The Study Of The Structures Of Solutions Of Several Kinds Of Fractional Differential Equations And Systems

The idea of derivatives of noninteger order was initially appeared in a letter from Leib-nizs to L'Hospital in 1695, So far there have been more than 300 years. for many years studies of the theory of fractional order were mainly constraint to the field of pure theoretical mathematics, One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional derivatives. Another difficulty is that fractional deriva-tives have no evident geometrical interpretation because of their nonlocal character. There are several definitions of a fractional derivative. The two most commonly used definitions are the Riemann-Liouville and Caputo. Each definition uses Riemann-Liouville fractional integration and derivatives of whole order. The difference between the two definitions is in the order of evaluation.In recent decades, many researchers found that derivatives of noninteger order are very suitable for the description of various practical problems. In the field of physics, chemistry and engineering, etc, such as rheology, damping laws, diffusion process can be elegantly modeled with the help of the fractional derivatives. In many mathematics documents, We can find the wide application of fractional differential equations. the nonlinear oscillation of earthquake can be modeled with fractional derivatives; and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of con-tinuum traffic flow; Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in Ref; differential equations with fractional order have recently proved to be valuable tools to the modeling in continuum, statistical me-chanics and Financial mathematics; Malthusian growth and the Poissonian birth process can be modeled with fractional differential equations, and so on.This paper is divided into five parts.the first part is the introduction, Introducing the research background of fractional derivatives, and paper need related definitions and proper- ties, and our research work and sturcture of the paper are studied.The second part mainly research the structure of solutions of some fractional differen-tial equations. the research of this part are as follow:1. we obtain the exact solution of the kind of equation via the approach of separating variables.2. we establish the structural theorem of solutions of invariable coecient linear homogeneous fractional differential equa-tions and give the fundamental solutions'system. The result generalizes the corresponding result of invariable coecient linear integer-order differential equations. The above research is difficult in building the fundamental system of solutions. In addition, we obtain solution-s of a type of invariable coecient nonhomogeneous fractional differential equations via the operator method.3. the problem on solving system of linear fractional differential equations with constant eoefficients is studied, first, we investigates the fundamental system of solu-tions of the kind of equations by using Jordan canonical matrix. Then, solving system of linear fractional differential equations with constant coefficients by the method of undeter-mined coefficients. The results contain the solution of linear first-order differential equations with constant coefficients.4. We construct the family of approximation functions by using the Euler dogleg method, as well as prove the existence of solutions of system of a-order fractional differential equations with time- fractional derivative by using the fixed point the-orem. In general, to make the solution of system of differential equations has its uniqueness, on the condition that nonhomogeneous satisfy Lipschitz continuity, We give a more general condition via proof by contradiction,which make solutions of system of a-order fractional differential equations satisfy uniqueness.The third part mainly research methods of solving some syetems of fractional differ-ential equations. In this part we do the following work:1. So far, people have used a lot of methods for solving systems of fractional differential equations with time- fractional derivative such as the Adomain decomposition method, the generalized differential transfor-m method, the homotopy perturbation method, the modified homotopy perturbation method, the homotopy analysis method, the variational iteration method and the separating variables method. these methods are applied to the system of fractional differential equations with space- and time- fractional derivative, the difficulty lay on each equation of the system con-tains different order of the space- fractional derivative and the time- fractional derivative.2. we mainly research methods on solving some syetems of fractional differential equations by using the variational iteration method, so far, the variational iteration method only solve fractional differential equations with integer-order, so this method has its limitation, it can not be widely used for fractional differential equations. we mainly research the method on solving fractional differential equations without integer-order, it overcomes the shortcom-ings of the traditional sense of solving fractional differential equations via this method, and we analysis the numerical result of the approximate solution and the exact solution,which are obtained by this method. In addition, we solve the system of fractional differential equations with space- and time- fractional derivative via the time direction and the space direction, The research of many scholars' focus on finding more effective the method on solving the system of fractional differential equations.The fourth part focuses on the blow-up solution of several kinds of systems of time-fractional differential equations. We obtains system of the integral equations which is equiv-alent to system of nonlinear partial differential equations with time- fractional derivative, and proves the local existence of solutions to the system of the integral equations. Secondly, we investigates the blowing-up solutions to the a nonlinear system of fractional differential equations by making use of Holder's inequality and obtains solution of system to blow up in a finite time,as well as gives an upper bound on the blow-up time.In the last part, fractional heat conduction equation is studied. we introduce the defini-tion and its properties of Guy.J's fractional derivative operator, establish a Banach function space with fractional derivative. Nature of fractional heat conduction equation which is par-allelled with a classical heat conduction equation is studied. In fact, the majority theory of integral calculus is built on the foundation which is to meet the division of integral operation on the basis of fractional integrator, thus some traditional methods and techniques can not be applied to the fractional differential equation, That increases the difficulty of researching fractional differential equations. While finding more effective definition of fractional deriva-tive operator is the objective of many scholars who study the fractional differential calculus.