Research On Oscillation Of Fractional Partial Differential Equations

The theories and applications of differential equations permeate almost all the fields.Many specialists pay more attention to the research on qualitative theory of differential equations.As a part of the qualitative theory of differential equations,the oscillation of fractional differential equations become hot topic.Not only the integral and fractional differential equations,but also the fractional partial differential equation draw the people's attention.On the basis of the former research,We will study three kinds of fractional partial differential equations by using generalized Riccati transformation,integral averaging method and operator method.Some conclusions will be provided.This paper includes the following four parts:Chapter 1 In the preference,we mainly introduce some basic definition and properties about fractional derivative and integral,several lemmas also will be shown.Chapter 2 Motivated by the previous methods,we will study the oscillation of the equation by new methods:Chapter 3 Based on the conditions of [1],we will study the following equations,and take an forced term into the above equation,some sufficient conditions for oscillation will be obtained:Chapter 4 The oscillation properties of the following nonlinear fractional partial differential equation are presented:with the boundary condition:where ? is a bounded domain in Rnwith piecewise smooth boundary ??,? ?(0,1)is a constant,R+=(0,?).D?+,tu(x,t)is the Riemann-Liouville fractional derivative of order ? of u with respect to t,? is the Laplacian in Rn.N is the unit exterior normal vector to ??,and g(x,t)is a nonnegative continuous function on ?? × R_+.