. Now we define , which approximates

Using standard methods for solving initial problems, we may assume that for ,

for a sufficiently small ε. From (2.13) we can see that satisfies the condition (1.9) when and therefore whenever , we have

where there is a positive constant for which, cf. (1.10),

Here

As in the above section, we have the following lemma.

Lemma 3.1For any eigenvalueof the problem, we can findand sufficiently smallεsuch thatfor, where

, are the solutions of the inequalities

Moreover,

Let . Then (3.15) and (3.19) imply

Therefore, θ, m must be chosen so that for ,

Let be an eigenvalue and be its approximation. Thus and . From (3.23) we have . Now we estimate the error for an eigenvalue . Finally, we have the following estimate.

Theorem 3.2Letbe an eigenvalue of the problem. For sufficient largeN, we have the following estimate:

Moreover, whenand.

In the following section, we have taken , where , in order to avoid the first singularity of .

This section includes three detailed worked examples illustrating the above technique accompanied by comparison with the sinc-method derived in [12]. It is clearly seen that the Hermite interpolations method gives remarkably better results. The first two examples are computed in [12] with the classical sinc-method where . But in the last example, where eigenvalues cannot be computed concretely, . By and we mean the absolute errors associated with the results of the classical sinc-method and our new method (Hermite interpolations), respectively. We indicate in these examples the effect of the amplitude error in the method by determining enclosure intervals for different values of ε. We also indicate the effect of the parameters m and θ by several choices. Each example is exhibited via figures that accurately illustrate the procedure near to some of the approximated eigenvalues. More explanations are given below. Recall that and are defined by

respectively. Recall also that the enclosure intervals and are determined by solving

respectively. We would like to mention that MATHEMATICA has been used to obtain the exact values for the three examples where eigenvalues cannot be computed concretely. MATHEMATICA is also used in rounding the exact eigenvalues, which are square roots.

Example 1

The boundary value problem

is a special case of the problem when , , and . Here the characteristic function is

The function will be

As is clearly seen, eigenvalues cannot be computed explicitly. Five tables indicate the application of our technique to this problem and the effect of ε, θ and m (Tables 1, 2, 3, 4 and 5). By exact, we mean the zeros of computed by Mathematica.

Figures 1 and 2 illustrate the comparison between and for different values of m and θ. Figures 3 and 4, for , and , illustrate the enclosure intervals for and , respectively. Also, Figures 5 and 6 illustrate the enclosure intervals for and , respectively, but for , .

Example 2

The Dirac system

is a special case of the problem treated in the previous section with , , and . The characteristic function is

The function will be

As in the previous example, Figures 7, 8, 9, 10, 11 and 12 illustrate the results of Tables 6, 7, 8, 9 and 10. Figures 7 and 8 illustrate the comparison between and for different values of m and θ. Figures 9 and 10, for , and , illustrate the enclosure intervals for and , respectively. Also, Figures 11 and 12 illustrate the enclosure intervals for and , respectively, but for , .

Example 3

The boundary value problem

is a special case of the problem when , , and . Here the characteristic function is

where and are Airy functions and , respectively, and and are derivatives of Airy functions. The function will be

Figures 13, 14 and Tables 11, 12 illustrate the applications of the method to this problem.

The author declares that he has no competing interests.

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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