Geodesic
A note on two dimensional Bratu problem
Kragujevac Journal of Mathematics, Tome 29 (2006) no. 1.

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

In this paper, we examined the Bratu problem $$-\Delta U = łambda exp[U(x,y)],\quad x,y \in D, $$ $$U(x,y) = 0,\quad x,y \in \partial D, $$ in 2-dimensions, where $ \Delta$ is the Laplace operator. The non linear equation is solved using various methods including finite difference method, weighted residual method and analytical method. Both the near exact solution and weighted residual solution, provide the upper and the lower branch solutions while the finite difference method only give the lower branch solution. 
@article{KJM_2006_29_1_a4,
     author = {S. A. Odejide and Y. A. S. Aregbesola},
     title = {A note on two dimensional {Bratu} problem},
     journal = {Kragujevac Journal of Mathematics},
     pages = {49 - 56},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {2006},
     zbl = {1121.35315},
     url = {http://geodesic.mathdoc.fr/item/KJM_2006_29_1_a4/}
}
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S. A. Odejide; Y. A. S. Aregbesola. A note on two dimensional Bratu problem. Kragujevac Journal of Mathematics, Tome 29 (2006) no. 1. http://geodesic.mathdoc.fr/item/KJM_2006_29_1_a4/