Numerical investigation of spiral structure solutions of a nonlinear elliptic problem

Authors:
Milena Dimova;Stefka Dimova
Affiliations:
Institute of Mathematics and Informatics, Bulgarian Acad. Sci., Sofia, Bulgaria;Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria
Venue:
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
Year:
2010

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Abstract

The nonlinear elliptic problem considered arises when investigating a class of self-similar solutions of a reaction-diffusion equation. We focus our study on the solutions of spiral structure. The proposed approach is based on the continuous analog of the Newton's method and on the Galerkin finite element method. To reveal solutions of spiral structure appropriate initial approximations are used. The last ones are expressed by the confluent hypergeometric function 1F1(a, b; z). Algorithms for accurate, fast and reliable computation of its values for broad ranges of the parameters a and b and of the variable z are worked out. A detailed numerical analysis of the evolution of the spiral structure solutions with respect to the medium parameters, including critical values, is carried out.