A Study Of Existence And Uniqueness Of Solutions Of Several Classes Fractional Differential Equations

So far，the fractional calculus has300years old. As a relatively young discipline，it’s onlyconcerned about in the field of mathematics for a very long time. With the rapid development ofhigh-tech，it gets involved in other areas more and more，especially in the wide application ofcontrol theory，viscoelastic theory，electronic chemicals，fractal theory and so on. See reference[1].Fractional calculus is a promotion of the integer calculus，but the former is wider used.Switched from integer to any fraction，even irrational numbers，it has attracted the strongattention and research of the engineering and technical personnel. Accompanied the perfectconclusions and results have sprung up，with the depth and difficulty are constantly increasing.Based on the previous study，I mainly done the following aspects:(1) The general expressions of both linear and nonlinear solutions of the δ,between2and3, order fractional differential two-point boundary value condition problems are obtained，andeigenfunctions under the different values of the parameters λ are given by Laplace transform.From the perspective of algebra，we account for that the structure of solution of the nonlinearequation in the case of fractional is still feasible.(2) The existence and uniqueness theorem of the solution of the δ,between2and3, orderfractional differential equation three-point boundary value condition problem is verified by fixedpoint theory.(3) The numerical solution of fractional differential equation is studied by the GLapproximation of difference method. With the Matlab software, a specific numerical example isgiven, its accuracy is verified by the analytical solution.(4) Between n-1and n order multi-point boundary value problem of fractionaldifferential equations，which has a boundary value mixed numbers derivative，is researched.Upper and lower solutions are used to deduce the existence and uniqueness conclusion，furtherderivation of existence conditions of a positive solution and a negative solution.