Department of Mathematical SciencesIn this thesis, several efficient numerical methods are proposed to solve initial value problems and boundary value problems of fractional di???erential equations.
For fractional initial value problems, we propose a new type of the predictorevaluate-corrector-evaluate method based on the Caputo fractional derivative operator.
Furthermore, we propose a new type of the Caputo fractional derivative operator that does not have a di???erential form of a solution. However, with some fractional orders, there are problems that a solution blows up and the scheme has a low convergence. Thus, we identify new treatments for these values. Then, we can expect a significant improvement for all fractional orders. The advantages and improvements are shown by testing various numerical examples.
For fractional BVPs, we propose an explicit method that dramatically reduces the computational time for solving a dense matrix system. Moreover, by adopting
high-order predictor-corrector methods which have uniform convergence rates O(h2) or O(h3) for all fractional orders [8], we propose a second-order method and a third-order method by using the Newton???s method and the Halley method, respectively. We show its advantage by testing various numerical examples.clos
