Abstract
In this paper, we apply a sincGaussian technique to compute approximate values of the eigenvalues of SturmLiouville problems which contain an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than that of the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.
MSC: 34L16, 94A20, 65L15.
Keywords:
sampling theory; SturmLiouville problems; transmission conditions; sincGaussian; sinc method; truncation and amplitude errors1 Introduction
By a sampling theorem we mean a representation of a certain function in terms of its
values at a discrete set of points. In communication theory, it means a reconstruction
of a signal (information) in terms of a discrete set of data. This has several applications,
especially in the transmission of information. If the signal is bandlimited, the
sampling process can be done via the celebrated WhittakerKotel’nikovShannon (WKS)
sampling theorem [13]. By a bandlimited signal with band width τ,
The WKS sampling theorem is a fundamental result in information theory. It states
that any
where
and the series converges absolutely and uniformly on any finite interval of ℝ. Expansion
(1.2) is used in several approximation problems which are known as sinc methods; see,
e.g., [47]. In particular the sincmethod is used to approximate eigenvalues of boundary value
problems; see, for example, [814]. The sincmethod has a slow rate of decay at infinity, which is as slow as
The speed of convergence of the series in (1.4) is determined by the decay of
where
In [18] Schmeisser and Stenger extended the operator (1.5) to the complex domain ℂ. For
where
where ϕ is a nondecreasing, nonnegative function on
where
The amplitude error arises when the exact values
Let
where
It is well known that many topics in mathematical physics require the investigation of the eigenvalues and eigenfunctions of SturmLiouville type boundary value problems. Therefore, the Sturmian theory is one of the most actual and extensively developing fields of theoretical and applied mathematics. Particularly, in recent years, highly important results in this field have been obtained for the case when the eigenparameter appears not only in the differential equation but also in the boundary conditions. The literature on such results is voluminous, and we refer to [2327] and corresponding bibliography cited therein. In particular, [24,26,28,29] contain many references to problems in physics and mechanics. Our task is to use formula (1.7) to compute the eigenvalues numerically of the differential equation
with boundary conditions
and transmission conditions
where μ is a complex spectral parameter;
The eigenvalue problem (1.14)(1.18) when
Our goal in this paper is to improve the results presented in Tharwat et al.[14] with the least conditions. In this paper we use the sincGaussian sampling formula
(1.7) to compute eigenvalues of (1.14)(1.18) numerically. As is expected, the new
method reduced the error bounds remarkably; see the examples at the end of this paper.
Also here, we use the same idea but the unknown part
2 Treatment of the eigenvalue problem (1.14)(1.18)
In this section we derive approximate values of the eigenvalues of the eigenvalue problem (1.14)(1.18). Recall that the problem (1.14)(1.18) has a denumerable set of real and simple eigenvalues, cf.[30]. Let
denote the solution of (1.14) satisfying the following initial conditions:
Since
According to [30], see also [3440], the function
where
Differentiating (2.3) and (2.4), we obtain
where
Define
In the following, we make use of the known estimates [41]
where
As in [14], we split
where
and
Then the function
where
and
Then
where
The samples
Therefore we get, cf. (1.12),
Now let
Let
The curves
It is worthwhile to mention that the simplicity of the eigenvalues guarantees the
existence of approximate eigenvalues, i.e., the
Theorem 2.1Let
where the interval
Proof Replacing μ by
where we have used
Since
3 Examples
This section includes two examples illustrating the sincGaussian method. All examples are computed in [14] with the classical sincmethod. It is clearly seen that the sincGaussian method gives remarkably better results. 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 N and h by several choices. We would like to mention that MATHEMATICA has been used to obtain the exact values for these examples where eigenvalues cannot computed concretely. MATHEMATICA is also used in rounding the exact eigenvalues, which are square roots. Each example is exhibited via figures that accurately illustrate the procedure near to some of the approximated eigenvalues. More explanations are given below.
Example 1 Consider the boundary value problem
Here
The characteristic function is
The function
As is clearly seen, eigenvalues cannot be computed explicitly. Tables 1, 2, 3 indicate the application of our technique to this problem and the effect of ε. By exact we mean the zeros of
Table 1. The approximation
Table 2. Absolute error
Table 3. For
Figures 1 and 2 illustrate the enclosure intervals dominating
Figure 1. The enclosure interval dominating
Figure 2. The enclosure interval dominating
Figure 3. The enclosure interval dominating
Figure 4. The enclosure interval dominating
Example 2 Consider the boundary value problem
where
The function
The characteristic determinant of the problem is
where
Table 4. The approximation
Table 5. Absolute error
Table 6. For
Here Figures {5, 6}, {7, 8} illustrate the enclosure intervals dominating
Figure 5. The enclosure interval dominating
Figure 6. The enclosure interval dominating
Figure 7. The enclosure interval dominating
Figure 8. The enclosure interval dominating
4 Conclusion
With a simple analysis, and with values of solutions of initial value problems computed
at a few values of the eigenparameter, we have computed the eigenvalues of discontinuous
SturmLiouville problems which contain an eigenparameter appearing linearly in two
boundary conditions, with a certain estimated error. The method proposed is a shooting
procedure, i.e., the problem is reformulated as two initial value ones, due to the interior discontinuity,
of size two and a missdistance is defined at the right end of the interval of integration
whose roots are the eigenvalues to be computed. The unknown part
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have equal contributions to each part of this article. All the authors read and approved the final manuscript.
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130065D1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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