Research On Singular Boundary Value Problems Of The Fractional Differential Equations

As an extension and expansion of the integer-order calculus, fractional calculus is a study of any order (the order of real or complex orders) differential, integral operator characteristics and application of theoretical mathematical problems. Its development is almost synchronized with the integer order calculus. Fractional calculus is an important branch of mathematical analysis, which has a wide range of theoretical and practical research value. In recent decades, fractional differential equations are increasingly being used to describe the optical and thermal systems, rheology and materials and mechani-cal systems, signal processing and system identification, control and robotics, and other fields of application problems, see[1-10]. Fractional calculus theory also received exten-sive attention of more and more domestic and foreign scholars, especially the fractional differential equations which abstract from the real problems become a research hotspot for many mathematicians. As the fractional differential equations appear in a growing number of scientific fields, the theoretical study of fractional differential equations and numerical analysis are particularly urgent. The thesis contains three chapters.In the first chapter, we study the existence of solutions for singular boundary value problem of the fractional differential equations where 2<α≤ 3 is a real number,f:[0,1]×(0,+∞)×[0,+∞)â†'[0,+∞) is continu-ous, Dα0+is the Riemann-Liouville fractional derivative. By the properties of the Green function and Leray-Schauder nonlinear alternative theorem, the fixed-point theorem, some new existence criterias for the problem(1.1) are established.In 2009, Nickolai Kosmatov[21] studied the existence of solutions for boundary value problem of the nonlinear fractional differential equations which function/is dependent of u’ by the fixed-point theorem. But this chapter has the different boundary value problem which caused the difference of Green function, so we choose a different way to study it.In the second chapter, we study the existence of solutions for boundary value prob-lem of the fractional differential equations which function f is sign-changing where 2<α≤ 3 is a real number,f:[0,1]×[0,+∞)â†'(-∞,+∞) is continuous, Dα0+is the Riemann-Liouville fractional derivative. By researching on the properties of the Green function and the fixed-point theorem, some new existence criterias for the problem (2.1) are established.In the third chapter, we study the existence of solutions for boundary value problem of the fractional differential equations which function f with parameter is sign-changing where 2<α≤3 is a real number,f:[0,1] x [0,+∞)â†' (-∞,+∞) is continuous, Dα0+is the Riemann-Liouville fractional derivative. By the properties of the Green function and the fixed-point theorem, some new existence criterias for the problem (3.1) are established.Though in the second and third chapter, function f has nothing to do with u’, function f is sign-changing which increases the difficulty of the problem. In these two chapters, the function f is different in the right range, so we choose differential methods to study them.The thesis not only prove each theorem, but also give examples to support the main theorems.