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In this paper we state the sufficient conditions for the existence and uniqueness of positive eigenvectors for a class of nonlinear operators associated with M-matrices. We also show how to construct a convergent iterative process for finding these eigenvectors. The details of numerical implementation of this algorithm for some spectral methods of discretization of elliptic partial differential equations are also discussed. Some results of numerical experiments for the Gross-Pitaevskii Equation with non-separable potentials in a rectangular domain are given in the end of the paper. [ABSTRACT FROM AUTHOR]

This paper presents a numerical scheme that avoids iterations to solve the nonlinear partial differential equation system for pricing American puts with constant dividend yields. Upon applying a front-fixing technique to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the discrete nonlinear scheme. In the comparison with the solutions from articles that cover zero dividend and constant dividend yields cases, our results are found accurate. The current method is conditionally stable since the Euler scheme is used, the convergency property of the scheme is shown by numerical experiments. [ABSTRACT FROM AUTHOR]