Approximate Analytical Solution of the Bratu Boundary-Value Problem

Three new approaches to the solution of the Bratu problem are presented. The first approach realizes the idea of successive differentiating the initial equation of this problem with expansion of the sought for function at the symmetry point of a space. The second approach is associated with the additional integration of the differential Bratu equation, and it represents a hybrid integral method. The third approach is based on the combined application of the successive differentiation of the Bratu equation, the hybrid integral method, the expansion of the sought for function at two points of the space, and an additional integral relation. The indicated approaches are characterized by the simplicity of the calculations required for their realization, and they reduce the solution of the Bratu problem to the solution of a system of linear algebraic equations. The numerical results of solving this problem demonstrate the high efficiency of the approaches proposed, providing the obtaining of the solutions whose accuracy exceeds the accuracy of the analogous solutions, obtained on the basis of the known numerical and numerical-analytical methods, by three to five orders of magnitude. This is especially true for the third approach that allows one to obtain two classical solutions of the Bratu problem fairly simply and with a very high accuracy. It is shown that the obtaining of an approximate solution of the Bratu problem with this approach calls for a small number of series terms, and the solutions obtained converge quickly and approximate the problem highly exactly.