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Bifurcation analysis of the spectrum of two-dimensional thermal structures evolving with blow-up

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Abstract

The article considers the self-similar solution of the nonlinear heat-conduction equation with a three-dimensional source evolving under blow-up conditions. The self-similar problem is a boundary-value problem for a nonlinear elliptical equation, which has a nonunique solution. The eigenfunction spectrum of the self-similar problem is investigated in the two-dimensional space. The problem is solved by Newton’s iterative method on a grid. Newton’s method is implemented using several alternative initial approximations. The eigenfunctions are continued in a parameter and their bifurcations are investigated. Several scenarios are identified for the evolution of the two-dimensional structures when the parameter is changed. The evolution of the eigenfunctions depends on the class and type of the particular structure. New types of structures are identified and the eigenfunctions are systematized.

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Authors
  1. E. S. Kurkina
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  2. I. M. Nikol’skii
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Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 50–75, 2005.

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Kurkina, E.S., Nikol’skii, I.M. Bifurcation analysis of the spectrum of two-dimensional thermal structures evolving with blow-up. Comput Math Model 17, 320–340 (2006). https://doi.org/10.1007/s10598-006-0027-z

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