An analytic approach to solve multiple solutions of a strongly nonlinear problem

Based on a new kind of analytic method, namely the homotopy analysis method, an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using Gelfand equation as an example. Its validity is verified by comparing the approximation series with the known exact solution. And different from perturbation techniques, this approach is independent upon any small/large perturbation quantities. So, the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering.