The Theory Of Ulam Stability Of Several Types Of Classical And Fuzzy Differential Equations

This thesis mainly contains three parts.The first part contains Chapter 2.In this part,the Ulam stability of two types of linear differential equations of second order is established.The second part is Chapter 3.The main purpose of this part is to establish the Ulam stability of fuzzy differential equations under generalized differentiability.The third part is Chapter4.This part deals with the Ulam stability problem of linear fractional differential equations with constant coefficients and the Cauchy type problem of interval-valued fractional differential equations,respectively.The main research works of this thesis are described as follows:Chapter 1 is introduction and preliminaries.This chapter makes a general survey of this thesis and provides some necessary preliminaries on the algebraic structure and operations in the space of interval numbers and fuzzy numbers,the calculus of fuzzy number-valued functions,fuzzy Laplace transform and Riemann-Liouville fractional calculus of single-valued and intervalvalued functions.Chapter 2 is the Ulam stability of two types of linear differential equations of second order.Firstly,the Ulam stability results of a class of Banach space valued linear differential equations of seconder order are obtained.These results improve and extend the corresponding results of Chebyshev differential equations in the relevant literature.Secondly,the Ulam stability of a class of exact differential equations of second order is established by employing the integrating factor method,which can be directly applied to a special class of Cauchy-Euler equations of second order.Chapter 3 is the Ulam stability of fuzzy differential equations under generalized differentiability.Firstly,the Ulam stability results of first order linear fuzzy differential equations with constant coefficients are proved by the fuzzy Laplace transform.To obtain the more general results,by using the direct construction method,we establish the Ulam stability of first order linear fuzzy differential equations with constant positive or constant negative coefficients functions.Secondly,we obtain the Ulam stability of a general fuzzy differential equation by employing the fixed point theorem in a generalized complete metric space.Finally,we establish the Ulam stability of two types of linear partial fuzzy differential equations of first order based on the corresponding results of first order linear fuzzy differential equations.Chapter 4 is the Ulam stability of linear fractional differential equations and the Cauchy type problem of interval-valued fractional differential equations.Firstly,the Ulam stability of linear fractional differential equations with constant coefficients involving Liouville fractional derivatives is obtained by the Laplace transform method.Secondly,the relationship between the Cauchy type problem for interval-valued fractional differential equations with the RiemannLiouville gH-fractional derivative and the corresponding interval-valued integral equation is derived.Moreover,the existence and uniqueness theorem of the solutions to the previous intervalvalued integral equation is also established.Furthermore,the solution to the Cauchy type problem is obtained under suitable conditions.Chapter 5 is conclusion and outlook.The main contribution of this thesis is summarized,and the future research directions are clarified.