Theoretical and numerical considerations on Bratu-type problems

"In this paper we present an heuristic introduction to Bratu problem and we give some variants of Bratu's theorem (G. Bratu, Sur les \'equations int\'egrales non lin\'eaires, Bulletin Soc. Math. France, 42(1914), 113-142). Using the positivity of Green's function, the monotone iterations technique and the contraction principle, some generalizations of Bratu's result are also given. Numerical aspects are also considered."

On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.