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**An analytical and numerical study of the two-dimensional Bratu equation.**
*(English)*
Zbl 0649.65057

Bratu’s problem, which is the nonlinear eigenvalue equation \(\Delta u+\lambda \exp (u)=0\) with \(u=0\) on the walls of the unit square and \(\lambda\) as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmetry: because of invariance under the \(C_ 4\) rotation group and parity in both x and y, one can slash the size of the basis set by a factor of eight and reduce the CPU time by three orders of magnitude. Second, the pseudospectral method is an analytical as well as a numerical tool: the simple approximation \(\lambda\approx 3.2A \exp (- 0.64A)\), where A is the maximum value of u(x,y), is derived via collocation with but a single interpolation point, but is a quantitatively acurate for small and moderate A.

Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions - five decimal places - on a microcomputer in BASIC. Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu’s equations fall off exponentially rather than algebraically with \(\nu\) until \(\nu\) is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in the limit \(\nu\to \infty\), are so weak as to be irrelevant, and replacing Bratu’s problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more.

Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions - five decimal places - on a microcomputer in BASIC. Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu’s equations fall off exponentially rather than algebraically with \(\nu\) until \(\nu\) is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in the limit \(\nu\to \infty\), are so weak as to be irrelevant, and replacing Bratu’s problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more.

### MSC:

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |

### Keywords:

Bratu’s problem; nonlinear eigenvalue equation; Chebyshev pseudospectral methods; Bratu’s equations; corner singularities; Chebyshev coefficients; corner branch points### Software:

PLTMGC
Full Text:
DOI

### References:

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