Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λeu = 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 λc and a unique solution when λ = λc. Numerical solutions to the Bratu-Gelfand problem at the critical value of λc = 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

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