On Computing of Eigenvalues of Differential Equations Q = λ P with Eigenparameter in Boundary Conditions

: Problem statement: Our purpose of this study is to use sinc methods to compute approximately the eigenval-ues of second-order operator pencil of the form Q-λ P. Approach: Where Q is second order self adjoint differential operator and P is a first order and λ∈ C is an eigenvalue parameter. Results: The eigenparameter appears in the boundary conditions linearly. Using computable error boundswe obtain eigenvalue enclosures in a simple way. Conclusion/Recommendations: We give some numerical examples and makecompanions with existing results.


INTRODUCTION
The aim of the present study is to compute the eigenvalues numerically of a differential operator of the form Q-λP approximately by the sinc method, where Q and P are self-adjoint differential operators of the second and first order respectively. By the sinc method we mean the use of the Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem, (Shannon, 1949;Whittaker, 1915;Zayed, 1993). The WKS states that if f(λ) is entire in λ of exponential type σ, σ>0, which belongs to L2(R) where restricted to , then f(λ) can be reconstructed via the sampling representation: n n f ( ) f sin c( n ), ∞ =−∞ π ⎛ ⎞ λ = σλ − π λ∈ ⎜ ⎟ σ ⎝ ⎠ ∑ (1.1) Series (1.1) converges absolutely on and uniformly on and on compact subsets of (Butzer et al., 2001;Stenger, 1993). The space of all such f is the Paley-Wiener space of band limited functions with band width σ which will be denoted by 2 PW σ . The nodes n n ∈ π ⎧ ⎫ ⎨ ⎬ σ ⎩ ⎭ are called the sampling points and the sinc functions are: sin c( n ) n , ( n ) sin c( n ) : n 1, σλ − π π ⎧ λ ≠ ⎪ σλ − π σ ⎪ σλ − π = ⎨ π ⎪ λ = ⎪ σ ⎩ (1.2) n ∈ Theorem (1.1) is used extensively in approximating solutions and eigenvalues of boundary value problems, (Boumenir, 2000a;2000b;Lund and Bowers, 1992;Stenger, 1981;. One type of error is associated with sinc-based methods, truncation error. An estimate for the truncation error is established by Jagerman (1966) , for some integer k>0, then for N , N / ∈ λ < π σ , we have: We are concerned with the computation of eigenvalues of the boundary-value problem: where, p, q, ρ and w are real-valued functions on [0, 1], p −1 , r, q, w∈L 1 (0, 1), p≥0, ρ∈AC[0, 1], the set of all absolutely continuous functions on [0, 1], q is essentially bounded from below and γ, δ∈[0,π). This problem has been studied in its general form in the comprehensive study of (Langer et al., 1966) as a linear pencil Q-λP, where Q is a second-order operator and P is a first-order operator. Problem (1.6-1.8) differs from classical second-order eigenvalue problems in several respects. First, the operator in the left-hand side of (1.6) is not the identity operator multiplied by the eigenparameter, but a first order operator. Also, the eigenvalue parameter appears linearly in the boundary conditions. Illustrative examples and tables are included in the last section. It is worthy to mention that the sampling scheme is used to approximate eigenvalues for different types of boundary value problems in (Boumenir, 1999;2000a;2000b;Chanane, 1999;2005).
Lemma 2: For λ ∈ , the following estimates hold:  The rest of the proof can be accomplished as in the previous lemma.

The method and error bounds:
In this section we derive the method of computing eigenvalues of problem (2.1-2.3) numerically.
The basic idea of the scheme is to split Δ(λ) into two parts: Then, from Lemma 2.1 and Lemma 2.2, we have the following lemma.   (3.8) The number θ will be specified latter. The number 1 is the smallest positive integer that suites our investigation as is seen in the next lemma.
and by calculating the integrals we obtain (3.11). What we have just proved is that F θ,m (λ) belongs to the Paley-Wiener space Hence, F θ,m (λ) can be recovered from its values at the points n n ,n π λ = ∈ σ via the sampling expansion: and approximate F θ,m (θ) by its truncated series F θ,m,N (λ), where: , the truncation error is given for (N 1) 1 4 ( / ) and θ is chosen sufficiently small for which ⎢θλ⎢<π. Let λ * be an eigenvalue, that is: Then it follows that: and so: and in particular λ * ∈ I N (λ * ). Now, we summarize the above idea in the following lemma, (Boumenir, 2000a).

CONCLUSION
In this study, we have used the regularized sampling method introduced recently (Chadan and Sabatier, 1989) to compute the eigenvalues of secondorder operator pencil of the form Q-λP, where Q is second order self adjoint differential operator and P is a first order and λ∈C is an eigenvalue parameter. We recall that this method constitutes an improvement upon the method based on Shannon's sampling theory introduced in (Boumenir, 1999) since it uses a regularization avoiding any multiple integration. The method allows us to get higher order estimates of the eigenvalues at a very low cost. We have presented two examples to illustrate the method and compared the computed eigenvalues with the exact ones when they are available. In these examples we observed, in Tables 1-4, that λ k,N and the exact solution λ k are all inside the enclosure interval [a -, a + ], and also we illustrated, in Fig. 1-4, a slight different between Δ(λ) and Δ N (λ) for different values of N, m and θ.