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Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem
Author(s) -
Buckmire Ron
Publication year - 2004
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.10093
Subject(s) - mathematics , discretization , eigenvalues and eigenvectors , boundary value problem , partial differential equation , mathematical analysis , boundary (topology) , scheme (mathematics) , quantum mechanics , physics
The boundary value problem Δ u + λ e u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ c , no solutions for λ > λ c and a unique solution when λ = λ c . Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004