| dc.description.abstract | Fractional derivatives (FDs) or derivatives of arbitrary order have attracted considerable interest in the past few decades, and almost every field of science and engineering has applications of fractional derivatives. Since fractional derivatives have such property as being non-local, it can be extremely challenging to find analytical solutions for fractional optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop approximate or numerical solutions for such problems. The primary focus of this thesis is to develop numerical schemes to solve optimization problems in fractional orders. Numerical methods for integer order problems of Variational Calculus, using the Euler-Lagrange equation, have already been well established. A Fractional Variational Calculus Problem (FVCP) is a problem in which either the objective functional or the constraints or both contain at least one fractional derivative term. There is a critical need to develop numerical algorithms for solving FVCPs. The main contributions of this thesis is to develop formulations and solution methods for various fractional order optimization problems, including fractional optimal control problems, linear functional minimization problems and isoperimetric problems in fractional orders. The FDs are defined in terms of the Riemann-Liouville or Caputo definitions. Numerical schemes have been developed to obtain the numerical results for various problems. For each scheme, the rate of convergence and the convergence errors are analyzed to ensure that the algorithm yields stable results. Various fractional orders of derivatives are considered and as the order approaches the integer value of I, the numerical solution recovers the analytical result of the corresponding integer order problem. | en_US |
