(621h) Numerical Methods for a Case of Fractional Isoperimetric Problem

An isoperimetric problem is a problem of constrained optimization and it is one case of study in Calculus of Variations.  When the objective function or the constraint equations or both contain fractional order derivatives, the problem becomes a fractional isoperimetric problem.  Unlike integer order isoperimetric problems that can be solved analytically, analytical solutions for fractional isoperimetric problems either do not exist or are extremely complex.   This work proposes numerical methods for a case of fractional isoperimetric problem.  The fractional derivatives are defined in term of Riemann-Liouville derivatives and with zero terminal conditions, these are the same as Caputo derivatives.  An iterative numerical scheme is developed in which the Grünwald–Letnikov (GL) approach is used to approximate the fractional derivatives. The spatial domain and the objective function are both approximated by three types of GL definitions.  These three methods all show good convergence as the integration step sizes decrease. The convergence errors of these methods are obtained and compared. As the order of the derivatives approaches the integer value of 1, the numerical results recover the analytical solution of the corresponding integer order isoperimetric problem.