No abstract

An algorithm for the solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is developed. The numerical technique is based on a perturbation of the coefficients of a differential equation combined with the Adomian decomposition method for the nonlinear term of the equation. The proposed approach provides an exponential convergence rate dependent on the index of the trial eigenvalue and on the transmission coefficient. Numerical examples support the theory.

-We develop a functional-discrete method with a high order of accuracy to find a numerical solution of an eigenvalue transmission problem. It allows to approximate the trial eigenvalue with any desired accuracy. This approach has no restriction on the number of eigenvalues, an approximation to which can be found. The convergence rate is proved as in the case of the geometric series. It is shown that depending on the data of the original problem, two kinds of eigenvalue sequences may exist. For the first one, the convergence rate increases as the ordinal number of the trial eigenvalue increases. For the second one, the convergence rate is the same for all eigenvalues and does not depend on the ordinal number of the trial eigenvalue. Based on the asymptotic behavior of the eigenvalues of the basic problem and the functional-discrete method, a qualitative result on the arrangement of eigenvalues of the original problem is established. A number of numerical examples are given to support the theory.2000 Mathematics Subject Classification: 34B24; 65L15.

517.983.27 Using the functional discrete approach and Adomian polynomials, we propose a numerical algorithm for an eigenvalue problem with a potential that consists of a nonlinear autonomous part and a linear part depending on an independent variable. We prove that the rate of convergence of the algorithm is exponential and improves as the order number of an eigenvalue increases. We investigate the mutual influence of the piecewise-constant approximation of the linear part of the potential and the nonlinearity on the rate of convergence of the method. Theoretical results are confirmed by numerical data.