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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.

65N25Numerical methods for eigenvalue problems (BVP of PDE)
65N06Finite difference methods (BVP of PDE)
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
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