Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

Abstract

The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.

1. Introduction

The mathematical modeling of many practical problems in mechanics and other areas of mathematical physics requires solutions of boundary value problems (see, for instance, [1–7]). 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 [8–15].

Sampling theory is one of the most powerful results in signal analysis. It is of great need in signal processing to reconstruct (recover) a signal (function) from its values at a discrete sequence of points (samples). If this aim is achieved, then an analog (continuous) signal can be transformed into a digital (discrete) one and then it can be recovered by the receiver. If the signal is band-limited, the sampling process can be done via the celebrated Whittaker, Shannon, and Kotel’nikov (WSK) sampling theorem [16–18]. By a band-limited signal with band width 𝜎, 𝜎>0, that is, the signal contains no frequencies higher than 𝜎/2𝜋 cycles per second (cps), we mean a function in the Paley-Wiener space 𝐵²_𝜎 of the entire functions of the exponential type at most 𝜎 which are 𝐿²(ℝ)-functions when restricted to ℝ. Assume that 𝑓(𝑡)∈𝐵²_𝜎⊂𝐵²_2𝜎. Then 𝑓(𝑡) can be reconstructed via the Hermite-type sampling series 𝑓(𝑡)=^∞_𝑛=−∞𝑓𝑛𝜋𝜎𝑆²_𝑛(𝑡)+𝑓^′𝑛𝜋𝜎sin(𝜎𝑡−𝑛𝜋)𝜎𝑆_𝑛(𝑡), (1) where 𝑆_𝑛(𝑡) is the sequences of sinc functions 𝑆_𝑛(𝑡)()∶=sin𝜎𝑡−𝑛𝜋(𝜎𝑡−𝑛𝜋),𝑡≠𝑛𝜋𝜎,1,𝑡=𝑛𝜋𝜎. (2) Series (1) converges absolutely and uniformly on ℝ (cf. [19–22]). Sometimes, series (1) is called the derivative sampling theorem. Our task is to use formula (1) to compute the eigenvalues numerically of differential equation −𝔶^′′𝑥,𝜇+𝑞(𝑥)𝔶𝑥,𝜇=𝜇²𝔶,[)∪(],𝑥,𝜇𝑥∈−1,00,1 (3) with boundary conditions ℒ₁(𝔶)∶=𝛼^′₁𝜇²−𝛼₁𝔶−𝛼−1,𝜇_′2𝜇²−𝛼₂𝔶^′ℒ−1,𝜇=0,₂(𝔶)∶=𝛽^′₁𝜇²+𝛽₁𝔶1,𝜇−𝛽^′₂𝜇²+𝛽₂𝔶^′1,𝜇=0 (4)(5) and transmission conditions ℒ₃(𝔶)∶=𝛾₁𝔶0⁻,𝜇−𝛿₁𝔶0⁺ℒ,𝜇=0,₄(𝔶)∶=𝛾₂𝔶^′0⁻,𝜇−𝛿₂𝔶^′0⁺,𝜇=0, (6) where 𝜇 is a complex spectral parameter; 𝑞(𝑥) is a given real-valued function, which is continuous in [−1,0) and (0,1] and has a finite limit 𝑞(0^±)=lim_𝑥→0^±𝑞(𝑥); 𝛾_𝑖, 𝛿_𝑖, 𝛼_𝑖, 𝛽_𝑖, 𝛼^′_𝑖, and 𝛽^′_𝑖 (𝑖=1,2) are real numbers; 𝛾_𝑖≠0, 𝛿_𝑖≠0 (𝑖=1,2); 𝛾₁𝛾₂=𝛿₁𝛿₂; and 𝛼det^′1𝛼₁𝛼^′₂𝛼₂𝛽>0,det^′1𝛽₁𝛽^′₂𝛽₂>0. (7) The eigenvalue problem (3)–(6) will be denoted by Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) when (𝛼^′₁,𝛼^′₂)≠(0,0)≠(𝛽^′₁,𝛽^′₂). It is a Sturm-Liouville problem which contains an eigenparameter 𝜇 in two boundary conditions, in addition to an internal point of discontinuity.

This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1) (cf. [23]). 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 ℛ_𝑁𝑓(𝑡)∶=𝑓(𝑡)−𝑓_𝑁(𝑡),𝑁∈ℤ⁺,𝑡∈ℝ, (8) where 𝑓_𝑁(𝑡) is the truncated series 𝑓_𝑁(𝑡)=_{|𝑛|≤𝑁}𝑓𝑛𝜋𝜎𝑆²_𝑛(𝑡)+𝑓^′𝑛𝜋𝜎sin(𝜎𝑡−𝑛𝜋)𝜎𝑆_𝑛(𝑡). (9) It is proved in [23] that if 𝑓(𝑡)∈𝐵²_𝜎 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 ℛ_𝑁𝑓(𝑡)≤𝒯_{𝑁,𝑘,𝜎}(𝑡)𝜂∶=^𝑘,𝜎ℰ_𝑘|sin𝜎𝑡|²√3(𝑁+1)^𝑘×1(𝑁𝜋−𝜎𝑡)^3/2+1(𝑁𝜋+𝜎𝑡)^3/2+𝜂_𝑘,𝜎𝜎ℰ_𝑘+𝑘ℰ_𝑘−1|sin𝜎𝑡|²𝜎(𝑁+1)^𝑘×1√+1𝑁𝜋−𝜎𝑡√,𝑁𝜋+𝜎𝑡 (10) where the constants ℰ_𝑘 and 𝜂_𝑘,𝜎 are given by ℰ_𝑘∶=^∞_−∞𝑡^𝑘𝑓(𝑡)²𝜂𝑑𝑡,_𝑘,𝜎𝜎∶=^𝑘+1/2𝜋^𝑘+11−4^−𝑘. (11) The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be 𝒜(𝑡)=𝜀,𝑓_{∞𝑛=−∞}𝑓𝑛𝜋𝜎−𝑓𝑛𝜋𝜎𝑆²_𝑛(𝑡)+𝑓^′𝑛𝜋𝜎−𝑓^′𝑛𝜋𝜎×sin(𝜎𝑡−𝑛𝜋)𝜎𝑆_𝑛(𝑡),𝑡∈ℝ, (12) where 𝑓(𝑛𝜋/𝜎) and 𝑓^′(𝑛𝜋/𝜎) are approximate samples of 𝑓(𝑛𝜋/𝜎) and 𝑓^′(𝑛𝜋/𝜎), respectively. Let us assume that the differences 𝜀_𝑛∶=𝑓(𝑛𝜋/𝜎)−𝑓(𝑛𝜋/𝜎), 𝜀^′_𝑛∶=𝑓^′(𝑛𝜋/𝜎)−𝑓^′(𝑛𝜋/𝜎), 𝑛∈ℤ, are bounded by a positive number 𝜀; that is, |𝜀_𝑛|,|𝜀^′_𝑛|≤𝜀. If 𝑓(𝑡)∈𝐵²_𝜎 satisfies the natural decay conditions 𝜀_𝑛≤𝑓𝑛𝜋𝜎,𝜀^′_𝑛≤𝑓^′𝑛𝜋𝜎,𝑓(𝑡≤ℳ)^𝑓|𝑡|^𝛼+1,𝑡∈ℝ−{0}, (13)(14)0<𝛼≤1, then, for 0<𝜀≤min{𝜋/𝜎,𝜎/𝜋,1/√𝑒}, we have [23] 𝒜𝜀,𝑓_∞≤4𝑒^1/4𝜎()×√𝛼+13𝑒(1+𝜎)+(𝜋/𝜎)𝐴+ℳ_𝑓𝜌(𝜀)+𝜎+2+log(2)ℳ_𝑓1𝜀log𝜀, (15) where 𝐴∶=3𝜎𝜋𝑓(0)+ℳ_𝑓𝜎𝜋^𝛼,𝜌(𝜀)1∶=𝛾+10log𝜀, (16) and 𝛾∶=lim_𝑛→∞[∑^𝑛_𝑘=1(1/𝑘)−log𝑛]≅0.577216 is the Euler-Mascheroni constant.

The classical [24] sampling theorem of WKS for 𝑓∈𝐵²_𝜎 is the series representation 𝑓(𝑡)=^∞_𝑛=−∞𝑓𝑛𝜋𝜎𝑆_𝑛(𝑡),𝑡∈ℝ, (17) where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ (cf. [24–26]). Series (17), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, for example, [8–13, 15, 27, 28].

The use of (17) in numerical analysis is known as the sinc method established by Stenger et al. (cf. [29–31]). In [9, 15, 28], the authors applied (17) and the regularized sinc method to compute eigenvalues of different boundary value problems with a derivation of the error estimates as given by [32, 33]. In [34], the authors used Hermite-type sampling series (1) to compute the eigenvalues of Dirac system with an internal point of discontinuity. In [14], Tharwat proved that Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) has a denumerable set of real and simple eigenvalues.

In [35], we compute the eigenvalues of the problem Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) numerically by using sinc-Gaussian technique. The main aim of the present work is to compute the eigenvalues of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) numerically by using Hermite interpolations with an error analysis. This method is based on sampling theorem and 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 [34]. Also, in this work, by using computable error bounds we obtain eigenvalue enclosures in a simple way which not have been proven in [35].

Notice that due to Paley-Wiener’s theorem 𝑓∈𝐵²_𝜎 if and only if there is 𝑔(⋅)∈𝐿²(−𝜎,𝜎) such that 𝑓(𝑡)=1√2𝜋^𝜎−𝜎𝑔(𝑥)𝑒^𝑖𝑥𝑡𝑑𝑥. (18) Therefore 𝑓^′(𝑡)∈𝐵²_𝜎; that is, 𝑓^′(𝑡) also has an expansion of the form (17). However, 𝑓^′(𝑡) can be also obtained by term-by-term differentiation formula of (17) 𝑓^′(𝑡)=^∞_𝑛=−∞𝑓𝑛𝜋𝜎𝑆^′_𝑛(𝑡) (19) (see [24, 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 (17) and (19), respectively.

In the next section, we derive the Hermite interpolation technique to compute the eigenvalues of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) with error estimates. The last section contains three worked examples with comparisons accompanied by figures and numerics with Lagrange interpolation method.

2. Treatment of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿)

In this section we derive approximate values of the eigenvalues of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿). Recall that Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) has denumerable set of real and simple eigenvalues (cf. [14]). Let 𝔶=𝔶𝑥,𝜇¹[)𝔶𝑥,𝜇,𝑥∈−1,0₂𝑥,𝜇,𝑥∈(0,1] (20) denote the solution of (3) satisfying the following initial conditions: 𝔶₁−1,𝜇𝔶₂0⁺𝔶,𝜇_′1−1,𝜇𝔶^′₂0⁺=𝜇,𝜇²𝛼^′₂−𝛼₂𝛾₁𝛿₁𝔶₁0⁻𝜇,𝜇₂𝛼^′₁−𝛼₁𝛾₂𝛿₂𝔶^′₁0⁻,𝜇. (21) Since 𝑦(⋅,𝜇) satisfies (4), (6), then the eigenvalues of problem (3)–(6) are the zeros of the characteristic determinant (cf. [14]) Γ𝜇∶=𝛽^′₁𝜇²+𝛽₁𝔶₂1,𝜇−𝛽^′₂𝜇²+𝛽₂𝔶^′₂1,𝜇. (22) According to [14], see also [36–38], function Γ(𝜇) is an entire function of 𝜇 where zeros are real and simple. We aim to approximate Γ(𝜇) and hence its zeros, that is, the eigenvalues by the use of the sampling theorem. The idea is to split Γ(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 using (1) to get the approximate Γ(𝜇) and then compute the approximate zeros. Using the method of variation of parameters, solution 𝑦(⋅,𝜇) satisfies the Volterra integral equations (cf. [14]) 𝔶₁=𝑥,𝜇−𝛼₂+𝜇²𝛼^′₂𝜇(−cos𝑥+1)−𝛼₁+𝜇²𝛼^′₁1𝜇𝜇(+𝔗sin𝑥+1)₁𝔶₁,𝔶𝑥,𝜇₂=𝛾𝑥,𝜇¹𝛿₁𝔶₁0⁻+𝛾,𝜇cos𝜇𝑥₂𝛿₂𝔶^′₁0⁻,𝜇sin𝜇𝑥𝜇+𝔗₂𝔶₂𝑥,𝜇, (23) where 𝔗₁ and 𝔗₂ are the Volterra operators 𝔗₁𝔶₁𝑥,𝜇∶=^𝑥₋₁sin𝜇(𝑥−𝑡)𝜇𝑞(𝑡)𝔶₁𝔗𝑡,𝜇𝑑𝑡,₂𝔶₂𝑥,𝜇∶=^𝑥₀sin𝜇(𝑥−𝑡)𝜇𝑞(𝑡)𝔶₂𝑡,𝜇𝑑𝑡. (24) Differentiating (23) we obtain 𝔶^′₁𝑥,𝜇=−−𝛼₂+𝜇²𝛼^′₂𝜇(−𝜇sin𝑥+1)−𝛼₁+𝜇²𝛼^′₁𝜇(+𝔗cos𝑥+1)₁𝔶₁,𝔶𝑥,𝜇_′2𝛾𝑥,𝜇=−¹𝛿₁𝜇𝔶₁0⁻+𝛾,𝜇sin𝜇𝑥₂𝛿₂𝔶^′₁0⁻+𝔗,𝜇cos𝜇𝑥₂𝔶₂𝑥,𝜇, (25) where 𝔗₁ and 𝔗₂ are the Volterra-type integral operators 𝔗₁𝔶₁𝑥,𝜇∶=^𝑥₋₁cos𝜇(𝑥−𝑡)𝑞(𝑡)𝔶₁𝔗𝑡,𝜇𝑑𝑡,₂𝔶₂𝑥,𝜇∶=^𝑥₀cos𝜇(𝑥−𝑡)𝑞(𝑡)𝔶₂𝑡,𝜇𝑑𝑡. (26) Define 𝜗_𝑖(⋅,𝜇) and 𝜗_𝑖(⋅,𝜇), 𝑖=1,2, to be 𝜗_𝑖𝑥,𝜇∶=𝔗_𝑖𝔶_𝑖𝑥,𝜇,𝜗_𝑖𝑥,𝜇∶=𝔗_𝑖𝔶_𝑖𝑥,𝜇. (27) In the following, we will make use of the known estimates: |cos𝑧|≤𝑒^|ℑ𝑧|,sin𝑧𝑧≤𝑐₀|𝑧|𝑒1+^|ℑ𝑧|, (28) where 𝑐₀ is some constant (we may take 𝑐₀≃1.72). For convenience, we define the constants 𝑞₁∶=⁰₋₁𝑞(𝑡)𝑑𝑡,𝑞₂∶=¹₀𝑞(𝑡)𝑐𝑑𝑡,₁∶=max𝛼₁,𝛼₂,𝛼^′₁,𝛼^′₂,𝑐₂∶=exp𝑐₀𝑞₁,𝑐₃∶=1+𝑐₀𝑐₂𝑞₁,𝑐₄∶=1+𝑐₀𝛾₁𝛿₁𝑐₃+𝛾₂𝛿₂𝑐₀1+𝑐₃𝑞₁,𝑐₅∶=exp𝑐₀𝑞₂,𝑐₆∶=1+𝑐₀𝑞₂𝑐₅. (29)

As in [15] we split Γ(𝜇) into two parts via Γ𝜇∶=𝒢𝜇+𝒮𝜇, (30) where 𝒢(𝜇) is the known part 𝒢𝜇∶=𝛽^′₁𝜇²+𝛽₁×𝜇²𝛼^′₂−𝛼₂𝛾₁𝛿₁cos²𝛾𝜇−²𝛿₂sin²𝜇−𝜇²𝛼^′₁−𝛼₁𝛾₁𝛿₁+𝛾₂𝛿₂cos𝜇sin𝜇𝜇+𝛽^′₂𝜇²+𝛽₂×𝜇²𝛼^′₂−𝛼₂𝛾₁𝛿₁+𝛾₂𝛿₂+𝜇𝜇cos𝜇sin𝜇₂𝛼^′₁−𝛼₁𝛾₂𝛿₂cos²𝛾𝜇−¹𝛿₁sin²𝜇 (31) and 𝒮(𝜇) is the unknown one 𝒮𝜇𝛾∶=¹𝛿₁×𝛽^′₁𝜇²+𝛽₁cos𝜇+𝛽^′₂𝜇²+𝛽₂𝜇sin𝜇×𝜗₁0⁻+𝛽,𝜇_′1𝜇²+𝛽₁𝜗₂+𝛾1,𝜇₂𝛿₂𝛽^′₁𝜇²+𝛽₁sin𝜇𝜇−𝛽^′₂𝜇²+𝛽₂×𝜗cos𝜇₁0⁻−𝛽,𝜇_′2𝜇²+𝛽₂𝜗₂1,𝜇. (32) Then function 𝒮(𝜇) is entire in 𝜇 for each 𝑥∈[0,1] for which (cf. [15]) 𝒮𝜇≤ℳ1+𝜇²²𝑒^2|ℑ𝜇|,𝜇∈ℂ, (33) where ℳ∶=𝑐₁𝑐1+𝑐₀²𝑞₁𝑐₀𝑐₂𝛾₁𝛿₁+𝑐₃𝛾₂𝛿₂+𝑐₁𝑐₄𝑐𝑞₂𝑐₆+𝑐₀𝑐₅,𝑐∶=max𝛽₁,𝛽₂,𝛽^′₁,𝛽^′₂. (34) The analyticity of 𝒮(𝜇) as well as estimate (33) is 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 𝑚∈ℤ⁺, 𝑚>5, be fixed. Let ℱ_𝜃,𝑚(𝜇) be the function ℱ_𝜃,𝑚𝜇∶=sin𝜃𝜇𝜃𝜇^𝑚𝒮𝜇,𝜇∈ℂ. (35) The regularizing factor has been introduced in [9], in the context of the regularized sampling method, which was used in [9–13] to compute the eigenvalues of several classes of Sturm-Liouville problems. More specifications on 𝑚, 𝜃 will be given latter on. Then ℱ_𝜃,𝑚(𝜇), see [15], is an entire function of 𝜇 which satisfies the estimate ℱ_𝜃,𝑚𝜇≤ℳ𝑐^𝑚₀1+𝜇²²1+𝜃𝜇^𝑚𝑒^{|ℑ𝜇|(2+𝑚𝜃)},𝜇∈ℂ. (36) Moreover, 𝜇^𝑚−5ℱ_𝜃,𝑚(𝜇)∈𝐿²(ℝ) and ℰ_𝑚−5ℱ_𝜃,𝑚∶=^∞_−∞𝜇^𝑚−5ℱ_𝜃,𝑚𝜇²𝑑𝜇≤2𝜈₀ℳ𝑐^𝑚₀, (37) where 𝜈₀1∶=𝜃^2𝑚−1×1+2𝑚−14𝜃²4𝑚³−12𝑚²++11𝑚−3144𝜃⁴280𝜃⁴Γ[2𝑚−9]+20𝜃²Γ[][]×()2𝑚−7+Γ2𝑚−5Γ[2𝑚]⁻¹. (38) What we have just proved is that ℱ_𝜃,𝑚(𝜇) belongs to the Paley-Wiener space 𝐵²_𝜎 with 𝜎=2+𝑚𝜃. Since ℱ_𝜃,𝑚(𝜇)∈𝐵²_𝜎⊂𝐵²_2𝜎, then we can reconstruct the functions ℱ_𝜃,𝑚(𝜇) via the following sampling formula: ℱ_𝜃,𝑚𝜇=^∞_𝑛=−∞ℱ_𝜃,𝑚𝑛𝜋𝜎𝑆²_𝑛𝜇+ℱ^′_𝜃,𝑚𝑛𝜋𝜎sin𝜎𝜇−𝑛𝜋𝜎𝑆_𝑛𝜇. (39)

Let 𝑁∈ℤ⁺, 𝑁>𝑚, and approximate ℱ_𝜃,𝑚(𝜇) by its truncated series ℱ_{𝜃,𝑚,𝑁}(𝜇), where ℱ_{𝜃,𝑚,𝑁}𝜇∶=^𝑁_𝑛=−𝑁ℱ_𝜃,𝑚𝑛𝜋𝜎𝑆²_𝑛𝜇+ℱ^′_𝜃,𝑚𝑛𝜋𝜎sin𝜎𝜇−𝑛𝜋𝜎𝑆_𝑛𝜇. (40) Since all eigenvalues are real, then from now on we restrict ourselves to 𝜇∈ℝ. Since 𝜇^𝑚−5ℱ_𝜃,𝑚(𝜇)∈𝐿²(ℝ), the truncation error (cf. (10)) is given for |𝜇|<𝑁𝜋/𝜎 by ℱ_𝜃,𝑚𝜇−ℱ_{𝜃,𝑚,𝑁}𝜇≤𝑇_{𝑁,𝑚−5,𝜎}𝜇, (41) where 𝒯_{𝑁,𝑚−5,𝜎}𝜇𝜂∶=^{𝑚−5,𝜎}ℰ_𝑚−5sin𝜎𝜇²√3(𝑁+1)^𝑚−5×1𝑁𝜋−𝜎𝜇^3/2+1𝑁𝜋+𝜎𝜇^3/2+𝜂_{𝑚−5,𝜎}𝜎ℰ_𝑚−5+(𝑚−5)ℰ_𝑚−6sin𝜎𝜇²𝜎(𝑁+1)^𝑚−5×1+1𝑁𝜋−𝜎𝜇.𝑁𝜋+𝜎𝜇 (42) The samples {ℱ_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁 and {ℱ^′_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁, in general, are not known explicitly. So we approximate them by solving numerically 8𝑁+4 initial value problems at the nodes {𝑛𝜋/𝜎}^𝑁_𝑛=−𝑁.

Let {ℱ_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁 and {ℱ^′_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁 be the approximations of the samples of {ℱ_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁 and {ℱ^′_𝜃,𝑚(𝑛𝜋/𝜎)}^𝑁_𝑛=−𝑁, respectively. Now we define ℱ_{𝜃,𝑚,𝑁}(𝜇), which approximates ℱ_{𝜃,𝑚,𝑁}(𝜇): ℱ_{𝜃,𝑚,𝑁}𝜇∶=^𝑁_𝑛=−𝑁ℱ_𝜃,𝑚𝑛𝜋𝜎𝑆²_𝑛𝜇+ℱ^′_𝜃,𝑚𝑛𝜋𝜎×sin𝜎𝜇−𝑛𝜋𝜎𝑆_𝑛𝜇,𝑁>𝑚. (43) Using standard methods for solving initial problems, we may assume that for |𝑛|<𝑁ℱ_𝜃,𝑚𝑛𝜋𝜎−ℱ_𝜃,𝑚𝑛𝜋𝜎ℱ<𝜀,_′𝜃,𝑚𝑛𝜋𝜎−ℱ^′_𝜃,𝑚𝑛𝜋𝜎<𝜀, (44) for a sufficiently small 𝜀. From (36) we can see that ℱ_𝜃,𝑚(𝜇) satisfies the condition (14) when 𝑚>5, and therefore whenever 0<𝜀≤min{𝜋/𝜎,𝜎/𝜋,1/√𝑒} we have ℱ_{𝜃,𝑚,𝑁}𝜇−ℱ_{𝜃,𝑚,𝑁}𝜇≤𝒜(𝜀),𝜇∈ℝ, (45) where there is a positive constant 𝑀_ℱ𝜃,𝑚 for which (cf. (15)) 𝒜(𝜀)∶=2𝑒^1/4𝜎×√()+𝜋3𝑒1+𝜎𝜎𝐴+ℳ_ℱ𝜃,𝑚𝜌(𝜀)+𝜎+2+log(2)ℳ_ℱ𝜃,𝑚1𝜀log𝜀. (46) Here 𝐴∶=3𝜎𝜋ℱ_𝜃,𝑚(0+𝜎)𝜋ℳ_ℱ𝜃,𝑚,𝜌(𝜀)1∶=𝛾+10log𝜀. (47) In the following we use the technique of [27], see also [15], to determine enclosure intervals for the eigenvalues. Let 𝜇^∗ be an eigenvalue; that is, Γ𝜇^∗=𝒢𝜇^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_𝜃,𝑚𝜇^∗=0. (48) Then it follows that 𝒢𝜇^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗=sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗−sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_𝜃,𝑚𝜇^∗=sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗−sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗−sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_𝜃,𝑚𝜇^∗ (49) and so 𝒢𝜇^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗≤sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝑇_{𝑁,𝑚−5,𝜎}𝜇^∗+𝒜(𝜀). (50) Since 𝒢𝜇^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗ (51) is given and sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝑇_{𝑁,𝑚−5,𝜎}𝜇^∗+𝒜(𝜀) (52) has computable upper bound, we can define an enclosure for 𝜇^∗, by solving the following system of inequalities: −sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇^∗(𝜀)𝜇+𝒜≤𝒢^∗+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇^∗≤sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇^∗+𝒜(𝜀). (53) Its solution is an interval containing 𝜇^∗ over which the graph 𝒢(𝜇^∗)+(sin𝜃𝜇^∗/𝜃𝜇^∗)^−𝑚ℱ_{𝜃,𝑚,𝑁}(𝜇^∗) is squeezed between the graphs −sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇^∗+𝒜(𝜀),sin𝜃𝜇^∗𝜃𝜇^∗−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇^∗+𝒜(𝜀). (54) Using the fact that ℱ_{𝜃,𝑚,𝑁}𝜇⟶ℱ_𝜃,𝑚𝜇, (55) uniformly over any compact set, and since 𝜇^∗ is a simple root, we obtain for large 𝑁 and sufficiently small 𝜀𝜕𝒢𝜇+𝜕𝜇sin𝜃𝜇𝜃𝜇^−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇≠0, (56) in a neighborhood of 𝜇^∗. Hence the graph of 𝒢𝜇+sin𝜃𝜇𝜃𝜇^−𝑚ℱ_{𝜃,𝑚,𝑁}𝜇 (57) intersects the graphs −sin𝜃𝜇𝜃𝜇^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇+𝒜(𝜀),sin𝜃𝜇𝜃𝜇^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇+𝒜(𝜀), (58) at two points with abscissae 𝑎₋(𝜇^∗,𝑁,𝜀)≤𝑎₊(𝜇^∗,𝑁,𝜀), and the solution of the system of inequalities (53) is the interval 𝐼_𝜀,𝑁∶=𝑎₋𝜇^∗,𝑁,𝜀,𝑎₊𝜇^∗,𝑁,𝜀 (59) and in particular 𝜇^∗∈𝐼_𝜀,𝑁. Summarizing the above discussion, we arrive at the following lemma which is similar to that of [27] for Sturm-Liouville problems.

Lemma 1. For any eigenvalue 𝜇^∗, one can find 𝑁₀∈ℤ⁺ and sufficiently small 𝜀 such that 𝜇^∗∈𝐼_𝜀,𝑁 for 𝑁>𝑁₀. Moreover 𝑎₋𝜇^∗,𝑁,𝜀,𝑎₊𝜇^∗,𝑁,𝜀⟶𝜇^∗𝑎𝑠𝑁⟶∞,𝜀⟶0. (60)

Proof. Since all eigenvalues of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿) are simple, then for large 𝑁 and sufficiently small 𝜀 we have (𝜕/𝜕𝜇)(𝒢(𝜇)+(sin𝜃𝜇/𝜃𝜇)^−𝑚ℱ_{𝜃,𝑚,𝑁}(𝜇))>0, in a neighborhood of 𝜇^∗. Choose 𝑁₀ such that 𝒢𝜇+sin𝜃𝜇𝜃𝜇^−𝑚ℱ_{𝜃,𝑚,𝑁0}𝜇=±sin𝜃𝜇𝜃𝜇^−𝑚𝒯_{𝑁0,𝑚−5,𝜎}𝜇+𝒜(𝜀) (61) has two distinct solutions which we denote by 𝑎₋(𝜇^∗,𝑁₀,𝜀)≤𝑎₊(𝜇^∗,𝑁₀,𝜀). The decay of 𝒯_{𝑁,𝑚−5,𝜎}(𝜇)→0 as 𝑁→∞ and 𝒜(𝜀)→0 as 𝜀→0 will ensure the existence of the solutions 𝑎₋(𝜇^∗,𝑁,𝜀) and 𝑎₊(𝜇^∗,𝑁,𝜀) as 𝑁→∞ and 𝜀→0. For the second point we recall that ℱ_{𝜃,𝑚,𝑁}(𝜇)→ℱ_𝜃,𝑚(𝜇) as 𝑁→∞ and 𝜀→0. Hence by taking the limit we obtain 𝒢𝑎₊𝜇^∗,∞,0+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_𝜃,𝑚𝑎₊𝜇^∗𝒢𝑎,∞,0=0,₋𝜇^∗,∞,0+sin𝜃𝜇^∗𝜃𝜇^∗−𝑚ℱ_𝜃,𝑚𝑎₋𝜇^∗,∞,0=0. (62) That isΓ(𝑎₊)=Γ(𝑎₋)=0. This leads us to conclude that 𝑎₊=𝑎₋=𝜇^∗ since 𝜇^∗ is a simple root.

Let Γ_𝑁(𝜇)∶=𝒢(𝜇)+(sin𝜃𝜇/𝜃𝜇)^−𝑚ℱ_{𝜃,𝑚,𝑁}(𝜇). Then (41) and (45) imply Γ𝜇−Γ_𝑁𝜇≤sin𝜃𝜇𝜃𝜇^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇(𝜀),𝜇<+𝒜𝑁𝜋𝜎, (63) and 𝜃 is chosen sufficiently small for which |𝜃𝜇|<𝜋. Therefore 𝜃, 𝑚 must be chosen so that for |𝜇|<𝑁𝜋/𝜎𝑚>5,𝜃∈(0,1),𝜃𝜇<𝜋. (64) Let 𝜇^∗ be an eigenvalue and let 𝜇_𝑁 be its approximation. Thus Γ(𝜇^∗)=0 and Γ_𝑁(𝜇_𝑁)=0. From (63) we have |Γ_𝑁(𝜇^∗)|≤|sin𝜃𝜇^∗/𝜃𝜇^∗|^−𝑚(𝒯_{𝑁,𝑚−5,𝜎}(𝜇^∗)+𝒜(𝜀)). Now we estimate the error |𝜇^∗−𝜇_𝑁| for an eigenvalue 𝜇^∗.

Theorem 2. Let 𝜇^∗ be an eigenvalue of Π(𝑞,𝛼,𝛽,𝛼^′,𝛽^′,𝛾,𝛿). For sufficiently large 𝑁 one has the following estimate: 𝜇^∗−𝜇_𝑁<sin𝜃𝜇_𝑁𝜃𝜇_𝑁^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇_𝑁+𝒜(𝜀)inf_{𝜁∈𝐼𝜀,𝑁}Γ^′(𝜁.) (65)

Proof. Since Γ(𝜇_𝑁)−Γ_𝑁(𝜇_𝑁)=Γ(𝜇_𝑁)−Γ(𝜇^∗), then from (63) and after replacing 𝜇 by 𝜇_𝑁 we obtain Γ𝜇_𝑁−Γ𝜇^∗≤sin𝜃𝜇_𝑁𝜃𝜇_𝑁^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇_𝑁+𝒜(𝜀). (66) Using the mean value theorem yields that for some 𝜁∈𝐽_𝜀,𝑁∶=[min(𝜇^∗,𝜇_𝑁),max(𝜇^∗,𝜇_𝑁)]𝜇^∗−𝜇_𝑁Γ^′(𝜁)≤sin𝜃𝜇_𝑁𝜃𝜇_𝑁^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇_𝑁+𝒜(𝜀),𝜁∈𝐽_𝜀,𝑁⊂𝐼_𝜀,𝑁. (67) Since thlarge 𝑁inf_{𝜁∈𝐼𝜀,𝑁}|Γ^′(𝜁)|>0 and we get (65).

3. Numerical Examples

This section includes three detailed worked examples illustrating the above technique. By 𝐸_𝑆 and 𝐸_𝐻 we mean the absolute errors associated with the results of the classical sinc method [9, 15] and our new method (Hermite interpolations), respectively. The first two examples are computed in [15] with the classical sinc method. We indicate in these two 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. Also, ine eigenvalues are simple, then for sufficiently the following two examples, we observe that the exact solutions 𝜇_𝑘 and the zeros of Γ(𝜇) are all inside the interval [𝑎₋,𝑎₊]. In the third example, we compare our new method with the classical sinc method [9]. We would like to mention that MATHEMATICA has been used to obtain the exact values for the two examples where eigenvalues cannot be computed concretely. MATHEMATICA is also used in rounding the exact eigenvalues, which are square roots. Both numerical results and the associated figures prove the credibility of the method.

Recall that 𝑎_±(𝜇) are defined by 𝑎_±𝜇=Γ_𝑁𝜇±sin𝜃𝜇𝜃𝜇^−𝑚𝒯_{𝑁,𝑚−5,𝜎}𝜇(𝜀),𝜇<+𝒜𝑁𝜋𝜎. (68) Recall also that the enclosure interval 𝐼_𝜀,𝑁∶=[𝑎₋,𝑎₊] is determined by solving 𝑎_±𝜇=0,𝜇<𝑁𝜋𝜎. (69)

Example 1. Consider the boundary value problem [15] −𝔶^′′𝑥,𝜇+𝑞(𝑥)𝔶𝑥,𝜇=𝜇²𝔶,[)∪(],𝜇𝑥,𝜇𝑥∈−1,00,1₂𝔶−1,𝜇+𝔶^′−1,𝜇=0,𝜇²𝔶1,𝜇−𝔶^′𝔶01,𝜇=0,₋,𝜇−𝔶0⁺,𝜇=0,𝔶^′0⁻,𝜇−𝔶^′0⁺,𝜇=0. (70) Here 𝛽^′₁=𝛽₂=𝛼^′₁=𝛼₂=1, 𝛽₁=𝛽^′₂=𝛼₁=𝛼^′₂=0, 𝛾₁=𝛿₁=2, 𝛾₂=𝛿₂=1/2, and 𝑞(𝑥)=[),(].−1,𝑥∈−1,0−2,𝑥∈0,1 (71) The characteristic function is Γ𝜇=11+𝜇²2+𝜇²×sin1+𝜇²−2+𝜇²𝜇⁴−𝜇²−1×cos2+𝜇²−𝜇²3+2𝜇²sin2+𝜇²−1+𝜇²cos1+𝜇²×−2𝜇²2+𝜇²cos2+𝜇²+𝜇⁴−𝜇²−2sin2+𝜇². (72) The function 𝒦(𝜇) will be 𝒦𝜇=−𝜇1+𝜇²sin2𝜇. (73)

The application of Hermite interpolations method and sinc method [15] to this problem and the effect of 𝜃 and 𝑚 at 𝑁=20 are indicated in Tables 1 and 2. In Tables 3 and 4, we display the maximum absolute error of 𝜇_𝑘−𝜇_𝑘,𝑁, using Hermite interpolations method and sinc method [15] with various choices of 𝜃 and 𝑚 at 𝑁=20. From these tables, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method [15].

Table 1 

Comparing the exact, sinc, and Hermite solutions at 𝑁=20, 𝑚=8, and 𝜃=1/6.

Table 2 

Comparing the exact, sinc, and Hermite solutions at 𝑁=20, 𝑚=12, and 𝜃=1/4.

Table 3 

Absolute errors |𝜇_𝑘−𝜇_𝑘,𝑁| for 𝑁=20, 𝑚=8, and 𝜃=1/6.

Table 4 

Absolute errors |𝜇_𝑘−𝜇_𝑘,𝑁| for 𝑁=20, 𝑚=12, and 𝜃=1/4.

Tables 5 and 6 list the exact solutions 𝜇_𝑘 for two choices of 𝑚 and 𝜃 at 𝑁=20 and different values of 𝜀. It is indicated that the solutions 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for all values of 𝜀.

Table 5 

For 𝑁=20, 𝑚=8, and 𝜃=1/6, the exact solutions 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

Table 6 

For 𝑁=20, 𝑚=12; and 𝜃=1/4, 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

For 𝑁=20, 𝑚=8, and 𝜃=1/6, Figures 1 and 2 illustrate the enclosure intervals dominating 𝜇₁ for 𝜀=10⁻⁵ and 𝜀=10⁻¹⁰, respectively. The middle curve represents Γ(𝜇), while the upper and lower curves represent the curves of 𝑎₊(𝜇), 𝑎₋(𝜇), respectively. We notice that when 𝜀=10⁻¹⁰ all three curves are almost identical. Similarly, Figures 3 and 4 illustrate the enclosure intervals dominating 𝜇₂ for 𝜀=10⁻⁵, 𝜀=10⁻¹⁰, respectively.

Figure 1 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=8, 𝜃=1/6, and 𝜀=10−5.

Figure 2 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=8, 𝜃=1/6, and 𝜀=10−10.

Figure 3 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=8, 𝜃=1/6, and 𝜀=10−5.

Figure 4 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=8, 𝜃=1/6, and 𝜀=10−10.

As in Table 6, for 𝑁=20, 𝑚=12, and 𝜃=1/4, Figures 5 and 6 illustrate the enclosure intervals dominating 𝜇₃ for 𝜀=10⁻⁵ and 𝜀=10⁻¹⁰, respectively, and Figures 7 and 8 illustrate the enclosure intervals dominating 𝜇₄ for 𝜀=10⁻⁵, 𝜀=10⁻¹⁰, respectively.

Figure 5 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=12, 𝜃=1/4, and 𝜀=10−5.

Figure 6 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=12, 𝜃=1/4, and 𝜀=10−10.

Figure 7 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=12, 𝜃=1/4, and 𝜀=10−5.

Figure 8 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=20, 𝑚=12, 𝜃=1/4, and 𝜀=10−10.

Example 2. Consider the boundary value problem −𝔶^′′𝑥,𝜇+𝑞(𝑥)𝔶𝑥,𝜇=𝜇²𝔶,[)∪(],𝔶𝑥,𝜇𝑥∈−1,00,1−1,𝜇+𝜇²𝔶^′𝔶−1,𝜇=0,1,𝜇+𝜇²𝔶^′𝔶01,𝜇=0,₋,𝜇−𝔶0⁺,𝜇=0,𝔶^′0⁻,𝜇−𝔶^′0⁺,𝜇=0, (74) where 𝛼₁=𝛽₁=1, 𝛼^′₂=𝛽^′₂=−1, 𝛽^′₁=𝛽₂=𝛼₂=𝛼^′₁=0, 𝛾₁=𝛿₁=3, 𝛾₂=𝛿₂=1/3, and 𝑞(𝑥)=[),(].−2,𝑥∈−1,0𝑥,𝑥∈0,1 (75) The function 𝒦(𝜇) will be 𝒦𝜇=1+𝜇⁶sin2𝜇𝜇. (76)

The application of Hermite interpolations method and sinc method [15] to this problem and the effect of 𝜃 and 𝑚 at 𝑁=40 are indicated in Tables 7 and 8. In Tables 9 and 10, we display the maximum absolute error of 𝜇_𝑘−𝜇_𝑘,𝑁, using Hermite interpolations method and sinc method [15] with various choices of 𝜃 and 𝑚 at 𝑁=40. Form these tables, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method [15].

Table 7 

Comparing the exact, sinc, and Hermite solutions at 𝑁=40, 𝑚=12, and 𝜃=1/14.

Table 8 

Comparing the exact, sinc, and Hermite solutions at 𝑁=40, 𝑚=16, and 𝜃=1/12.

Table 9 

Absolute errors |𝜇_𝑘−𝜇_𝑘,𝑁| for 𝑁=40, 𝑚=12, and 𝜃=1/14.

Table 10 

Absolute errors |𝜇_𝑘−𝜇_𝑘,𝑁| for 𝑁=40, 𝑚=16, and 𝜃=1/12.

Tables 11 and 12 list the exact solutions 𝜇_𝑘 for two choices of 𝑚 and 𝜃 at 𝑁=40 and different values of 𝜀. It is indicated that the solutions 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for all values of 𝜀.

Table 11 

For 𝑁=40, 𝑚=12, and 𝜃=1/14, 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

Table 12 

For 𝑁=40, 𝑚=16, and 𝜃=1/12, 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

For 𝑁=40, 𝑚=12, and 𝜃=1/14, Figures 9 and 10 illustrate the enclosure intervals dominating 𝜇₁ for 𝜀=10⁻⁶ and 𝜀=10⁻¹², respectively. Similarly, Figures 11 and 12 illustrate the enclosure intervals dominating 𝜇₂ for 𝜀=10⁻⁶, 𝜀=10⁻¹², respectively.

Figure 9 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=12, 𝜃=1/14, and 𝜀=10−6.

Figure 10 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=12, 𝜃=1/14, and 𝜀=10−12.

Figure 11 

𝑎+,Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=12, 𝜃=1/14, and 𝜀=10−6.

Figure 12 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=12, 𝜃=1/14, and 𝜀=10−12.

For 𝑁=40, 𝑚=16, and 𝜃=1/12, Figures 13 and 14 illustrate the enclosure intervals dominating 𝜇₁ for 𝜀=10⁻⁷ and 𝜀=10⁻¹², respectively, and Figures 15 and 16 illustrate the enclosure intervals dominating 𝜇₂ for 𝜀=10⁻⁷, 𝜀=10⁻¹², respectively.

Figure 13 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=16, 𝜃=1/12, and 𝜀=10−7.

Figure 14 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=16, 𝜃=1/12, and 𝜀=10−12.

Figure 15 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=16, 𝜃=1/12, and 𝜀=10−7.

Figure 16 

𝑎+, Γ(𝜇), and 𝑎− with 𝑁=40, 𝑚=16, 𝜃=1/12, and 𝜀=10−12.

Example 3. Consider the continuous boundary value problem [9] −𝔶^′′𝑥,𝜇=𝜇²𝔶[],=𝜇𝑥,𝜇,𝑥∈0,1−𝔶0,𝜇²+𝑑𝔶^′0,𝜇,𝔶1,𝜇=𝜇²𝔶^′1,𝜇, (77) where 𝑑=−4𝜋², 𝛽₁=𝛽^′₂=1, 𝛼₂=𝑑, 𝛼₁=𝛽₂=𝛽^′₁=0, 𝛼₂=𝑑, and 𝛼^′₂=−1. The exact characteristic function is Γ𝜇=1+4𝜋²𝜇⁴−𝜇⁶sin𝜇𝜇−2𝜇²−4𝜋²cos𝜇, (78) where zero is not an eigenvalue. The application of Hermite interpolations method and sinc method [9] to this problem is indicated in Table 13. From this table, it is shown that the proposed method is significantly more accurate than that based on the sinc method [9].

Table 13 

Comparing the exact, sinc, and Hermite solutions at 𝑁=40, 𝑚=2.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-065-D1433. The authors, therefore, acknowledge with thanks DSR technical and financial support.