FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions

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.

Exponentially Convergent Functional-discrete Method for Eigenvalue Transmission Problems with a Discontinuous Flux and the Potential as a Function in the Space L_1

“…As it was in the linear case described in section 2, zero approximation (λ ni (x), i = 1, 2) can be found using formulas (11) and (12). Applying the same approach as in section 2, we determine u (11), we obtain the following expression for λ…”

“…can be found using formulas (11) and (12). Applying the same approach as in section 2, we determine u (j+1) ni (x) (i = 1, 2) according to formulas (13), (44), (45).…”

“…ni (x), i = 1, 2) are determined by formulas (12). We see that in this case sin λ can be determined from the second equation of (14), that is,…”

“…Let us compute the approximations to the six least eigenvalues of the problem. From formulas (11) and (12) we obtain the zero approximations for the FD-method's algorithm:…”

“…Linear eigenvalue transmission problems were investigated in [3,4,16]. The numerical treatment of transmission problems are given in [14,15,12,13,19,18,5,23,9,10] et al Particularly, papers [14,15,12,13,19,18] are devoted to the development of numerical functional-discrete exponentially convergent methods for the eigenvalue transmission problem with discontinuous solutions and continuous flux.…”