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Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation
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.
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. . 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  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, , where and is the Euler-Mascheroni constant.
The classical  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  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 , the authors used Hermite-type sampling series (1) to compute the eigenvalues of problem numerically. In , see also , 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  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  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:
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
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 , where only truncation error analysis is considered to determine enclosure intervals for the eigenvalues, see also . 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 .
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:
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. . 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