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

Al-Harbi Al-Harbi

doi:10.3844/jmssp.2011.28.36

Abstract

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-λP. 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.

Highlights

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
We indicate the effect of the parameters m and by several choices
We summarize the above idea in the following lemma, (Boumenir, 2000a)

Summary

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. 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: v(x,λ) ≤ exp(( Iλ + I λ2 + λ )x). Lemma 3: The function S(λ) is entire in λ for each x∈[0, 1] and the following estimates hold: S(λ) ≤ c3 exp(( Iλ + I λ2 + λ )). Combining the estimates sin z ≤ c0 e Iz , where z 1+ z c0 1.72 , cf (Chadan and Sabatier, 1989) and (3.4), we obtain: Fθ,m (λ). Lemma 4: Fθ,m(λ) is an entire function of λ which satisfy the estimates: c3c0m (1 + θ λ )m exp(( Iλ (1 + mθ) +.

Since λ

CONCLUSION