Study On The Solutions Of Several Kinds Of Fractional Differential Equations And Their Related Problems

With the rapid development of fractional calculus,the existence of solutions fractional differential equations and its solutions have been widely concerned and studied.Nowadays,fractional differential equations have been successfully applied to many scientific fields,such as anomalous diffusion,chemical physics,chaos and turbulence,automatic control theory,viscoelastic mechanics and non-Newtonian fluid mechanics.its wide application and importance are self-evident.This paper deals with the exact solution and dynamic properties of fractional telegraph equation and the existence of solutions for boundary value problems of two kinds of Riemann-Liouville fractional differential equations,its specific contents are as follows:In the second chapter,the exact solution fractional telegraph equations is studied.Firstly,the integer order telegraph equation is transformed into fractional telegraph equation by scale transformation,Secondly,using the method combing the separation of variables with homogeneous equilibrium principle,various exact solutions of fractional telegraph equations and their sub-equations are obtained,it includes the exact solution in the form of exponential function and trigonometric function in the part of space variable or time variable,Finally,the mechanical properties and evolution phenomena of these exact solutions are discussed,and the three-dimensional coordinate map of the exact solution evolution with time and space is drawn by using Maple software.In the third chapter,the existence of positive solutions for a class of fractional boundary value problems with Riemann-Liouville type is discussed.Mainly through the Green function and the definition of completely continuous operators,the existence problem of the positive solution of the boundary value problem is transformed into the existence problem of the positive fixed point of the operator.Then combining the spectral radius of the operator,the inequality conditions for nonlinear terms are given,and then the fixed point exponent theorem is used to prove that there are two positive solutions to the boundary value problem.In the fourth chapter,the existence of solutions for a class of Riemann-Liouville type fractional boundary value problems is studied.Similar to the third chapter,the existence problem of solutions of boundary value problem is transformed into the existence problem of fixed points of operators.Then,under the condition of the inequality suitable for the nonlinear term,the first eigenvalue construction condition combining with correlation operator,and then using the knowledge of the topological degree theory,the existence of nontrivial solution for the boundary value problem is established.