Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions
Tharwat et al. Boundary Value Problems 2013, 2013:132
http://www.boundaryvalueproblems.com/content/2013/1/132
RESEARCH
Open Access
Approximation of eigenvalues of
discontinuous Sturm-Liouville problems with
eigenparameter in all boundary conditions
Mohammed M Tharwat1* , Ali H Bhrawy1,2 and Abdulaziz S Alofi1
*
Correspondence:
1
Department of Mathematics,
Faculty of Science, King Abdulaziz
University, Jeddah, Saudi Arabia
2
Permanent address: Department of
Mathematics, Faculty of Science,
Beni-Suef University, Beni-Suef,
Egypt
Abstract
In this paper, we apply a sinc-Gaussian technique to compute approximate values of
the eigenvalues of Sturm-Liouville 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; Sturm-Liouville problems; transmission conditions;
sinc-Gaussian; sinc method; truncation and amplitude errors
1 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 band-limited, the sampling process can be
done via the celebrated Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem [–].
By a band-limited signal with band width τ , τ > , we mean a function in the Paley-Wiener
space
Bτ := f entire, f (λ) ≤ Ceτ |λ| , f (λ) dλ .
R
(.)
The WKS sampling theorem is a fundamental result in information theory. It states that
any f ∈ Bτ can be reconstructed from its sampled values f (xk ), where xk = kπ/τ and k ∈ Z,
by the formula
f (x) =
f (xk ) sinc(τ x/π – k),
x ∈ R,
(.)
k∈Z
where
⎧
⎨ sin π x , x ∈ R \ {},
πx
sinc(x) :=
⎩,
x = ,
(.)
© 2013 Tharwat et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Tharwat et al. Boundary Value Problems 2013, 2013:132
http://www.boundaryvalueproblems.com/content/2013/1/132
Page 2 of 15
and the series converges absolutely and uniformly on any finite interval of R. Expansion
(.) is used in several approximation problems which are known as sinc methods; see, e.g.,
[–]. In particular the sinc-method is used to approximate eigenvalues of boundary value
problems; see, for example, [–]. The sinc-method has a slow rate of decay at infinity,
which is as slow as O(|x– |). There are several attempts to improve the rate of decay. One
of the interesting ways is to multiply the sinc-function in (.) by a kernel function; see,
e.g., [–]. Let h ∈ (, π/τ ] and γ ∈ (, π – hτ ). Assume that ∈ Bγ such that () = ,
then for f ∈ Bτ we have the expansion, []
f (x) =
∞
f (nh) sinc h– πx – nπ h– x – n .
(.)
n=–∞
The speed of convergence of the series in (.) is determined by the decay of |(x)|. But
the decay of an entire function of exponential type cannot be as fast as e–c|x| as |x| −→ ∞,
for some positive c []. In [], Qian has introduced the following regularized sampling
formula. For h ∈ (, π/τ ], N ∈ N and r > , Qian defined the operator []
(Gh,N f )(x) =
n∈ZN
x – nh
,
f (nh)Sn h– πx G √
rh
(x)
x ∈ R,
(.)
where G(t) := exp(–t ), which is called the Gaussian function, Sn (h– πx) := sinc(h– πx –
nπ), ZN (x) := {n ∈ Z : |[h– x] – n| ≤ N} and [x] denotes the integer part of x ∈ R; see also
[, ]. Qian also derived the following error bound. If f ∈ Bτ , h ∈ (, π/τ ] and a :=
min{r(π – hτ ), (N – )/r} ≥ , then [, ]
√
f (x) – (Gh,N f )(x) ≤ τ π f
π a
√
/
πa + e/r e–a
,
x ∈ R.
(.)
In [] Schmeisser and Stenger extended the operator (.) to the complex domain C. For
τ > , h ∈ (, π/τ ] and ω := (π – hτ )/, they defined the operator []
(Gh,N f )(z) :=
f (nh)Sn
n∈ZN (z)
πz
G
h
√
ω(z – nh)
,
√
Nh
(.)
where ZN (z) := {n ∈ Z : |[h– z + /] – n| ≤ N} and N ∈ N. Note that the summation
limits in (.) depend on the real part of z. Schmeisser and Stenger [] proved that if f is
an entire function such that
f (ξ + iη) ≤ φ |ξ | eτ |η| ,
ξ , η ∈ R,
(.)
where φ is a non-decreasing, non-negative function on [, ∞) and τ ≥ , then for h ∈
(, π/τ ), ω := (π – hτ )/, N ∈ N, |z| < N , we have
f (z) – (Gh,N f )(z)
e–ωN
βN h– z ,
≤ sin h– πz φ | z| + h(N + ) √
πωN
z ∈ C,
(.)
Tharwat et al. Boundary Value Problems 2013, 2013:132
http://www.boundaryvalueproblems.com/content/2013/1/132
Page 3 of 15
where
e–ωt
eωt
βN (t) := cosh(ωt) + √
+
.
+
πωN[ – (t/N) ] eπ (N–t) – eπ (N+t) –
eωt
/N
(.)
The amplitude error arises when the exact values f (nh) of (.) are replaced by the approximations
f (nh). We assume that
f (nh) are close to f (nh), i.e., there is ε > sufficiently
small such that
f (nh) < ε.
sup f (nh) –
(.)
n∈Zn (z)
Let h ∈ (, π/τ ), ω := (π – hτ )/ and N ∈ N be fixed numbers. The authors in [] proved
that if (.) holds, then for |z| < N , we have
(Gh,N f )(z) – (Gh,N
f )(z) ≤ Aε,N (z),
(.)
where
Aε,N (z) = εe–ω/N ( +
√
N/ωπ) exp (ω + π)h– |z| .
(.)
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. 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 [–] and corresponding bibliography cited
therein. In particular, [, , , ] contain many references to problems in physics and
mechanics. Our task is to use formula (.) to compute the eigenvalues numerically of the
differential equation
x ∈ –, ) ∪ (, ,
(.)
L (y) := α μ – α y(–, μ) – α μ – α y (–, μ) = ,
(.)
L (y) := β μ + β y(, μ) – β μ + β y (, μ) = ,
(.)
–y (x, μ) + q(x)y(x, μ) = μ y(x, μ),
with boundary conditions
and transmission conditions
L (y) := γ y – , μ – δ y + , μ = ,
(.)
L (y) := γ y – , μ – δ y + , μ = ,
(.)
where μ is a complex spectral parameter; q(x) is a given real- (...truncated)
