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Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors
Author(s) -
Ward Michael J.,
Keller Joseph B.
Publication year - 1991
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm19918511
Subject(s) - eigenvalues and eigenvectors , boundary value problem , mathematics , nonlinear system , mathematical analysis , bifurcation , constant (computer programming) , asymptotic analysis , simple (philosophy) , method of matched asymptotic expansions , differential equation , asymptotic expansion , physics , quantum mechanics , philosophy , epistemology , computer science , programming language
Nonlinear eigenvalue problems are considered for partial differential equations and boundary conditions of the for The problems are perturbed by deleting a small subdomain D ε from D and imposing a boundary condition on the surface of the resulting hole, or else by changing the constant b in the boundary condition to a different constant ε −1 k on a small part of ∂D . In both cases the perturbed solution is constructed for ε small by the method of matched asymptotic expansions. Particular attention is paid to the calculation of λ c ( ε ), the critical value of λ at which the number of solutions changes by two, for example from two to none. This value of λ represents a simple fold point in the bifurcation diagram of ‖ u ‖ versus λ . The analysis is applied to chemical reactors with F ( x,u )= e u , in which case λ c is the critical value of the Frank‐Kamenetskii parameter. The asymptotic results are compared with exact and numerical results for some special cases, and fair agreement is found. Some previous asymptotic calculations are found to be incorrect.