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

Let σ > 0 and let $$ be the Paley-Wiener space of all L²(ℝ), entire functions of exponential type σ. Assume that $$. Then f(t) can be reconstructed via the Hermite-type sampling series as where S_n(t) is the sequences of sinc functions as follows:

Series (1) converges absolutely and uniformly on ℝ, 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 f_N(t) is the truncated series as follows: It is proved in [22] that if $$ and f(t) is sufficiently smooth in the sense that there exists k ∈ ℤ⁺ such that t^kf(t) ∈ L²(ℝ), then for t ∈ ℝ, |t | < Nπ/σ, we have where the constants E_k and ξ_k,σ are given by The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be where $$ and $$ are approximate samples of f(nπ/σ) and f^′(nπ/σ), respectively. Let us assume that the differences $$, $$, and n ∈ ℤ are bounded by a positive number ɛ, that is, $$. If $$ satisfies the natural decay conditions 0 < ν ≤ 1, then for $$, we have, [22], where and $$ is the Euler-Mascheroni constant.

The classical [23] sampling theorem of Whittaker, Kotelnikov, and Shannon (WKS) for $$ is the series representation as follows: where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ cf. [23–25]. Series (12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems, see for example, [17, 26–29]. The use of (12) in numerical analysis is known as the sinc method established by Stenger et al., cf. [30–32]. The aim of this paper is to investigate the possibilities of using Hermite interpolations rather than Lagrange interpolations, to compute the eigenvalues numerically. Notice that, due to Paley-Wiener′s theorem [33] $$ if and only if there is g(·) ∈ L²(−σ, σ) such that Therefore, $$, that is, f^′(t) also has an expansion of the form (12). However, f^′(t) 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 f(nπ/σ) will be used to compute both f(t) and f^′(t) according to (12) and (14), respectively.

Now, we consider the following differential equations: with the following boundary conditions: where μ is a complex spectral parameter, q(·) is assumed to be real valued and continuous on [0,1], and $$, i = 0,1 satisfying The eigenvalue problem (15)–(17) will be denoted by Π(q, a, b, a^′, b^′) when $$. It is a Sturm-Liouville problem when the eigenparameter μ appears linearly in both boundary conditions. The classical problem when $$, which we denote by Π(q, a, b, 0,0) has a countable set of real and simple eigenvalues with ∞ as the only possible limit point, [34, 35]. In [14], the authors used Hermite-type sampling series (1) to compute the eigenvalues of problem Π(q, a, b, 0,0) numerically. In [36], see also [37], Annaby and Tharwat proved that Π(q, a, b, a^′, b^′) has a denumerable set of real and simple eigenvalues with ∞ as the limit point using techniques similar of those established in [38–40], where also sampling theorems have been established. Similar results are established in [38] for the problem when the eigenparameter appears in one condition, that is, when $$, $$ or equivalently when $$ and $$. These problems will be denoted by Π(q, a, b, 0, b^′), Π(q, a, b, a^′, 0), respectively. The aim of the present work is to compute the eigenvalues of Π(q, a, b, a^′, b^′), Π(q, a, b, 0, b^′), and Π(q, a, b, a^′, 0) 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 Π(q, a, b, a^′, b^′) and Π(q, a, b, 0, b^′) with error estimates, respectively. The last section involves some illustrative examples.

In this section, we derive approximate values of the eigenvalues of Π(q, a, b, a^′, b^′). Let y(·, μ) denote the solution of (15) satisfying the following initial conditions: Thus, y(·, μ) satisfies the boundary condition (16). The eigenvalues of the problem Π(q, a, b, a^′, b^′) are the zeros of the function as follows: These zeros are real and simple. The function Δ(μ) is an entire function of μ. We aim to approximate Δ(μ) and hence its zeros, that is, the eigenvalues by the use of the Hermite Interpolation. The idea is to split Δ(μ) 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 Δ(μ) and then compute the approximate zeros. Using the method of variation of constants, the solution y(x, μ) satisfies Volterra integral equation as follows: where T is the Volterra operator defined by Differentiating (21), we get where $$ is the Volterra operator Define f(·, μ) and g(·, μ) to be In the following, we will make use of the estimates [44] as follows: where c₀ is some constant (we may take c₀≃1.72). For convenience, we define the constants by From (21) and (25), we get

Now we split Δ(μ) into two parts via where 𝒢(μ) is known part and 𝒮(μ) is unknown part Then, from Lemma 1, we have the following lemma.

The analyticity of 𝒮(μ) and estimate (39) are not adequate to prove that 𝒮(μ) lies in a Paley-Wiener space. To solve this problem, we will multiply 𝒮(μ) by a regularization factor. Let θ ∈ (0,1) and let m ∈ ℤ⁺, m > 4 be fixed. Let ℱ_θ,m(μ) be the function More specifications on m, θ will be given later on. Then we have the next lemma.

What we have just proved is that ℱ_θ,m(μ) belongs to the Paley-Wiener space $$ with σ = 1 + mθ. Since $$, then we can reconstruct the functions ℱ_θ,m(μ) via the following sampling formula: Let N ∈ ℤ⁺, N > m and approximate ℱ_θ,m(μ) by its truncated series ℱ_θ,m,N(μ), where Since all eigenvalues are real, then from now on we restrict ourselves to μ ∈ ℝ. Since μ^m−4ℱ_θ,m(μ) ∈ L²(ℝ), the truncation error, cf. (5), is given for |μ | < Nπ/σ by where The samples $$ and $$, in general, are not known explicitly. So we approximate them by solving numerically 8N + 4 initial value problems at the nodes $$. Let $$ and let $$ be the approximations of the samples of $$ and $$, respectively. Now we define $$, which approximates ℱ_θ,m,N(μ) as Using standard methods for solving initial problems, we may assume that for |n | < N, for a sufficiently small ɛ. From (42) we can see that ℱ_θ,m(μ) satisfies the condition (9) when m > 4 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 $$ has computable upper bound, we can define an enclosure for μ^* by solving the following system of inequalities: Its solution is an interval containing μ^*, and over which the graph $$ is squeezed between the graphs as follows: Using the fact that uniformly over any compact set and since μ^* is a simple root, we obtain the following for large N and sufficiently small ɛ: in a neighborhood of μ^*. Hence, the graph of $$ intersects the graphs −|sin θμ/θμ|^−m(T_N,m−4,σ(μ) + 𝒜(ɛ)) and |sin θμ/θμ|^−m(T_N,m−4,σ(μ) + 𝒜(ɛ)) at two points with abscissae a₋(μ^*, N, ɛ) ≤ a₊(μ^*, N, ɛ) and the solution of the system of inequalities (60) is the interval and in particular μ^* ∈ I_ɛ,N. Summarizing the above discussion, we arrive at the following lemma which is similar to that of [26].

Let $$. Then (50) and (54) imply and θ is chosen sufficiently small for which |θμ | < π. Therefore, θ, m must be chosen so that for |μ | < Nπ/σ Let μ^* be an eigenvalue and let μ_N be its approximation. Thus, Δ(μ^*) = 0 and $$. From (68) we have $$. Now we estimate the error |μ^* − μ_N| for an eigenvalue μ^*.

This section includes briefly a treatment similarly to that of the previous section for the eigenvalue problem Π(q, a, b, 0, b^′) introduced in Section 1. Notice that condition (18) implies that the analysis of problem Π(q, a, b, 0, b^′) is not included in that of Π(q, a, b, a^′, b^′). Let ψ(·, μ) denote the solution of (15) satisfying the following initial conditions: Thus, ψ(·, μ) satisfies the boundary condition (16). The eigenvalues of the problem Π(q, a, b, 0, b^′) are the zeros of the function as follows: Recall that Π(q, a, b, a^′, b^′) has denumerable set of real and simple eigenvalues, cf. [38]. Using the method of variation of constants, the solution ψ(x, μ) satisfies Volterra integral equation as follows: where T is the Volterra operator defined in (22). Differentiating (75), we get where $$ is the Volterra operator defined in (24). Define h₁(·, μ) and h₂(·, μ) to be As in the preceding section we split Ω(μ) into where 𝒦(μ) is the known part and 𝒰(μ) is the unknown one Then, as in the previous section, 𝒰(μ) is entire in μ for each x ∈ [0,1] for which where $$.

Let θ ∈ (0,1) and let m be as in the previous section, but m > 2. Define ℛ_m,θ(μ) to be Hence, and μ^m−2ℛ_m,θ(μ) ∈ L²(ℝ) with where Thus, ℛ_m,θ(μ) belongs to the Paley-Wiener space $$ with σ = 1 + mθ. Since $$, then we can reconstruct the functions ℛ_θ,m(μ) via the following sampling formula: Let N ∈ ℤ⁺, N > m, and approximate ℛ_θ,m(μ) by its truncated series ℛ_θ,m,N(μ), where Since all eigenvalues are real, then from now on we restrict ourselves to μ ∈ ℝ. Since μ^m−2ℛ_θ,m(μ) ∈ L²(ℝ), the truncation error, cf. (5), is given for |μ | < Nπ/σ by where The samples $$ and $$, in general, are not known explicitly. So we approximate them by solving numerically 4N + 2 initial value problems at the nodes $$. Let $$ and let $$ be the approximations of the samples of $$ and $$, respectively. Now we define $$, which approximates ℛ_θ,m,N(μ)

Using standard methods for solving initial problems, we may assume that for |n | < N for a sufficiently small ɛ. From (83) we can see that ℛ_θ,m(μ) satisfies the condition (9) when m > 2 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.

Let $$. Then (88) and (92) imply and θ is chosen sufficiently small for which |θμ | < π. Therefore, θ, m must be chosen so that for |μ | < Nπ/σ Let μ^* be an eigenvalue and μ_N be its approximation. Thus Ω(μ^*) = 0 and $$. From (98) we have $$. Now we estimate the error |μ^* − μ_N| for an eigenvalue μ^*. Finally we have the following estimate.

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 ɛ. 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 a_±(μ) and b_±(μ) are defined by respectively. Recall also that the enclosure intervals I_ɛ,N : = [a₋, a₊] and ℐ_ɛ,N : = [b₋, b₊] 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.