Abstract and Applied Analysis

Volume 2013 (2013), Article ID 412028, 14 pages

http://dx.doi.org/10.1155/2013/412028

Research Article

Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation

M. M. Tharwat1,2 and S. M. Al-Harbi³

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

²Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

³Department of Mathematics, University College, Umm Al-Qura University, P.O. Box 8140, Makkah, Saudi Arabia

Received 5 August 2013; Accepted 21 September 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 M. M. Tharwat and S. M. Al-Harbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

Introduction

The mathematical modeling of many practical problems in mechanics and other areas of mathematical physics requires solutions of boundary value problems (see, [1–7]) and fractional differential equations (see, [8–13]). It is well known that many topics in mathematical physics require the investigation of the eigenvalues and eigenfunctions of Sturm-Liouville type boundary value problems. The literature on computing eigenvalues of various types of Sturm-Liouville problems is little and we refer to [14–17].

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

where is the sequences of sinc functions as follows:

Series (1) converges absolutely and uniformly on , cf. [18–21]. Sometimes, series (1) is called the derivative sampling theorem. Our task is to use (1) to compute eigenvalues of Sturm-Liouville problems with eigenvalue parameter in boundary conditions numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1), cf. [22]. 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

where is the truncated series as follows:

It is proved in [22] 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 , , and are bounded by a positive number , that is, . If satisfies the natural decay conditions

, then for , we have, [22],

where

and is the Euler-Mascheroni constant.

The classical [23] sampling theorem of Whittaker, Kotelnikov, and Shannon (WKS) for is the series representation as follows:

where the convergence is absolute and uniform on and it is uniform on compact sets of cf. [23–25]. Series (12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems, see for example, [17, 26–29]. The use of (12) in numerical analysis is known as the sinc method established by Stenger et al., cf. [30–32]. 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 [33] if and only if there is such that

Therefore, , that is, also has an expansion of the form (12). However, can also be obtained by term-by-term differentiation formula of (12) as follows:

see [23, page 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 (12) and (14), respectively.

Now, we consider the following differential equations:

with the following boundary conditions:

where is a complex spectral parameter, is assumed to be real valued and continuous on , and , satisfying

The eigenvalue problem (15)–(17) will be denoted by when . It is a Sturm-Liouville problem when the eigenparameter appears linearly in both boundary conditions. The classical problem when , which we denote by has a countable set of real and simple eigenvalues with as the only possible limit point, [34, 35]. In [14], the authors used Hermite-type sampling series (1) to compute the eigenvalues of problem numerically. In [36], see also [37], Annaby and Tharwat proved that has a denumerable set of real and simple eigenvalues with as the limit point using techniques similar of those established in [38–40], where also sampling theorems have been established. Similar results are established in [38] for the problem when the eigenparameter appears in one condition, that is, when , or equivalently when and . These problems will be denoted by , , 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 [41–43]. In Sections 2 and 3 we derive the Hermite interpolation technique to compute the eigenvalues of and with error estimates, respectively. The last section involves some illustrative examples.

2. Treatment of

In this section, we derive approximate values of the eigenvalues of . Let denote the solution of (15) satisfying the following initial conditions:

Thus, satisfies the boundary condition (16). The eigenvalues of the problem are the zeros of the function as follows:

These zeros are real and simple. The function is an entire function of . We aim to approximate and hence its zeros, that is, the eigenvalues by the use of the Hermite Interpolation. The idea is to split into two parts, one is known and the other is unknown, but lies in a Paley-Wiener space. Then we approximate the unknown part to get the approximate and then compute the approximate zeros. Using the method of variation of constants, the solution satisfies Volterra integral equation as follows:

where is the Volterra operator defined by

Differentiating (21), we get

where is the Volterra operator

Define and to be

In the following, we will make use of the estimates [44] as follows:

where is some constant (we may take ). For convenience, we define the constants by

From (21) and (25), we get

Lemma 1. For , , the following estimates hold:

Proof. We divide into two parts and and estimate each of them. Indeed, for and we have

Moreover, , ,

Combining (31) and (32), we obtain , ,

Applying Gronwall's inequality, cf. for example, [34, page 51], yields ,

from which we get

Then from (25) and (29), we obtain the estimate (30).

Now we split into two parts via

where is known part

and is unknown part

Then, from Lemma 1, we have the following lemma.

Lemma 2. The function is entire in and the following estimate holds:

Proof. Since

then from (29) and (30) we get (39).

The analyticity of and estimate (39) 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 let , be fixed. Let be the function

More specifications on will be given later on. Then we have the next lemma.

Lemma 3. is an entire function of which satisfies the estimates

Moreover, and

where

Proof. Since is entire, then also is entire in . Combining the estimates and (39), we obtain

leading to (42). Therefore, we get

That is, . Moreover, we get

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. (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 let be the approximations of the samples of and , respectively. Now we define , which approximates as

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

for a sufficiently small . From (42) we can see that satisfies the condition (9) when and therefore whenever we have

where there is a positive constant for which, cf. (10),

Here

In the following we use the technique of [26], where only truncation error analysis is considered to determine enclosure intervals for the eigenvalues, see also [41]. Let be an eigenvalue; 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 as follows:

Using the fact that

uniformly over any compact set and since is a simple root, we obtain the following for large and sufficiently small :

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 (60) is the interval

and in particular . Summarizing the above discussion, we arrive at the following lemma which is similar to that of [26].

Lemma 4. For any eigenvalue , we can find and sufficiently small such that   for . Moreover, we get

Proof. Since all eigenvalues of are simple, then for large and sufficiently small 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 (50) and (54) imply

and is chosen sufficiently small for which . Therefore, must be chosen so that for

Let be an eigenvalue and let be its approximation. Thus, and . From (68) we have . Now we estimate the error for an eigenvalue .

Theorem 5. Let be an eigenvalue of . For sufficient large we have the following estimate:

Proof. Since , then from (68) and after replacing by , we obtain

Using the mean value theorem yields that for some ,

Since the eigenvalues are simple, then for sufficiently large and we get (70).

3. The Case of

This section includes briefly a treatment similarly to that of the previous section for the eigenvalue problem introduced in Section 1. Notice that condition (18) implies that the analysis of problem is not included in that of . Let denote the solution of (15) satisfying the following initial conditions:

Thus, satisfies the boundary condition (16). The eigenvalues of the problem are the zeros of the function as follows:

Recall that has denumerable set of real and simple eigenvalues, cf. [38]. Using the method of variation of constants, the solution satisfies Volterra integral equation as follows:

where is the Volterra operator defined in (22). Differentiating (75), we get

where is the Volterra operator defined in (24). Define and to be

As in the preceding section we split into

where is the known part

and is the unknown one

Then, as in the previous section, is entire in for each for which

where .

Let and let be as in the previous section, but . Define to be

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. (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 let 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 . From (83) we can see that satisfies the condition (9) when and therefore whenever we have

where there is a positive constant for which, cf. (10), and

Here

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

Lemma 6. For any eigenvalue of the problem , we can find and sufficiently small such that for , where

are the solutions of the inequalities

Moreover, we get

Let . Then (88) and (92) imply

and is chosen sufficiently small for which . Therefore, must be chosen so that for

Let be an eigenvalue and be its approximation. Thus and . From (98) we have . Now we estimate the error for an eigenvalue . Finally we have the following estimate.

Theorem 7. Let be an eigenvalue of the problem . For sufficient large we have the following estimate

4. Numerical Examples

This section includes two detailed worked examples illustrating the above technique. Examples 1 and 2 computed in [27, 45] with the classical sinc method, where only truncation error analysis is considered, respectively. It is clearly seen that our new method (Hermite interpolations) gives remarkably better results than in [27, 45], see also [41–43]. We indicate in these examples the effect of the amplitude error in the method by determining enclosure intervals for different values of . We also indicate the effect of the parameters and 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 [27]

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

The function will be

As is clearly seen, the eigenvalues cannot be computed explicitly. Tables 1, 2, and 3 indicate the application of our technique to this problem and the effect of , and . By exact we mean the zeros of computed by Mathematica.

Figures 1 and 2 illustrate the enclosure intervals dominating for , , , and and respectively. The middle curve represents , while the upper and lower curves represent the curves of , respectively. We notice that when , the two curves are almost identical. Similarly, Figures 3 and 4 illustrate the enclosure intervals dominating for , , , and respectively.

Table 1: With , the approximation and the exact solution for different choices of and .

Exact
1.0493258679653497
3.5207214555369464
6.505146961583527
9.578576417519093

        θ = 1/9
1.0493258487568435
3.5207214557864277
6.5051469611825405
9.578576417536349

        θ = 1/12
1.0493291624256957
3.5207212860735546
6.5051469673141336
9.578576393056661

        θ = 1/5
1.0493258679653554
3.5207214555369624
6.505146961583516
9.57857641751921

        θ = 1/8
1.049325865990155
3.520721455899672
6.505146961464727
9.578576417559768

Table 2: Absolute error .

    θ = 1/9
1.92085 × 10⁻⁸
2.49481 × 10⁻¹⁰
4.00987 × 10⁻¹⁰
1.72555 × 10⁻¹¹

    θ = 1/12
3.29446 × 10⁻⁶
1.69463 × 10⁻⁷
5.73061 × 10⁻⁹
2.44624 × 10⁻⁸

    θ = 1/5
5.77316 × 10⁻¹⁵
1.59872 × 10⁻¹⁴
1.15463 × 10⁻¹⁴
1.1724 × 10⁻¹³

    θ = 1/8
1.97519 × 10⁻⁹
3.62725 × 10⁻¹⁰
1.188 × 10⁻¹⁰
4.0675 × 10⁻¹¹

Table 3: For , , and , the exact solution are all inside the interval [, ] for different values of ε.

Exact
1.0493258679653497
3.5207214555369464
6.505146961583527
9.578576417519093

,
[1.04294069, 1.05557896]
[3.51981844, 3.52162396]
[6.50375768, 6.50653831]
[9.55222305, 9.60712093]

,
[1.04932561, 1.049326118]
[3.52072141, 3.52072149]
[6.50514690, 6.50514702]
[9.57857529, 9.57857754]

, , , .

Figure 1: The enclosure interval dominating for , , , and .

Figure 2: The enclosure interval dominating for , , , and .

Figure 3: The enclosure interval dominating for , , , and .

Figure 4: The enclosure interval dominating for , , , and .

Example 2. The boundary value problem [45]

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

After computing , we obtain

As is clearly seen, the eigenvalues cannot be computed explicitly. As in the previous example, Figures 5, 6, 7, and 8 illustrate the results of Tables 4, 5, 6, and 7.

Table 4: With , the approximation and the exact solution for different choices of θ.

Exact
0.4828692021748484
1.966318052350425
4.827089429919572
7.919684444168381

         θ = 1/32
0.48286920221045176
1.96631805234574
4.827089429919605
7.919684444168366

         θ = 1/35
0.4828692337692527
1.966318047624416
4.8270894299720776
7.91968444416245

Table 5: Absolute error .

    θ = 1/32
3.56034 × 10⁻¹¹
4.68492 × 10⁻¹²
3.28626 × 10⁻¹⁴
1.5099 × 10⁻¹⁴

    θ = 1/35
3.15944 × 10⁻⁸
4.72601 × 10−9
5.25047 × 10⁻¹¹
5.93126 × 10⁻¹²

Table 6: The approximation and the exact solution for , and .

exact

0.4828692021748484698568637
0.4828692021748484678442680
2.012596 × 10⁻¹⁸

1.966318052350424642326091
1.9663180523504246423320204
5.93 × 10⁻²¹

4.8270894299195722717631337
4.8270894299195722717463715
1.6762 × 10⁻²⁰

7.9196844441683813942255769
7.9196844441683813942260057
4.29 × 10⁻²²

Table 7: For , and , the exact solution are all inside the interval [, ] for different values of ε.

Exact
0.4828692021748484
1.966318052350425
4.827089429919572
7.919684444168381

,, ε = 10⁻⁵
[0.47918888, 0.48651557]
[1.96592879, 1.96670680]
[4.82707252, 4.82710633]
[7.91968171, 7.919687175]

, ε = 10⁻¹⁰
[0.48284084, 0.48289756]
[1.96631794, 1.96631815]
[4.82708919, 4.82708966]
[7.919684437, 7.919684450]

, , , .

Figure 5: The enclosure interval dominating for , , , and .

Figure 6: The enclosure interval dominating for , , , and .

Figure 7: The enclosure interval dominating for , , , and .

Figure 8: The enclosure interval dominating for , , , and .

Acknowledgment

This research has been supported by a grant from the Institute of Scientific Research at Umm AL-Qura University, Saudi Arabia.

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