Approximation of eigenvalues of boundary value problems

Abstract

In the present paper we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of discontinuous Dirac systems, which contain an eigenvalue parameter in one boundary condition, with transmission conditions at the 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 the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper showing that this method gives us better results. MSC: 34L16; 94A20; 65L15

Computation of Spectral Parameter of Discontinuous Dirac Systems with a Gaussian Multiplier

“…In the following, we study some properties of the eigenvalues of the problems (12)-(15) which need in our method; see [26,28]. For functions u( ), which defined on [−1, 0) ∪ (0, 1] and has finite limit u(±0) := lim → ±0 u( ), by u (1) ( ) and u (2) ( ), we denote the functions…”

“…are independent on variable ∈ Γ ( = 1, 2) and ( , ) and ( , ) are entire functions of the parameter for each ∈ Γ ( = 1, 2). Taking into account (26) and 28, a short calculation gives…”

“…Integrating the above equation through [−1, 0] and [0, 1], and taking into account the initial conditions (23), (26), and (28), we obtain…”

“…Also, the authors introduced some examples illustrating the sinc-Gaussian method accompanied by comparison with the sinc-method which explained that the sinc-Gaussian method gives remarkably better results; see also [24,25]. Tharwat and Al-Harbi [26] computed the eigenvalues of discontinuous Dirac system but with eigenparameter in one boundary condition. In [14,27] Tharwat et al computed the eigenvalues of discontinuous Dirac system approximately, with eigenparameter appears in any of boundary conditions, by Hermite interpolations and regularized sinc-methods, respectively.…”

“…Then we approximate the unknown part using (5) and obtain better results. We would like to mention that the papers in computing eigenvalues by sinc-Gaussian method are few; see [22][23][24][25][26]. In Sections 2 and 3, we discuss some properties of the eigenvalues of the boundary value problems (12)- (15) and also, we derive the sinc-Gaussian technique to compute the eigenvalues of (12)-(15) with error estimates.…”