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¡¡¡¡Fractional order calculus is the theory of arbitrary order differential and integral£¬ it is unified with the integer order differential and integral calcu-lus£¬ is the development of classical calculus£¬ fractional calculus as a descrip-tion of classical physics and related discipline theory analytic mathematical tools have been widely accepted£¬ but when people study complex systems and complex phenomena£¬ the classical integer order differential and integral equation description for the systems will encounter a series of problems£¬therefore£¬ there is an urgent looking forward to having a kind of mathema-tical tools available and can be based on the basic principle of the complex system modeling. Fractional-order differential equations are very suitable for describing materials and processes with memory and heritability£¬ and their description of complex systems has the advantages of simple model-ing£¬ clear physical meaning of parameters and accurate description.¡¡¡¡In recent decades£¬ fractional differential equations have been increasing-ly used to describe problems in optical and thermal systems£¬ theological and material and mechanical systems£¬ signal processing and system identi-fication£¬ control and robotics and their applications. Fractional differentialequations have been investigated by more and more scholars from around the world. With the emergence of fractional-order differential equation model in more and more scientific fields£¬ the theoretical analysis of fractional-order differential equations is particularly urgent.¡¡¡¡The first chapter is mainly about some definitions£¬ lemmas and prepa-ration theorems£¬ preparing for the presentation of follow-up results.¡¡¡¡In chapter 2£¬ after a brief introduction£¬ in section 2£¬ nonlinear frac-tional differential equations involving different Riemann-Liouville fractional derivatives are presented; in sections 3 and 4£¬ three-point boundary value problem and multi-point boundary value problem under different conditions are discussed; in section 5£¬ integral boundary value problem for nonlinear fractional differential equations on an unbounded domain is presented; sec-tion 6 is about the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments.Section 7 is concerned with the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional dif-ferential equation with integral boundary conditions. In section 8£¬ under certain nonlinear growth conditions of the nonlinearity£¬ the existence of so-lutions for a nonlinear Hadamard type fractional differential equation with strip condition and p-Laplacian operator is studied. In section 9£¬ the unique-ness£¬ existence and nonexistence of solutions of the fractional turbulent fiow model are discussed.¡¡¡¡Chapter 3 is about nonlinear fractional integro-differential equation.Section 2 is about initial value problem for nonlinear neutral fractional integro-differential equation with nonlinear term depending on lower order derivative. In the third section£¬ the existence of minimal nonnegative so-lution for a class of nonlinear fractional integro-differential equations on semi-infinite intervals in Banach spaces is discussed. Section 4 is concerned with the existence of solutions for nonlinear fractional differential equations of Volterra type with nonlocal fractional integro-differential boundary con-ditions on an infinite interval. In section 5£¬ a Hadamard type fractional integro-differential equation on infinite intervals is considered.¡¡¡¡Nonlinear impulsive fractional differential equation is the focus of Chap-ter 4. Nonlinear impulsive fractional differential equations with anti-periodic boundary conditions are discussed in section 2. Section 3 is about nonlinear impulsive fractional differential equations with nonlocal integral boundary condition. Section 4 is concerned with nonlinear Langevin equa-tion with two different fractional orders and impulses.¡¡¡¡Chapter 5 is devoted to the system of nonlinear fractional differentialequation. In Section 2£¬ the existence results and the monotone iterativetechnique for systems of nonlinear fractional differential equations are p-resented. Section 3 is about the existence of an extremal solution for a nonlinear system involving the right-handed Riemann-Liouville fractional derivative with nonlocal coupled integral boundary conditions. Section 4 is about the existence of solutions for a coupled system of nonlinear neutral fractional differential equations£¬ which involves Riemann-Liouville deriva-tives of different fractional orders. Section 5 in the chapter is concerned with a coupled system of nonlinear fractional differential equations with multi-point fractional boundary conditions on an unbounded domain.¡¡¡¡At the end of each chapter£¬ there are short notes and remarks indicating the source of the content.