On Exact and Numerical Solutions of the One-Dimensional Planar Bratu Problem

The nonlinear eigenvalue problem Δ+λ = 0 in the unit square with = 0 on the boundary is often referred to as “the Bratu problem” or “Bratu's problem.” The Bratu problem in 1-dimensional planar coordinates, ″ + λ = 0 with (0) = (1) = 0 has two known, bifurcated, exact solutions for values of λ < λ and no solutions for λ > λ. The value of λ is simply 8( − 1) where is the fixed point of the hyperbolic cotangent function. Numerical approximations to the exact solution of the one-dimensional planar Bratu problem are computed using various numerical methods. Of particular interest is the application of nonstandard finite-difference schemes known as Mickens finite differences to solve the problem. In addition, standard finite-differences, Boyd collocation and Adomian polynomial decomposition are employed to generate numerical solutions to this Bratu problem and the results compared