Existence Of Solutions For Several Classes Of Singular Boundary Value Problems Of Fractional Differential Equations And Their Applications

Tracing back to the 1920 s, singular boundary value problem was raised up by two renowned physicians when they were trying to establish a model of the electromotive force of an atom. The model they built was a second-order singular ordinary equation boundary value problem. Gradually, it became an independent research area in that the method and means used when dealing with singular boundary value problem was special. Singular boundary value problems play an important part in various fields of science of nature, such as electrically conducting solids, electrical potential theory, circular membrance theory and so on.Fractional models compared with integral ones are more flexible, that’s to say problems described by fractional models usually turn to be more precise and accurate. Recently, taking a glimpse of the numerous scientific achievements by researchers among mathematics, physics, control theory, chaos and turbulence, biology and medical science and so forth, we find it no doubt that fractional differential equation boundary value problem as a latest topic has already become one of the hottest inernational researches on differential equation theories. Some even speak highly of fractional differential calculus as the twenty-first century calculus.Nonlinear functional analysis theory is considered as the most important method and significant basis of investigating fractional differential equation boundary value problems of the contemporary and modern era. Series of fixed point theorems are deduced from it. It takes significant part in telling the existence multiplicity or uniqueness of solutions for fractional differential equations boundary value problems. Varying from the properties of the nonlinear items, the method and means adopted just depends. However, in terms of singular boundary problems, besides the above work, researchers have to take adequate measures to cancel its singularity. Simplification itself makes singular problems full of theoretical value. In addition, singular differential boundary value problems possess an extensive background of application. Easy to find out that singular fractional differential boundary value problem has not only theoretical value but also practical applications.In this thesis, we mainly discuss several problems as follows, the existence, uniqueness and multiplicity of solutions to some classes of boundary value problems for several kinds of singular fractional differential equations. After deep investigation, we get some interesting results under more generalized conditions. Some of them have already been pubilished on journals “Computers and Mathematics with Applications”, “Boundary Value Problems” and so on, and have been adopted by SCI.This thesis is devided into six chapters, mainly involves four kinds of problems. In chapter two, a class of fractional differential boundary value problems with nonlinear boundary conditions is discussed, it is generalized from integral order to fractional one. The boundary condition is much wider, it includes Dirichlet boundary value condition, integral boundary condition and multiple points boundary condition and so on. By using Guo-Krasnosel’skii’s fixed point theorem, several proficient conditions of existence of positive solutions are available. In chapter three, on the basis of chapter two, singularity is brought in. Via constructing approximating sequences, the singularity is removed. By means of Gatica-Oliker-Waltman fixed point theorem, we state its positive solution exists. Well worth to mention, this method is rarely used to discuss the existence of positive solutions of fractional differential equation boundary value problems. In chapter four, a real application model, Thomas-Fermi model, is considered. It is a singular fractional differential boundary value problem with a parameter. By virtue of Guo-Krasnosel’skii’s fixed point theorem and telling from the rang of the parameter, series of proficient conditions about whether the equation has a solution or not, when would it has more than one solution has been obtained. In chapter five, a class of singular fractional differential system catches our eyes. This system is also widely used in real applications, for instance, establishing a coupled circuit model, AIDS model and so on. With the help of Leray-Schauder nonlinear alternative theorem several proficient conditions of existence of positive solutions are established.