The Hyers-Ulam Stability Of Fractional Differential Equations

Fractional calculus is a traditional integer order calculus, although it has 300 years of history, until the end of nineteenth Century, the theory was gradually established by the Grunwall, Letnikov, Liouville and Riemann et al. In this 300 years, there is no research on the practical application to the physical mechanics, mathematicians have been studying the pure theory. Research has not taken a breakthrough. Until the 1990s fractional calculus theory is applied to many fields, especially in control theory, viscoelastic theory, electronic chemistry, fractal theory and so on. At the same time, a large number of research results have also greatly promoted the research progress of fractional calculus, some scholars have put into this new research field. In the use of fractional order model, a series of fractional differential equations have been appeared, so it is very important to study the fractional differential equations.However, Khalil defined a new well-behaved simple fractional derivative called the conformable fractional derivative, depending just on the basic limit definition of the derivative. Namely, for a function f:(0,∞)â†'R the conformable fractional derivative of order 0<α<1of f at x>0 was defined by If f is α-differentiable in some (0,a),a>0 and limxâ†'0+ f(α)(x) exists, then define f(α)(0)= limxâ†'0+ f(α)(x),In ths thesis two class of fractional differential equations are investigated. Mine results are the following:(1) In consideration of the Laplace transformation for a class of fractional order linear differential equations with Caputo derivative, And then discuss the stability of Hyers-Ulam.. For the nonlinear case, the Volterra integral equation is given, Hyers-Ulam stability of the equation is obtained by using fractional Gronwall inequality.(2) For the conformable fractional differential operator, the Gronwall inequality with the conformable score is proved by using the related properties, and the related inferences can be obtained by using the same method.(3) Hyers-Ulam stability of a class of differential equations is proved on the basis of conformable fractional differential operator and Gronwall inequality.