Theory And Applications Of The Generalized Fractional Calculus

Abstract:In this thesis, we study fractional calculus and generalized fractional calculus and their applications, including existence and uniqueness results of fractional boundary value prob-lem, fractional harmonic oscillator and fractional van der Pol oscillator defined by the first class generalized fractional derivative, generalized fractional diffusion-wave equation, fractional Burgers equation and advection-diffusion equation defined by the second generalized fractional derivative, and generalized fractional variational problems. It consists of seven chapters, which are described in detailed as follows:In Chapter1, the brief history of fractional calculus is introduced. We discuss the devel-opment of fractional calculus in recent forty years. The surveys of fractional boundary value problem, numerical method of fractional differential equation, generalized fractional calculus and generalized fractional variational problem are given.In Chapter2, the topological degree theory in Banach space is introduced. Several use-ful fixed point theorems are given. We introduce the Riemann-Liouville fractional integral and derivative, Caputo fractional derivative. The semigroup properties of fractional operators are proved.In Chapter3, the fractional boundary value problem is studied. The fractional boundary value problem with Caputo derivative and irregular boundary condition, integral boundary con-dition and anti-periodic boundary condition are investigated separately. By using Banach fixed point theorem, the existence and uniqueness of fractional boundary value problem is verified. By using the Leray-Schauder degree theorem, the existence of solution is proved. By using the Krasnoselskii fixed point theorem on the cone, the existence of positive solution is obtained.In Chapter4, the first class of generalized fractional calculus, named K-, A-and-B-operators are studied. The numerical solution of diffusion-wave equation with5-operator involving ex-ponential kernel is studied. The diffusion-wave equation behaviors like both super-diffusion and wave equations. The harmonic oscillator by A-and B-operators is calculated. The dynamics of generalized oscillator equations are investigated. With the smooth kernel in generalized fraction-al operators, the initial conditions are required to satisfy the kernel. Applying B-operator, we define three types of van der Pol oscillator. The van der Pol oscillators with exponential kernel and weak singular kernel are solved, respectively. The generalized van der Pol oscillators exhibit more interesting dynamics. In unforced van der Pol oscillator, the limit cycle is a single Jordan curve. While in unforced van der Pol oscillator with modified power kernel, we see the limit cy-cle may twist together. Moreover, the chaotic phenomenon appears in forced van der Pol system with certain force. However, in unforced van der Pol oscillator with modified singular kernel, many new dynamics can be observed.In Chapter5, the second class of generalized fractional calculus is introduced. The gen-eralized fractional integral and derivative are defined by using a scale function and a weight function, which extend many existing integer and fractional order operators. We deduce that semigroup properties of generalized fractional operators. As applications, the Burgers equation and advection-diffusion equation are studied. We investigate the influences of four classes of scale function, linear and nonlinear stretch scale functions, linear and nonlinear contracting s-cale functions on the diffusion processes of them. The weight function weights differently in the domain. In simulation, we find that the increasing weight function coincides the inner property of fractional operator while the decreasing weight function arises much different phenomenon in diffusion process. Finally, we deduce the analytical solution of linear fractional advection-diffusion equation with generalized fractional derivative, which plays an important role in ana-lyzing the scale and weight functions.In Chapter6, the fractional variational problem is studied. First of all, we introduce the vari-ational principle, review the necessary conditions of extreme of integer-order functional. Then the fractional variational problem and the fractional variational principle, including fractional Euler-Lagrange equation and transversality condition, are deduced. We discuss the fractional minimizing functional problem, fractional isoperimetric problem and fractional optimal control problem. The fractional variational problems with fixed boundary conditions and free bound-ary conditions are studied respectively. Finally, we study the generalized fractional variational problem, including generalized fractional minimizing functional problem with fixed and free boundary conditions. The generalized fractional calculus is discussed over an arbitrary convex domain. Using the finite difference method and orthogonal polynomial technique, the general-ized fractional variational problems are easily solved.In Chapter7, we conclude the whole thesis and discuss some possible future work in frac-tional calculus field.