A two-stage iterative method for solving a weakly nonlinear parametrized system. (English) Zbl 1016.65029
Author’s summary: We consider a parametrized system of weakly nonlinear equations which corresponds to a nonlinear elliptic boundary-value problem with zero source, homogeneous boundary conditions and a positive parameter in the linear term. Positive solutions of this system are of interest to us. A characterization of this positive solution is given. Such a solution is determined by the modified Newton-arithmetic mean method. This method is well suited for implementation on parallel computers. A theorem about the monotone convergence of the method is proved. An application of the method for solving a real practical problem related to the study of interacting populations is described.
Reviewer: B.Döring (Düsseldorf)
MSC:
| 65H10 | Numerical computation of solutions to systems of equations |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |
| 65Y05 | Parallel numerical computation |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
Keywords:
weakly nonlinear systems; modified Newton method; arithmetic mean method; monotone convergence; diffusion; eigenvalue problem; parallel computation; positive solutions; nonlinear elliptic boundary-value problem; interacting populationsReferences:
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