Some Problems In Fractional Differential Equations And Nonlinear Evolution Equations

This dissertation primarily considers the following problems of fractional differential equations and nonlinear partial differential equations.1. Construction of Mittag-Leffler type function and studies on its integral properties.2. Some studies on the Fractional differential forms and of fractional gradient, curl and divergence.3. Introduction of a kind of Darboux transform, and verification of its correctness by virtue of the Computational Symbolic Software Maple and display of its applications in the nonlinear evolution equations.Chapter one involves a survey of the fundamental knowledge and achievement obtained by foregoers on the subjects for the purpose of the later parts in this dissertation, including the introduction of the history and development of mathematics mechanization, brief review of the history and status in quo of the fractional calculus and soliton theory.Chapter two focuses on several problems of fractional differential equations and fractional differential forms:(1) Constructs a new type of Mittag-Leffler function, gives some of its important integral transforms such as Laplace transform, Mellin transform and Inverse-Fourier transform. And then emphasizes on its application on the fractional diffusion-wave equations.(2) Based on the ideas and work of fractional exterior derivative and fractional differential forms, compares the fractional differential forms them the standard ones. Furthermore, seeks the fractional gradient, curl and divergence in the fractional differential form space in order to meet the develop of theory in physics and finance.Chapter three of this dissertation is devoted to introducing the essentials of "AC=BD". Also, under the guidance of such idea we try to construct a kind of Darboux transform and apply to the (1+1)-dimensional higher Broer-Kaup(HBK) system.