Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

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.

Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

“…Although the analysis of integral transforms with nonlinear eigenvalue problems [29] is out of the scope of the present contribution, it is instructive to verify that the convection-diffusion problem given in Eq. (1), for any general form of the associated coefficients and source term, can be actually rewritten as a generalized diffusion problem with modified spatially variable coefficients.…”

“…Numerical solution of the ODE's system (6) for the general nonlinear situation here considered can be accomplished by welltested routines for initial value problems with automatic accuracy control [28], and then provide results for the transformed potentials along the t variable. However, it is well known that improved convergence is achieved by appropriate filtering of problem (2), particularly to reduce the importance of nonhomogeneous boundary conditions [7][8][9][10][11][12][13][14], and even the employment of eigenvalue problems with nonlinear boundary conditions, as more recently advanced [29]. In any case, it is clear from the above development that the convective term in the original formulation of problem (1) is never accounted for, even partially, in the proposed eigenvalue problem, and is directly transported to the modified nonlinear source term.…”

“…Clearly, the adoption of a more informative eigenfunction expansion basis, including at least part of the convective term contribution, while at the same time reducing the importance of the source term that incorporates the convective terms, significantly contributes to the convergence acceleration. The approach also opens new perspectives for future work, as research advances toward the integral transform solution of nonlinear problems through the adoption of nonlinear eigenvalue problems [29], when the exponential transformation can be employed to fully incorporate the nonlinear convective terms into a generalized nonlinear diffusion formulation.…”

Analysis of Mass Transfer in Hollow-Fiber Membrane Separator via Nonlinear Eigenfunction Expansions

“…Here, a recently introduced integral transforms approach (Cotta et al, 2016b), based on the adoption of nonlinear eigenvalue problems, will be further investigated. There is some computational advantage, as will be seen further ahead, in adopting a simpler eigenvalue problem formulation that does not account for the circular geometry and velocity profile information, but instead directly incorporates the nonlinear boundary condition information.…”

“…Then, such characteristic equation and boundary condition linear coefficients naturally lead to the eigenvalue problem choice to be employed in constructing the expansions. Recently, a variant in the GITT approach has been advanced, based on retaining the original nonlinear operator coefficients in the eigenvalue problem proposition (Cotta et al, 2016b). This methodology has been demonstrated in diffusion problems with nonlinear boundary conditions, which (Cotta et al, 2015(Cotta et al, , 2016bPontes et al, 2017) clearly illustrate the relative gains in convergence enhancement in comparison to other alternative convergence acceleration schemes, such as filtering and integral balances.…”