Oscillation Criteria Of Certain Fractional Partial Differential Equations

Fractional differential equations have been an important branch of the differential e-quation,at the same time,the appearance of the fractional partial differential equations is becoming more and more frequent and playing an important role in many fields,such as mechanical model,engineering,biological genetic,aerospace,finance,bioengineering options and so on.It is worth to consider that integer differential equations have been researched for a long time and have reduced a lot of methods to illustrate the sufficient conditions for oscillation of the solutions.Therefore we can refer integer differential equations when we deliberate the oscillation of the fractional partial differential equations.In recent years,the research of the fractional partial differential equations have caused the concern day by day,more and more scholars explore the oscillation and other properties,including existence as well as boundary value problem,and appears numerous thesises and monographs at the same time.In this paper,we study the sufficient conditions for oscillation criteria of frac-tional partial differential equationd by using the generalized Riccati substitution,fractional integral and Riemann-Liouville fractional derivative.According to the content,this article is divided into following four chapters.Chapter 1 introduce the background and main contents of this article.Chapter 2 investigate the oscillation criteria for the fractional partial differential equa-tion with the Robin boundary condition where ??(0,1)is a constant,D+?,t is the Riemann-Liouville fractional derivative of order? of u with respect t,? is a bounded domain in Rn with piecewise smooth boundary(?)?,? is the Laplacian operator and N is the unit exterior normal vector to(?)?,and g(x,t)is a nonnegative continuous function on(?)?×R+.We extend the connotation of function H,segment the section and discuss respectively in order to analyzed the oscillation of the equations.Chapter 3 investigate the oscillation criteria for the fractional partial differential equa-tion with the Robin boundary condition where ??(0,1)is a constant,D+,t? is the Riemann-Liouville fractional derivative of order a of u with respect t,? is a bounded domain in Rn with piecewise smooth boundary(?)?,? is the Laplacian operator and N is the unit exterior normal vector to(?)?,and g(x,t)is a nonnegative continuous function on(?)?×R+.In this chapter,we deal with the oscillation by using disproof,function H as well as Holder inequation.Chapter 4 summarize the main contents of the paper and make a series of vistas to the future orientation of the reaearch.