Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 3 May 2016
Abstract
Purpose
The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.
Design/methodology/approach
The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.
Findings
An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.
Originality/value
This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.
Keywords
Acknowledgements
The authors would like to acknowledge the partial financial support provided by CNPq and FAPERJ, sponsoring agencies in Brazil. The authors would also like to express their sincere gratitude for the kind and honoring invitation by the Editor-in-Chief of the IJNMHFF, Professor Roland Lewis, to participate in this special issue.
Citation
Cotta, R.M., Naveira-Cotta, C.P. and Knupp, D.C. (2016), "Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26 No. 3/4, pp. 767-789. https://doi.org/10.1108/HFF-08-2015-0309
Publisher
:Emerald Group Publishing Limited
Copyright © 2016, Emerald Group Publishing Limited