Numerical solutions of one-dimensional Gelfand equation with fractional Laplacian

In this paper, we discuss an efficient numerical method to obtain all solutions of fractional Gelfand equation with Dirichlet boundary condition. More precisely, we derive a good initial value motivated by the bifurcation curve of fractional Gelfand equation. It is obvious to see that the number of solutions depends on the value of parameter in fractional Gelfand equation. By collocation technique and finite difference method, numerical solutions can be found very quickly based on Newton iteration method with the aid of such initial guess. Numerical simulation for one-dimensional fractional Gelfand equation are provided, which demonstrates the accuracy and easy-to-implement of our algorithm.