Numerical Algorithms for Computing Eigenvalues of Discontinuous Dirac System Using Sinc-Gaussian Method

Abstract

The eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity are computed using the sinc-Gaussian method. The error analysis of this method for solving discontinuous regular Dirac system is discussed. It shows that the error decays exponentially in terms of the number of involved samples. Therefore, the accuracy of the new method is higher than the classical sinc-method. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Comparisons with the classical sinc-method are given.

1. Introduction

The mathematical modeling of many practical problems in mechanics and other areas of mathematical physics requires solutions of boundary value problems (see, e.g., [1–7]). Boundary value problems with discontinuity conditions arise in different branches of mathematics, mechanics, radio, electronics, geophysics, and other fields of natural science and technology. For example, discontinuous conditions inside an interval are connected with discontinuous or nonsmooth properties of media (see [8–11] and references there in).

Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. A fundamental result in information theory is the Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem [12–14]. It states that any 𝑓∈𝑃𝑊²_𝜎, where 𝑃𝑊²_𝜎 is the space of all entire functions of exponential type 𝜎>0 which lie in 𝐿²(ℝ) when restricted to ℝ, can be reconstructed from its sampled values {𝑓(𝑛𝜋/𝜎)∶𝑛∈ℤ} by the formula 𝑓(𝜆)=_𝑛∈ℤ𝑓𝑛𝜋𝜎sinc(𝜎𝜆−𝑛𝜋),𝜆∈ℂ, (1.1) where ()()sinc𝜎𝜆−𝑛𝜋∶=sin𝜎𝜆−𝑛𝜋(𝜎𝜆−𝑛𝜋),𝜆≠𝑛𝜋𝜎,1,𝜆=𝑛𝜋𝜎. (1.2) Series (1.1) converges absolutely and uniformly on compact subsets of ℂ, and uniform on ℝ, cf. [15]. Expansion (1.1) is used in several approximation problems which are known as sinc methods, see, for example, [16–19]. In particular the sinc-method is used to approximate eigenvalues of boundary value problems, see, for examples, [20–24]. The sinc-method has a slow rate of decay at infinity, which is as slow as 𝑂(|𝜆⁻¹|). There are several attempts to improve the rate of decay. One of the interesting ways is to multiply the sinc-function in (1.1) by a kernel function, see, for example, [25–27]. Let ℎ∈(0,𝜋/𝜎] and 𝛾∈(0,𝜋−ℎ𝜎). Assume that Φ∈𝑃𝑊²_𝛾 such that Φ(0)=1, then for 𝑓∈𝑃𝑊²_𝜎 we have the expansion [28], 𝑓(𝜆)=^∞_𝑛=−∞𝑓(𝑛ℎ)sincℎ⁻¹𝜋𝜆−𝑛𝜋Φℎ⁻¹𝜆−𝑛. (1.3) The speed of convergence of the series in (1.3) is determined by the decay of |Φ(𝜆)|. But the decay of an entire function of exponential type cannot be as fast as 𝑒^{−𝑐|𝑥|} as |𝑥|→∞, for some positive 𝑐, [28]. In [29], Qian has introduced the following regularized sampling formula. For ℎ∈(0,𝜋/𝜎], 𝑁∈ℕ and 𝑟>0, 𝐺_ℎ,𝑁𝑓(𝑥)=_{𝑛∈𝑍𝑁(𝑥)}𝑓(𝑛ℎ)𝑆_𝑛ℎ⁻¹𝜋𝑥𝐺𝑥−𝑛ℎ√2𝑟ℎ,𝑥∈ℝ, (1.4) where 𝐺(𝑡)∶=exp(−𝑡²), which is called the Gaussian function, 𝑆_𝑛(ℎ⁻¹𝜋𝑥)∶=sinc(ℎ⁻¹𝜋𝑥−𝑛𝜋), 𝑍_𝑁(𝑥)∶={𝑛∈ℤ∶|[ℎ⁻¹𝑥]−𝑛|≤𝑁} and [𝑥] denotes the integer part of 𝑥∈ℝ, see also [30, 31]. Qian also derived the following error bound. If 𝑓∈𝑃𝑊²_𝜎, ℎ∈(0,𝜋/𝜎] and 𝑎∶=min{𝑟(𝜋−ℎ𝜎),(𝑁−2)/𝑟}≥1, then 𝑓(𝑥)−𝐺_ℎ,𝑁𝑓(𝑥≤2√)𝜎𝜋𝑓₂𝜋²𝑎²√2𝜋𝑎+𝑒^3/2𝑟2𝑒^−𝑎2/2,𝑥∈ℝ. (1.5) In [28], Schmeisser and Stenger extended the operator (1.4) to the complex domain ℂ. For 𝜎>0, ℎ∈(0,𝜋/𝜎] and 𝜔∶=(𝜋−ℎ𝜎)/2, they defined the operator [28], 𝒢_ℎ,𝑁𝑓(𝜆)∶=_{𝑛∈ℤ𝑁(𝜆)}𝑓(𝑛ℎ)𝑆_𝑛𝜋𝜆ℎ𝐺√𝜔(𝜆−𝑛ℎ)√,𝑁ℎ (1.6) where ℤ_𝑁(𝜆)∶={𝑛∈ℤ∶|[ℎ⁻¹ℜ𝜆+1/2]−𝑛|≤𝑁} and 𝑁∈ℕ. Note that the summation limits in (1.6) depend on the real part of 𝜆. Schmeisser and Stenger, [28], proved that if 𝑓 is an entire function of exponential type 𝜎>0, then for ℎ∈(0,𝜋/𝜎), 𝜔∶=(𝜋−ℎ𝜎)/2, 𝑁∈ℕ, |ℑ𝜆|<𝑁, we have 𝑓(𝜆)−𝒢_ℎ,𝑁𝑓(𝜆)≤2sinℎ⁻¹𝜋𝜆𝑓_∞𝑒^−𝜔𝑁√𝛽𝜋𝜔𝑁^𝑁ℎ⁻¹ℑ𝜆,𝜆∈ℂ, (1.7) where 𝛽_𝑁(𝑡)∶=cosh(2𝜔𝑡)+2𝑒^{𝜔𝑡2/𝑁}√𝜋𝜔𝑁1−(𝑡/𝑁)²+12𝑒^2𝜔𝑡𝑒^{2𝜋(𝑁−𝑡)}+𝑒−1𝑒^{−2𝜔𝑡2𝜋(𝑁+𝑡)}.−1 (1.8)

The amplitude error arises when the exact values𝑓(𝑛ℎ) of (1.6) are replaced by the approximations 𝑓(𝑛ℎ). We assume that 𝑓(𝑛ℎ) are close to 𝑓(𝑛ℎ), that is, there is 𝜀>0, sufficiently small such that sup_{𝑛∈ℤ𝑛(𝜆)}𝑓(𝑛ℎ)−𝑓(𝑛ℎ)<𝜀. (1.9) Let ℎ∈(0,𝜋/𝜎), 𝜔∶=(𝜋−ℎ𝜎)/2 and 𝑁∈ℕ be fixed numbers. The authors in [9] proved that if (1.9) is held, then for |ℑ𝜆|<𝑁, we have 𝒢_ℎ,𝑁𝑓(𝜆)−𝒢_ℎ,𝑁𝑓(𝜆)≤𝐴_𝜀,𝑁(ℑ𝜆), (1.10) where 𝐴_𝜀,𝑁(ℑ𝜆)=2𝜀𝑒^{−𝜔/4𝑁}1+𝑁()ℎ𝜔𝜋exp𝜔+𝜋⁻¹|ℑ𝜆|. (1.11)

We are concerned with the computation of eigenvalues the Dirac system 𝑢^′₂(𝑥,𝜆)−𝑝₁(𝑥)𝑢₁(𝑥,𝜆)=𝜆𝑢₁(),𝑢𝑥,𝜆_′1(𝑥,𝜆)+𝑝₂(𝑥)𝑢₂(𝑥,𝜆)=−𝜆𝑢₂(),[)∪(],𝑥,𝜆𝑥∈−1,00,1 (1.12) with boundary conditions 𝑈₁(𝐮)∶=sin𝛼𝑢₁(−1,𝜆)+cos𝛼𝑢₂()𝑈−1,𝜆=0,₂(𝐮)∶=sin𝛽𝑢₁(1,𝜆)+cos𝛽𝑢₂(1,𝜆)=0, (1.13)(1.14) and transmission conditions 𝑈₃(𝐮)∶=𝑢₁(0⁻,𝜆)−𝛿𝑢₁0⁺𝑈,𝜆=0,₄(𝐮)∶=𝑢₂(0⁻,𝜆)−𝛿𝑢₂0⁺,𝜆=0, (1.15)(1.16) where 𝜆∈ℂ; the real valued function 𝑝₁(⋅) and 𝑝₂(⋅) are continuous in [−1,0) and (0,1], and have finite limits 𝑝₁(0^±)∶=lim_𝑥→0±𝑝₁(𝑥), 𝑝₂(0^±)∶=lim_𝑥→0±𝑝₂(𝑥); 𝛿∈ℝ, 𝛼,𝛽∈[0,𝜋) and 𝛿≠0.

In this paper we will use the sinc-Gaussian sampling formula (1.6) to compute eigenvalues of the Dirac system (1.12)–(1.16). As expected, the new method reduced the error bounds remarkably, see examples at the end of this paper. Special attention is given to the comparison of the numerical results obtained by the new method with those found by classical sinc-method. We would like to mention that works in direction of computing eigenvalues with the sinc-Gaussian, are few, see for example, [9]. Also papers in computing of eigenvalues with discontinuous are few, see [10, 32]. However, the computing of eigenvalues by sinc-Gaussian technique which has discontinuity conditions, do not exist as for as we know.

The paper is organized as follows: Section 2 contains some preliminary results and the approximated values of the eigenvalues of the Dirac system with discontinuous. The method with error estimates are contained in Section 3. The last section involves some illustrative examples for showing the high accuracy of the proposed technique.

2. The Approximated Eigenvalues of Dirac System

In this section we derive approximate values of the eigenvalues of problem (1.12)–(1.16). Recall that problem (1.12)–(1.16) has denumerable set of real and simple eigenvalues, compare with [33, 34]. Let 𝜙()=𝜙⋅,𝜆¹()𝜙⋅,𝜆₂(⋅,𝜆),𝜙_𝑖()=𝜙𝑥,𝜆^𝑖1()[),𝜙𝑥,𝜆,𝑥∈−1,0_𝑖2(𝑥,𝜆),𝑥∈(0,1],𝑖=1,2, (2.1) be the solution of (1.12) satisfying the following initial conditions: 𝜙₁₁(−1,𝜆)𝜙₁₂0⁺𝜙,𝜆₂₁(−1,𝜆)𝜙₂₂0⁺=,𝜆cos𝛼𝛿⁻¹𝜙₁₁(0⁻,𝜆)−sin𝛼𝛿⁻¹𝜙₂₁(0⁻,𝜆). (2.2) Since 𝜙(⋅,𝜆) satisfies (1.13), then the eigenvalues of the problem (1.12)–(1.16) are the zeros of the function Δ(𝜆)=−𝛿²sin𝛽𝜙₁₂(1,𝜆)+cos𝛽𝜙₂₂(1,𝜆). (2.3) Notice that both 𝜙(⋅,𝜆) and Δ(𝜆) are entire functions of 𝜆 and 𝜙(⋅,𝜆) satisfies the system of integral equations, see [32, 35], 𝜙₁₁(𝑥,𝜆)=cos(𝜆(𝑥+1)−𝛼)−𝒯_−1,1𝜙₁₁(𝑥,𝜆)−𝒯_−1,2𝜙₂₁(),𝜙𝑥,𝜆₂₁(𝑥,𝜆)=sin(𝜆(𝑥+1)−𝛼)+𝒯_−1,1𝜙₁₁(𝑥,𝜆)−𝒯_−1,2𝜙₂₁(),𝜙𝑥,𝜆₁₂()=1𝑥,𝜆𝛿𝜙₁₁(0⁻)()−1,𝜆cos𝜆𝑥𝛿𝜙₂₁(0⁻,𝜆)sin(𝜆𝑥)−𝒯_0,1𝜙₁₂(𝑥,𝜆)−𝒯_0,2𝜙₂₂(),𝜙𝑥,𝜆₂₂()=1𝑥,𝜆𝛿𝜙₁₁(0⁻)()+1,𝜆sin𝜆𝑥𝛿𝜙₂₁(0⁻)()+𝒯,𝜆cos𝜆𝑥_0,1𝜙₁₂(𝑥,𝜆)−𝒯_0,2𝜙₂₂(𝑥,𝜆), (2.4)(2.5)(2.6)(2.7) where 𝒯_−1,𝑖, 𝒯_−1,𝑖, 𝒯_0,𝑖 and 𝒯_0,𝑖, 𝑖=1,2, are the Volterra integral operators defined by 𝒯_−1,1𝜑(𝑥,𝜆)∶=^𝑥₋₁sin𝜆(𝑥−𝑡)𝑝₁(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_−1,2𝜑(𝑥,𝜆)∶=^𝑥₋₁sin𝜆(𝑥−𝑡)𝑝₂(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_−1,1𝜑(𝑥,𝜆)∶=^𝑥₋₁cos𝜆(𝑥−𝑡)𝑝₁(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_−1,2𝜑(𝑥,𝜆)∶=^𝑥₋₁cos𝜆(𝑥−𝑡)𝑝₂(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_0,1𝜑(𝑥,𝜆)∶=^𝑥₀sin𝜆(𝑥−𝑡)𝑝₁(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_0,2𝜑(𝑥,𝜆)∶=^𝑥₀sin𝜆(𝑥−𝑡)𝑝₂(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_0,1𝜑(𝑥,𝜆)∶=^𝑥₀cos𝜆(𝑥−𝑡)𝑝₁(𝑡)𝜑()𝒯𝑡,𝜆𝑑𝑡,_0,2𝜑(𝑥,𝜆)∶=^𝑥₀cos𝜆(𝑥−𝑡)𝑝₂(𝑡)𝜑(𝑡,𝜆)𝑑𝑡. (2.8)

For convenience, we define the constants 𝑐₁∶=⁰₋₁𝑝₁(𝑡)+𝑝₂(𝑡)𝑑𝑡,𝑐₂∶=𝑐₁exp𝑐₁,𝑐₃∶=¹₀𝑝₁(𝑡)+𝑝₂(𝑡)𝑑𝑡,𝑐₄∶=𝑐₂+2|𝛿|1+𝑐₂. (2.9) Define 𝑓_−1,𝑖(⋅,𝜆) and 𝑓_0,𝑖(⋅,𝜆), 𝑖=1,2, to be 𝑓_−1,1(𝑥,𝜆)∶=𝒯_−1,1𝜙₁₁(𝑥,𝜆)+𝒯_−1,2𝜙₂₁(),𝑓𝑥,𝜆_−1,2(𝑥,𝜆)∶=𝒯_−1,1𝜙₁₁(𝑥,𝜆)−𝒯_−1,2𝜙₂₁(),𝑓𝑥,𝜆_0,1(𝑥,𝜆)∶=𝒯_0,1𝜙₁₂(𝑥,𝜆)+𝒯_0,2𝜙₂₂(),𝑓𝑥,𝜆_0,2(𝑥,𝜆)∶=𝒯_0,1𝜙₁₂(𝑥,𝜆)−𝒯_0,2𝜙₂₂(𝑥,𝜆). (2.10)(2.11)

Lemma 2.1. The functions 𝑓_−1,1(𝑥,𝜆) and 𝑓_−1,2(𝑥,𝜆) are entire in 𝜆 for any fixed 𝑥∈[−1,0) and satisfy the growth condition 𝑓_−1,1(𝑥,𝜆),𝑓_−1,2(𝑥,𝜆)≤2𝑐₂𝑒^{|ℑ𝜆|(𝑥+1)},𝜆∈ℂ. (2.12)

Proof. Since 𝑓_−1,1(𝑥,𝜆)=𝒯_−1,1𝜙₁₁(𝑥,𝜆)+𝒯_−1,2𝜙₂₁(𝑥,𝜆), then from (2.4) and (2.5) we obtain 𝑓_−1,1(𝑥,𝜆)=𝒯_−1,1cos(𝜆(𝑥+1)−𝛼)+𝒯_−1,2sin(𝜆(𝑥+1)−𝛼)−𝒯_−1,1𝑓_−1,1()+𝒯𝑥,𝜆_−1,2𝑓_−1,2(𝑥,𝜆). (2.13) Using the inequalities |sin𝑧|≤𝑒^|ℑ𝑧| and |cos𝑧|≤𝑒^|ℑ𝑧| for 𝑧∈ℂ, leads for 𝜆∈ℂ to 𝑓_−1,1(𝑥,𝜆)≤𝒯_−1,1cos(𝜆(𝑥+1)−𝛼)+𝒯_−1,2(𝜆()+𝒯sin𝑥+1−𝛼)_−1,1𝑓_−1,1(𝑥,𝜆)+𝒯_−1,2𝑓_−1,2(𝑥,𝜆)≤𝑒^{|ℑ𝜆|(𝑥+1)𝑥}₋₁𝑝₁(𝑡)𝑓_−1,1(𝑡,𝜆)+𝑝₂(𝑡)𝑓_−1,2(𝑡,𝜆)𝑒^{−|ℑ𝜆|(𝑡+1)}𝑑𝑡+2𝑒^{|ℑ𝜆|(𝑥+1)𝑥}₋₁𝑝₁(𝑡)+𝑝₂(𝑡)𝑑𝑡≤2𝑐₁𝑒^{|ℑ𝜆|(𝑥+1)}+𝑒^{|ℑ𝜆|(𝑥+1)𝑥}₋₁𝑝₁(𝑡)𝑓_−1,1(𝑡,𝜆)+𝑝₂(𝑡)𝑓_−1,2(𝑡,𝜆)𝑒^{−|ℑ𝜆|(𝑡+1)}𝑑𝑡. (2.14) The above inequality can be reduced to 𝑒^{−|ℑ𝜆|(𝑥+1)}𝑓_−1,1(𝑥,𝜆)≤2𝑐₁+^𝑥₋₁𝑝₁(𝑡)𝑓_−1,1(𝑡,𝜆)+𝑝₂(𝑡)𝑓_−1,2(𝑡,𝜆)𝑒^{−|ℑ𝜆|(𝑡+1)}𝑑𝑡. (2.15) Similarly, we can prove that 𝑒^{−|ℑ𝜆|(𝑥+1)}𝑓_−1,2(𝑥,𝜆)≤2𝑐₁+^𝑥₋₁𝑝₁(𝑡)𝑓_−1,1(𝑡,𝜆)+𝑝₂(𝑡)𝑓_−1,2(𝑡,𝜆)𝑒^{−|ℑ𝜆|(𝑡+1)}𝑑𝑡. (2.16) Then from (2.12) and (2.15) and Lemma 3.1 of [34, pp. 204], we obtain (2.12).

In a similar manner, we will prove the following lemma for 𝑓_0,1(⋅,𝜆) and 𝑓_0,2(⋅,𝜆).

Lemma 2.2. The functions 𝑓_0,1(𝑥,𝜆) and 𝑓_0,2(𝑥,𝜆) are entire in 𝜆 for any fixed 𝑥∈(0,1] and satisfy the growth condition 𝑓_0,1(𝑥,𝜆),𝑓_0,2(𝑥,𝜆)≤2𝑐₃𝑐₄𝑒^{|ℑ𝜆|(𝑥+1)},𝜆∈ℂ. (2.17)

Proof. Since 𝑓_0,1(𝑥,𝜆)=𝒯_0,1𝜙₁₁(𝑥,𝜆)+𝒯_0,2𝜙₂₁(𝑥,𝜆), then from (2.6) and (2.7) we obtain 𝑓_0,1()=1𝑥,𝜆𝛿𝜙₁₁(0⁻,𝜆)𝒯_0,1()−1cos𝜆𝑥𝛿𝜙₂₁(0⁻,𝜆)𝒯_0,1sin(𝜆𝑥)−𝒯_0,1𝑓_−1,2()+1𝑥,𝜆𝛿𝜙₁₁(0⁻,𝜆)𝒯_0,2()+1sin𝜆𝑥𝛿𝜙₂₁(0⁻,𝜆)𝒯_0,2cos(𝜆𝑥)+𝒯_0,2𝑓_−1,2(𝑥,𝜆). (2.18) Then from (2.4) and (2.5) and Lemma 2.1, we get 𝑓_0,1()≤1𝑥,𝜆|𝛿|𝜙₁₁(0⁻,𝜆)𝒯_0,1(+1cos𝜆𝑥)|𝛿|𝜙₂₁(0⁻,𝜆)𝒯_0,1sin(𝜆𝑥)+𝒯_0,1𝑓_−1,2(+1𝑥,𝜆)|𝛿|𝜙₁₁(0⁻,𝜆)𝒯_0,2(+1sin𝜆𝑥)|𝛿|𝜙₂₁(0⁻,𝜆)𝒯_0,2cos(𝜆𝑥)+𝒯_0,2𝑓_−1,2(𝑐𝑥,𝜆)≤2₂+2|𝛿|1+𝑐₂𝑐₃𝑒^{|ℑ𝜆|(𝑥+1)}=2𝑐₃𝑐₄𝑒^{|ℑ𝜆|(𝑥+1)}. (2.19) Similarly, we can prove that 𝑓_0,2(𝑥,𝜆)≤2𝑐₃𝑐₄𝑒^{|ℑ𝜆|(𝑥+1)}. (2.20)

3. The Method and Error Analysis

In this section we derive the method of computing the eigenvalues of problem (1.12)–(1.16) numerically. We aim to approximate Δ(𝜆) and hence its zeros, that is, the eigenvalues. The idea is to split Δ(𝜆) into two parts, one is known and the other is unknown, but is an entire function of exponential type. Then we approximate the unknown part using (1.6) to get the approximate Δ(𝜆) and then compute the approximate zeros. Now, let us split Δ(𝜆) into Δ(𝜆)∶=𝒦(𝜆)+𝒰(𝜆), (3.1) where 𝒰(𝜆) is the unknown part involving integral operators 𝒰(𝜆)∶=𝛿²sin𝛽𝑓_0,1(1,𝜆)−𝛿²cos𝛽𝑓_0,2(1,𝜆)−𝛿sin𝜆−𝛽𝑓_−1,1(0⁻)𝑓,𝜆−𝛿cos𝜆+𝛽_−1,2(0⁻,𝜆), (3.2) and 𝒦(𝜆) is the known part 𝒦(𝜆)∶=−𝛿sin2𝜆−𝛼+𝛽. (3.3) Then, from Lemmas 2.1 and 2.2, we have the following result.

Lemma 3.1. The function 𝒰(𝜆) is entire in 𝜆 and the following estimate holds: 𝒰𝜇≤𝑀𝑒^2|ℑ𝜆|, (3.4) where 𝑀∶=4|𝛿|𝑐₂+|𝛿|𝑐₃𝑐₄. (3.5)

Proof. From (3.2), we have |𝒰(𝜆)|≤|𝛿|²sin𝛽𝑓_0,1(1,𝜆)+|𝛿|²cos𝛽𝑓_0,2(1,𝜆)+|𝛿|sin𝜆−𝛽𝑓_−1,1(0⁻+|𝛿𝑓,𝜆)|cos𝜆+𝛽_−1,2(0⁻,𝜆). (3.6) Using the inequalities |sin𝜆|≤𝑒^|ℑ𝜆| and |cos𝜆|≤𝑒^|ℑ𝜆| for 𝜆∈ℂ, and Lemmas 2.1 and 2.2 imply (3.4).

Then 𝒰(𝜆) is an entire function of exponential type 2. In the following we let 𝜆∈ℝ since all eigenvalues are real. Now we approximate the function 𝒰(𝜆) using the operator (1.6) where ℎ∈(0,𝜋/2] and 𝜔∶=(𝜋−2ℎ)/2 and then we obtain 𝒰(𝜆)−𝒢_ℎ,𝑁𝒰(𝜆)≤𝑇_ℎ,𝑁(𝜆), (3.7) where 𝑇_ℎ,𝑁(𝜆)∶=2sinℎ⁻¹𝜋𝜆‖𝒰‖_∞𝑒^−𝜔𝑁√𝛽𝜋𝜔𝑁^𝑁(0),𝜆∈ℝ. (3.8) The samples 𝒰(𝑛ℎ)=Δ(𝑛ℎ)−𝒦(𝑛ℎ), 𝑛∈ℤ_𝑁(𝜆) cannot be computed explicitly in the general case. We approximate these samples numerically by solving the initial-value problems defined by (1.12) and (2.2) to obtain the approximate values 𝒰(𝑛ℎ), 𝑛∈ℤ_𝑁(𝜆), that is, 𝒰(𝑛ℎ)=Δ(𝑛ℎ)−𝒦(𝑛ℎ). Accordingly we have the explicit expansion 𝒢_ℎ,𝑁𝒰(𝜆)∶=_{𝑛∈ℤ𝑁(𝜆)}𝒰(𝑛ℎ)𝑆_𝑛𝜋𝜆ℎ𝐺√𝜔(𝜆−𝑛ℎ)√.𝑁ℎ (3.9) Therefore we get, compare with (1.10), 𝒢_ℎ,𝑁𝒰(𝜆)−𝒢_ℎ,𝑁𝒰(𝜆)≤𝐴_𝜀,𝑁(0),𝜆∈ℝ. (3.10) Now let Δ_𝑁(𝜆)∶=𝒦(𝜆)+(𝒢_ℎ,𝑁𝑈)(𝜆). From (3.7) and (3.10) we obtain Δ(𝜆)−Δ_𝑁(𝜆)≤𝑇_ℎ,𝑁(𝜆)+𝐴_𝜀,𝑁(0),𝜆∈ℝ. (3.11) Let 𝜆^∗ be an eigenvalue and 𝜆_𝑁 be its desired approximation, that is, Δ(𝜆^∗)=0 and Δ_𝑁(𝜆_𝑁)=0. From (3.11) we have |Δ_𝑁(𝜆^∗)|≤𝑇_ℎ,𝑁(𝜆^∗)+𝐴_𝜀,𝑁(0). Now we define an enclosure interval 𝐼_𝜀,𝑁 for 𝜆^∗. Define the curves 𝑎_±(𝜆)=Δ_𝑁(𝜆)±𝑇_ℎ,𝑁(𝜆)+𝐴_𝜀,𝑁(0). (3.12) The curves 𝑎₊(𝜆), 𝑎₋(𝜆) trap the curve of Δ(𝜆) for suitably large 𝑁. Hence the closure interval is determined by solving 𝑎_±(𝜆)=0, gives an interval 𝐼_𝜀,𝑁∶=[𝑎₋,𝑎₊]. Next we estimate the error |𝜆^∗−𝜆_𝑁| for the eigenvalue 𝜆^∗.

Theorem 3.2. Let 𝜆^∗ be an eigenvalue of (1.12)–(1.16) and 𝜆_𝑁 be its approximation. Then, for 𝜆∈ℝ, one has the following estimate: 𝜆^∗−𝜆_𝑁<𝑇_ℎ,𝑁𝜆_𝑁+𝐴_𝜀,𝑁(0)inf_{𝜁∈𝐼𝜀,𝑁}Δ^′(𝜁,) (3.13) where the interval 𝐼_𝜀,𝑁 is defined above.

Proof. Replacing 𝜆 by 𝜆_𝑁 in (3.11) we obtain Δ𝜆_𝑁−Δ𝜆^∗<𝑇_ℎ,𝑁𝜆_𝑁+𝐴_𝜀,𝑁(0), (3.14) where we have used Δ_𝑁(𝜆_𝑁)=Δ(𝜆^∗)=0. Using the mean value theorem yields that for some 𝜁∈𝐽_𝜀,𝑁∶=[min(𝜆^∗,𝜆_𝑁),max(𝜆^∗,𝜆_𝑁)], 𝜆^∗−𝜆_𝑁Δ^′(𝜁)≤𝑇_ℎ,𝑁𝜆_𝑁+𝐴_𝜀,𝑁(0),𝜁∈𝐽_𝜀,𝑁⊂𝐼_𝜀,𝑁. (3.15) Since 𝜆^∗ is simple and 𝑁 is sufficiently large, then inf_{𝜁∈𝐼𝜀,𝑁}|Δ^′(𝜁)|>0 and we get (3.13).

4. Numerical Examples and Comparisons

This section includes two examples illustrating the sinc-Gaussian method. In the following examples, we consider 𝜆_𝑘,𝑁 being the kth root of 𝒦(𝜆)+(𝒢_ℎ,𝑁𝒰)=0. Also, it is observed that the approximation 𝜆_𝑘,𝑁 and the exact solution 𝜆_𝑘 are all inside the interval [𝑎₋,𝑎₊]. We indicate in these two examples the effect of the amplitude error in the proposed method by determining enclosure intervals for different values of 𝜀. All examples are computed in [32] by using the classical sinc method. We see that the sinc-Gaussian method gives remarkably better results.

Example 4.1. Consider the system 𝑢^′₂(𝑥)−𝑝(𝑥)𝑢₁(𝑥)=𝜆𝑢₁(𝑥),𝑢^′₁(𝑥)+𝑝(𝑥)𝑢₂(𝑥)=−𝜆𝑢₂(𝑥),𝑥∈[−1,0)∪(0,1], (4.1) subject to 𝑢₁(−1)+𝑢₂(−1)=0,𝑢₁(1)+𝑢₂(1)𝑢=0,₁(0⁻)−2𝑢₁0⁺=0,𝑢₂(0⁻)−2𝑢₂0⁺=0. (4.2) Here 𝑝₁(𝑥)=𝑝₂(𝑥)(𝑥)=𝑥=𝑝²[)(],,𝑥∈−1,0𝑥,𝑥∈0,1 (4.3)𝛼=𝛽=𝜋/4 and 𝛿=2.
Direct calculations give 𝒦(𝜆)[],Δ(𝜆)5=−2sin2𝜆=−2sin6+2𝜆, (4.4) therefore the eigenvalues are𝜆_𝑘=(6𝑘𝜋−5)/12, 𝑘∈ℤ.
Let 𝐸 and 𝐸_𝐺 denote the absolute errors associated with the results of classical sinc method and sinc-Gaussian method, respectively. In Table 1, we give comparison between the absolute error of sinc-Gaussian and the classical sinc-method.

Table 1 

The absolute errors of sinc-Gaussian and the classical sinc-method with 𝑁=20, ℎ=0.2, 𝜔=1.3708.

In Table 2, we observe that the approximation 𝜆_𝑘,𝑁 and the exact solution 𝜆_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

Table 2 

The approximation 𝜆_𝑘,𝑁 and the exact solution 𝜆_𝑘 are all inside the interval [𝑎₋, 𝑎₊] for different values of 𝜀.

Example 4.2. In this example we consider the system 𝑢^′₂(𝑥)−𝑝(𝑥)𝑢₁(𝑥)=𝜆𝑢₁(𝑥),𝑢^′₁(𝑥)+𝑝(𝑥)𝑢₂(𝑥)=−𝜆𝑢₂(𝑥),𝑥∈[−1,0)∪(0,1], (4.5) subject to √3𝑢₁(−1)+𝑢₂(−1)=0,𝑢₁(1)+√3𝑢₂(1)𝑢=0,₁(0⁻)−3𝑢₁0⁺=0,𝑢₂(0⁻)−3𝑢₂0⁺=0, (4.6) where 𝑝₁(𝑥)=𝑝₂(𝑥)(𝑥)=[)𝑥=𝑝𝑥,𝑥∈−1,0₂+1,𝑥∈(0,1], (4.7)𝛼=𝜋/3, 𝛽=𝜋/6 and 𝛿=3.
Direct calculations give 𝒦(𝜆)𝜋=3sin6,Δ(𝜆)=3−2𝜆25cos6+2𝜆−√53sin6+2𝜆, (4.8) therefore the eigenvalues are 𝜆_𝑘=((6𝑘+1)𝜋−5)/12, 𝑘∈ℤ.
Table 3 gives comparison between the absolute error of sinc-Gaussian and the classical sinc-method. Moreover, in Table 4 the approximation 𝜆_𝑘,𝑁 and the exact solution 𝜆_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

Table 3 

The absolute errors of sinc-Gaussian and the classical sinc-method with 𝑁=20, ℎ=0.3, 𝜔=1.2708.

Table 4 

The approximation 𝜆_𝑘,𝑁 and the exact solution 𝜆_𝑘 are all inside the interval [𝑎₋, 𝑎₊] for different values of 𝜀.

Acknowledgment

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