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Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. (English) Zbl 1048.65102

Summary: The boundary value problem \(\Delta u + \lambda e^u = 0\) 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 \(\lambda < \lambda_c\), no solutions for \(\lambda > \lambda_c\) and a unique solution when \(\lambda = \lambda_c\). Numerical solutions to the Bratu-Gelfand problem at the critical value of \(\lambda_c = 2\) 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 \(\lambda < 2\) are given, which illustrate the superiorityof the nonstandard scheme.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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