Abstract
The purpose of this paper is to present a new kind of analytical method, the so-called residual power series, to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. The present method is capable of calculating all branches of solutions simultaneously, even if these multiple solutions are very close and thus rather difficult to distinguish even by numerical techniques. To verify the computational efficiency of the designed proposed technique, two nonlinear models are performed, one of them arises in mixed convection flows and the other one arises in heat transfer, which both admits multiple solutions. Graphical results and tabulate data are presented and discussed quantitatively to illustrate the multiple solutions. The results reveal that the method is very effective, straightforward, and powerful for formulating these multiple solutions.