Research On Hyers-Ulam Stability Of Some Kinds Of Differential Equations

The research of Hyers-Ulam stability stems from the general functional equations proposed by S.Ulam:when is it true that a approximate solution of given equation must have a precise solution nearby. D.Hers used the direct method and solved part of Ulam’s problem in the framework of Banach space. T.Rassias introduced the concept of unbound-ed Cauchy difference and extended the Hyers’s theorem to the case of approximate linear mappings. After that,such problems were called Hyers-Ulam stability problem, which at-tracted a lot of mathematicians attention. Many an important result is obtained. For instance, S.Jung studied the Hyers-Ulam stability of the Jensen functional equations and the first order linear differential equations.This paper mainly studies the problem of Hyers-Ulam stability of the integer order differential equations and fractional order partial differential equations.The main content of this thesis is arranged as follows:In the first chapter, we first introduce briefly to the reality background and the significance of research of differential equations and Hyers-Ulam stability, then this paper content is arranged.In the second chapter, some preliminary knowledge of need is introduced.In the third chapter, we discuss the Hyers-Ulam stability of second order constant coefficient differential equations and partial differential equations.In the fourth chapter,it considers the Hyers-Ulam stability of fractional differential equation. The first section firstly introduces that fractional order mixed partial differ-ential equations is converted to a second order differential equation by using fractional transform. And then we discuss the Hyers-Ulam stability of the second order differential equation, so the original equation is obtained with Hyers-Ulam stability. The second and third sections we discuss the Hyers-Ulam stability of fractional order partial differential equations.Firstly, the fractional order partial differential equations is converted to a first order partial differential equation by using fractional transform. Secondly,the first order partial differential equation is converted to ordinary differential equation by variable trans-form, and we write a solution of the first order partial differential equation. Finally, we discuss the Hyers-Ulam stability of the first order partial differential equation,therefore the original equation is obtained with Hyers-Ulam stability.In the fifth chapter, this chapter is summary and outlook of this paper.