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Computation of eigenvalues of discontinuous dirac system using Hermite interpolation technique
Advances in Difference Equations volume 2012, Article number: 59 (2012)
We use the derivative sampling theorem (Hermite interpolations) to compute eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity numerically. We closely follow the analysis derived by Levitan and Sargsjan (1975) to establish the needed relations. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Numerical examples, illustrations and comparisons with the sinc methods are exhibited.
Mathematical Subject Classification 2010: 34L16; 94A20; 65L15.
Let σ > 0 and be the Paley-Wiener space of all L2(ℝ)-entire functions of exponential type type σ. Assume that . Then f(t) can be reconstructed via the sampling series
where S n (t) is the sequences of sinc functions
Series (1) converges absolutely and uniformly on ℝ (cf. [1–4]). Sometimes, series (1) is called the derivative sampling theorem. Our task is to use formula (1) to compute eigenvalues of Dirac systems numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1) (cf. ). 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
It is proved in  that if and f(t) is sufficiently smooth in the sense that there exists k ∈ ℤ+ such that tkf(t) ∈ L2(ℝ), then, for t ∈ ℝ, |t| < Nπ/σ, we have
where the constants E k and ξk,σare given by
The amplitude error occurs when approximate samples are used instead of the exact ones, which we can not compute. It is defined to be
where and are approximate samples of and , respectively. Let us assume that the differences are bounded by a positive number ε, i.e. If satisfies the natural decay conditions
0 < λ ≤ 1, then for we have, ,
and is the Euler-Mascheroni constant.
The classical  sampling theorem of Whittaker, Kotel'nikov and Shannon (WKS) for is the series representation
where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ (cf. [6–8]). Series (12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second order eigenvalue problems (see e.g. [9–13]). The use of (12) in numerical analysis is known as the sinc-method established by Stenger (cf. [14–16]). In [11, 12], the authors applied (12) and the regularized sinc method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [17, 18]. The regularized sinc method; a method which is based on (WKS) 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. The aim of this article 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  if and only if there is g(·)∈L2(-σ, σ) such that
Therefore i.e, f′(t) also has an expansion of the form (12). However, f′(t) can be also obtained by term-by-term differentiation formula of (12)
see [, p. 52] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples will be used to compute both f(t) and f′(t) according to (12) and (14), respectively. We would like to mention that works in direction of computing eigenvalues with the new method, Hermite interpolation technique, are few (see e.g. ). Also articles in computing of eigenvalues with discontinuous are few (see [20–22]). However the computing of eigenvalues by Hermite interpolation technique which has discontinuity conditions, do not exist as for as we know. The next section contains some preliminary results. The method with error estimates are contained in Section three. The last section involves some illustrative examples.
2 The eigenvalue problem
and transmission conditions
where λ ∈ ℂ; the real valued function r1(·) and r2(·) are continuous in [−1, 0) and (0, 1], and have finite limits and δ ≠ 0.
Let H be the Hilbert space
The inner product of H is defined by
where ⊤ denotes the matrix transpose,
Equation (15) can be written as
For functions u(x), which defined on [−1, 0) ⋃ (0, 1] and has finite limit by u(1)(x) and u(2)(x) we denote the functions
which are defined on Γ1 := [−1, 0] and Γ2 := 0 respectively.
In the following lemma, we will prove that the eigenvalues of the problem (15)-(19) are real.
Lemma 2.1 The eigenvalues of the problem (15)-(19) are real.
Proof. Assume the contrary that λ0 is a nonreal eigenvalue of problem (15)-(19). Let be a corresponding (non-trivial) eigenfunction. By (15), we have, for x ∈ [−1, 0) ⋃ ( 0, 1],
Integrating the above equation through [−1, 0) and (0, 1], we obtain
Then from (16), (17) and transmission conditions, we have respectively
Since it follows from the last three equations and (25), (26) that
Then u i (x) = 0, i =1, 2 and this is contradiction. Consequently, λ0 must be real.
Lemma 2.2 Let λ1 and λ2 be two different eigenvalues of the problem (15)-(19). Then the corresponding eigenfunctions u(x, λ1) and v(x, λ2) of this problem satisfy the following equality
Proof. By (15) we obtain
Integrating the above equation through [−1, 0) and (0, 1], and taking into account u(x, λ1) and v(x, λ2) satisfy (16)-(19), we obtain (28), where λ1≠λ2.
Now, we shall construct a special fundamental system of solutions of the Equation (15) for λ not being an eigenvalue. Let us consider the next initial value problem:
By virtue of Theorem 1.1 in  this problem has a unique solution which is an entire function of λ ∈ ℂ for each fixed x ∈ [−1, 0]. Similarly, employing the same method as in proof of Theorem 1.1 in , we see that the problem
has a unique solution which is an entire function of parameter λ for each fixed x ∈ [0.1].
Now the functions φi 2(x, λ) and χi 1(x, λ) are defined in terms of φi 1(x, λ) and χi 2(x, λ), i =1, 2, respectively, as follows: The initial-value problem,
has unique solution for each λ ∈ ℂ.
Similarly, the following problem also has a unique solution
Let us construct two basic solutions of the equation (15) as
Let W (φ, χ)(·, λ) denote the Wronskian of φ(·, λ) and χ(·, λ) defined in [, p. 194], i.e.,
It is obvious that the Wronskian
are independent of x ∈ Γi and are entire functions. Taking into account (34) and (36), a short calculation gives
for each λ ∈ ℂ.
Corollary 2.3 The zeros of the functions Ω1(λ) and Ω2(λ) coincide.
Then, we may introduce to the consideration the characteristic function Ω(λ) as
In the following lemma, we show that all eigenvalues of the problem (15)-(19) are simple.
Lemma 2.4 All eigenvalues of problem (15)-(19) are just zeros of the function Ω(λ). Moreover, every zero of Ω(λ) has multiplicity one.
Proof. Since the functions φ1(x, λ) and φ2(x, λ) satisfy the boundary condition (16) and both transmission conditions (18) and (19), to find the eigenvalues of the (15)-(19) we have to insert the functions φ1(x, λ) and φ2(x, λ) in the boundary condition (17) and find the roots of this equation.
By (15) we obtain for λ, µ ∈ ℂ, λ ≠ μ,
Integrating the above equation through [−1, 0) and (0, 1], and taking into account the initial conditions (30), (34) and (36), we obtain
Dividing both sides of (41) by (λ − µ) and by letting µ → λ, we arrive to the relation
We show that equation
has only simple roots. Assume the converse, i.e., Equation (43) has a double root λ∗, say. Then the following two equations hold
Combining (46) and (42), with λ = λ*, we obtain
It follows that φ1(x, λ*)=φ2(x, λ*)=0, which is impossible. This proves the lemma.
Here will be a sequence of eigen-vector-functions of (15)-(19) corresponding to the eigenvalues Since χ(·, λ) satisfies (17)-(19), then the eigenvalues are also determined via
Therefore is another set of eigen-vector-functions which is related by with
Where c n ≠ 0 are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the eigen-vector-functions to be real valued.
Since φ(·, λ) satisfies (16), then the eigenvalues of the problem (15)-(19) are the zeros of the function
where and are the Volterra integral operators defined by
For convenience, we define the constants
Define h−1,i(·, λ) and h0,i(·, λ), i = 1, 2, to be
Lemma 2.5 The functions h−1,1(x, λ) and h−1,2(x, λ) are entire in λ for any fixed x ∈ [−1, 0) and satisfy the growth condition
Proof. Since then from (51) and (52) we obtain Using the inequalities and leads for λ ∈ ℂ to
The above inequality can be reduced to
Similarly, we can prove that
Then from (58) and (59) and and Lemma 3.1 of [, pp. 204], we obtain (58).
In a similar manner, we will prove the following lemma for h0,1(·, λ) and h0,2(·, λ).
Lemma 2.6 The functions h0,1(x, λ) and h0,2(x, λ) are entire in λ for any fixed x ∈ (0, 1] and satisfy the growth condition
Proof. Since then from (53) and (54) we obtain
Then from (51) and (52) and Lemma 2.5, we get
Similarly, we can prove that
3 The numerical scheme
In this section we derive the method of computing eigenvalues of problem (15)-(19) numerically. The basic idea of the scheme is to split Ω(λ) into two parts a known part and an unknown one . Then we prove that has an expansion of the form (1). We then approximate in two stages. First by truncating the sampling expansion (4) and then by approximating the samples, using standard methods of solving ordinary differential equations. This produces both a truncation error and an amplitude error. We apply forms (4) and (7) to derive an estimate of the error of the technique. We first split Ω(λ) into two parts:
where is the unknown part involving integral operators
and is the known part
Then, from Lemmas 2.5 and 2.6, we have the following result.
Lemma 3.1 The function is entire in λ and the following estimate holds
Proof. From (63), we have
Using the inequalities and for λ ∈ ℂ, and Lemmas 2.5 and 2.6 imply (65).
Let θ ∈ (0, 1) and m ∈ ℤ+, m> 1 be fixed. Let be the function
Lemma 3.2 is an entire function of λ which satisfies the estimate
Proof. Since is entire, then also is entire in λ. Combining the estimates where c0 ≃ 1.72, cf. , and (65), we obtain
Therefore if λ ∈ ℝ we have
A direct and important result of Lemma 51 is that belongs to the Paley-Wienerz space with σ = 1+mθ. Since then we can reconstruct the functions via the following sampling formula
Let N ∈ ℤ+, N > m and approximate by its truncated series , where
Since all eigenvalues are real, then from now on we restrict ourselves to λ ∈ ℝ. Since the truncation error, cf. (5), is given for
The samples and in general, are not known explicitly. So we approximate them by solving numerically 2N + 1 initial value problems at the nodes Let and be the approximations of the samples of and respectively. Now we define which approximates
Using standard methods for solving initial problems, we may assume that for |n| < N,
for a sufficiently small ε. From Lemma 3.2 we can see that satisfies the condition (9) when m > 1 and therefore whenever we have
where there is a positive constant for which, cf. (10),
In the following we use the technique of  to determine enclosure intervals for the eigenvalues. Let λ∗ be an eigenvalue, that is
Then it follows that
Since is given and, has computable upper bound, we can define an enclosure for λ∗, by solving the following system of inequalities
Its solution is an interval containing λ*, and over which the graph is squeezed between the graphs
Using the fact that
uniformly over any compact set, and since λ* is a simple root, we obtain for large N and sufficiently small ε
in a neighborhood of λ*. Hence the graph of intersects the graphs and at two points with abscissa a _(λ*, N, ε) ≤ a + (λ*, N, ε) the interval
and in particular λ*∈I ε,N . Summarizing the above discussion, we arrive at the following lemma which is similar to that of  for Sturm-Liouville problems.
Lemma 3.3 For any eigenvalue λ*, we can find N0 ∈ ℤ+ and sufficiently small ε such that λ*∈I ε,N for N > N0. Moreover
Proof. Since all eigenvalues of (15)-(19) are simple, then for large N and sufficiently small ε we have in a neighborhood of λ*. Choose N0 such that
has two distinct solutions which we denote by a _(λ*,N0,ε) ≤ a+(λ*,N0,ε). The decay of TN,m-1,σ(λ)→0 as N → ∞ and as ε → 0 will ensure the existence of the solutions a _(λ*,N,ε) and a+(λ*,N,ε) as N → ∞ and ε → 0. For the second point we recall that as N → ∞ and as ε → 0. Hence by taking the limit we obtain
that is Ω(a+)=Ω(a-)=0. This leads us to conclude that a+ = a- = λ*, since λ* is a simple root.
Let Then (75) and (79) imply
and θ is chosen sufficiently small for which |θλ| < π. Let λ* be an eigenvalue and λ N be its approximation. Thus Ω(λ*) = 0 and From (85) we have Now we estimate the error |λ*-λ N |, for the eigenvalue λ*.
Lemma 3.4 Let λ* be an eigenvalue of (15)-(19). For sufficient large N we have the following estimate
Proof. Since then from (85) and after replacing λ by λ N we obtain