##
**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 |

### Keywords:

Mickens discretization; Bratu problem; Gelfand problem; nonstandard finite differences; comparison of methods; nonlinear eigenvalue problem; results results
PDFBibTeX
XMLCite

\textit{R. Buckmire}, Numer. Methods Partial Differ. Equations 20, No. 3, 327--337 (2004; Zbl 1048.65102)

Full Text:
DOI

### References:

[1] | Gelfand, Am Math Soc Transl Ser 2 29 pp 295– (1963) · Zbl 0127.04901 · doi:10.1090/trans2/029/12 |

[2] | Bratu, Bull Math Soc France 42 pp 113– (1914) |

[3] | Diffusion and heat exchange in chemical kinetics, Princeton University Press, Princeton, NJ, 1955. · doi:10.1515/9781400877195 |

[4] | An introduction to the study of stellar structure, Dover Publications, New York, 1957. |

[5] | Abbott, J Comput Appl Math 4 pp 19– (1978) |

[6] | and Numerical solution of boundary value problems for ordinary differential equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1995. · doi:10.1137/1.9781611971231 |

[7] | Jacobsen, J Differ Eq 184 pp 283– (2002) |

[8] | Mickens, Numer Methods Partial Differ Eq 2 pp 123– (1986) |

[9] | Mickens, J Differ Eq Appl 8 pp 823– (2002) |

[10] | Difference equation models of differential equations having zero local truncation errors, Differential equations (Birmingham, Ala., 1983), North-Holland Math. Stud., Vol. 92, North-Holland, Amsterdam-New York, 1984, pp. 445-449. |

[11] | The design of shock-free transonic slender bodies of revolution, Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY, 1994. |

[12] | A new finite-difference scheme for singular boundary value problems in cylindrical or spherical coordinates, and editors, Mathematics is for solving problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996, pp. 3-9. |

[13] | Buckmire, Numer Methods Partial Differ Eq 19 pp 380– (2003) |

[14] | Wanelik, J Math Phys 30 pp 1707– (1989) |

[15] | editor, Applications of nonstandard finite differences, World Scientific, Singapore, 2000. · doi:10.1142/4272 |

[16] | Nonstandard difference models of differential equations, World Scientific, Singapore, 1994. · Zbl 0810.65083 |

[17] | Mickens, J Franklin Instit 327 pp 143– (1990) |

[18] | Mickens, Math Comput Model 11 pp 528– (1988) |

[19] | and Applied partial differential equations, Oxford University Press, Oxford, October 1999. |

[20] | On numerical solutions of the one-dimensional planar Bratu problem, Preprint. Available at http://faculty.oxy.edu/ron/research. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.