Shooting Methods for Boundary Value Problems of Fractional Differential Equations

ABSTRACT

We consider two‐point boundary value problems of fractional differential equations (FDEs) including a Caputo derivative with a fractional order in the range (0,1) or (1,2). The numerical solution of a boundary value problem is investigated using shooting methods. This approach yields a nonlinear system of algebraic equations for the unknown initial values of the solution that satisfies the boundary conditions. Consequently, initial value problems of FDEs have to be integrated. The nonlinear systems can be solved by methods of Newton‐type. However, these methods often fail if the problem is sensitive with respect to the starting values of the iteration and the initial guess is not sufficiently close to the true solution. Alternatively, a residual can be reduced nearly to zero using minimisation schemes. For example, differential evolution represents a stochastic population‐based optimisation method, that can be used for such a task. Furthermore, we introduce a continuation method to achieve the convergence of iterative techniques, where the fractional order represents the homotopy parameter. We present results of numerical computations for test examples of FDEs.