Oscillation Of Differential Equations

The theory of partial differential equations can be applied to many fields, such as to biology, physics, chemistry, engineering, control theory, population growth, generic repression and cli-mate model. In the last few years, the fundamental theory of partial differential equations with deviating argument has undergone intensive development. However, the qualitative theory of this class of equations is still in an initial stage of development. Many studies have been done under the assumption that the state variables and system parameters change continuously. However, there are many natural phenomena such as shock. They are short time disturbance and their dura-tion is quite short. Hence we can assume that these disturbance act instantaneously, it is called impulses.In the last 50 years the oscillation theory of ordinary, functional, neutral, partial and impul-sive differential equations, and their discrete versions has attracted many researchers. Oscillation theory is an important research area in qualitative theory of differential equations and it is widely used in natural and social science. Recently, more and more researchers investigated the theory of fractional differential equations. However, the research of oscillation of fractional differential equations is still initial now, only a few researchers published papers.In Chapter 1, we introduce the background of partial differential equations, functional dif-ferential equations, impulsive differential equations, fractional calculus and differential equations and some books about these. We introduce background and results of oscillation of partial differ-ential equations and fractional differential equations as well.In Chapter 2, we study oscillation of two hyperbolic equations. In Section 1 we investigate nonlinear neutral hyperbolic equations with linear impulses and Neumann boundary condition and establish sufficient conditions. We use the method of proof by contradiction, and we use in-tegral averaging technique, Green formula and linearized conditions to transform partial differen-tial equations to impulsive differential inequality. Finally, we use Riccati transformation and the property of impulsive differential inequality to prove the theorem. In Section 2 we study a hy-perbolic equation with nonlinear impulses and boundary conditions (Dirichlet and Robin bound-ary conditions), and also use the same methods to prove the theorem. However, the difference is that we restrict the nonlinear impulses and transform them to linear impulses.In Chapter 3, we research oscillation of two parabolic equations. In Section 1 we study forced oscillation of nonlinear impulsive delay parabolic equations with Dirichlet, nonlinear and Robin boundary conditions. We also use the method of proof by contradiction, integral averaging technique and Green formula to transform it to impulsive differential inequality. Then transform it to differential inequality without impulses and establish sufficient conditions for oscillation of all solutions. We investigate a nonlinear impulsive parabolic equation with continuous distributed deviating arguments with three boundary conditions in Section 2. We also use the same method to transform the equation to impulsive differential inequality with continuous distributed deviat-ing arguments, then we transform it to impulsive differential inequality with discrete distributed deviating arguments.In Chapter 4, we discuss the oscillation of fractional differential equations. In Section 1, we introduce the definition of fractional calculus. In this thesis, we use Riemann-Liouville definition and we introduce some properties of Riemann-Liouville derivative. In Section 2, we discuss forced oscillation of fractional differential equation with damping term. We use the method of proof by contradiction, the properties of fractional calculus and prove the theorem directly. This method have not been used by other researchers. In the end of this section, we mention that the effect of delay term. In Section 3, we investigate fractional partial differential equations. We also use the method of proof by contradiction, integral averaging technique and Green formula to transform the fractional partial differential equations to fractional differential inequality and use the properties mentioned in Section 1 to complete the prove. We establish sufficient conditions for oscillation of all solutions.In Chapter 5, we summarize the research and results in this thesis and introduce the outlook.