FORCED OSCILLATION FOR A CLASS OF FRACTIONAL PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

We investigate the oscillation of class of time fractional partial differential equation of the form , ( ))) , ( ( ( j t             for ), [0,= , = ) , (     R G R t x where  is a bounded domain in N R with a piecewise smooth boundary (0,1) ,    is a constant,  t D ,  is the Riemann-Liouville fractional derivative of order  condition      R t x t x u t x t x u ) , ( 0, = ) , ( ) , ( ) , (  

, where  is a bounded domain in N R with a piecewise smooth boundary

1.INTRODUCTION
Fractional differential equations, that is differential equations involving fractional order derivatives seems to be a natural description of observed evolution phenomena of several real world problems. Recently studying fractional order differential systems turn out to be an active area of research. It is evident that in interdisciplinary fields many systems can be described by fractional differential equations [2, 9-12, 16,20]. The study of oscillation and other asymptotic properties of solutions of fractional order differential equations has attracted a good bit of attention in the past few years [4][5][6]8]. However, only a few results have appeared regarding the oscillatory behavior of fractional partial differential equations, see [1,13,[14][15]17] and the refrences cited there in.
Chen [3] studied the oscillation of the fractional differential equation is Liouville right-sided fractional derivative of order In [19], the authors considered non linearity term ) ( y g to self adjoint term in the class of fractional differential equation and derive the oscillation criteria for the following equation To the best of our knowledge, nothing is known regarding the oscillatory behavior for the following class of fractional partial differential equations with forced term of the form where  is a bounded domain in N R with a piecewise smooth boundary  is the Riemann-Liouville fractional derivative of order  of u with respect to t and  is the Laplacian operator in Our results established in this paper are infact improvement of results in [13] and [19]. These oscillation criteria generalize those of existing one. In what follows, we always assume without mentioning that we mean a non trivial function ) ; A solution ) , ( t x u of ) (E is said to be oscillatory in G if it is neither eventually positive nor eventually negative. Otherwise it is nonoscillatory. Equation ) (E is said to be oscillatory if all its solutions are oscillatory. The purpose of this paper is to establish some new oscillation criteria for ) (E by using a generalized Riccati technique and integral averaging method. Our results are essentially new.

2.PRELIMINARIES
In this section, we give the definitions of fractional derivatives and integrals and some notations which are useful throughout this paper. There are serveral kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left sided definition on the half-axis .
 R The following notations will be used for the convenience.
Definition 2.1. [8] The Riemann-Liouville fractional partial derivative of order provided the right hand side is pointwise defined on  R where  is the gamma funciton. on the half-axis  R is given by provided the right hand side is pointwise defined on . on the halfaxis  R is given by

3.OSCILLATION OF THE PROBLEM
We begin with the following theorem. Theorem 3.1. If the fractional differential inequality has no eventually positive solution, then every solution of is a nonoscillatory solution of ) (E and ). ( 1 B Without loss of genearality, we may assume Integrating (E) with respect to Using Green's formula and boundary condition By using Jensen's inequality and   2 H we get In view of (1), (10)- (12) and (9) yield is an eventually positive solution of (8). This contradicts the hypothesis and completes the proof. (14) hold. Futhermore, Assume that there exists a positive function ) ); ,

Theorem 3.2. Suppose that the conditions
is a non oscillatory solution of (8). Without loss of generality we may assume that V is an eventually positive solution of (8). Then there exists Therefore from (6) we have This contradicts (14).
Define the function W by the generalized Riccati substititution Using Lemma 2.2 and (20) in (19), we have Integrating both sides of the above inequality from Taking the limit supremum of both sides of the above inequality as which contradicts (15), and completes the proof.

Proof. Suppose that
is non oscillatory solution of (8) Without loss of generality we may assume that V is an eventually positive solution of (8). Then proceeding as in the proof Theorem 3.2, to get (21) which is a contradiction to (22).The proof is complete.
Next, we consider the case , < ) ( which yields that (14) does not hold. In this case, we have the following result.  Then every solution V of (8) is oscillatory or satisfies is nonoscillatory solution of (8) Without loss of generality we may assume that V is an eventually positive solution of (8). Then proceeding as in the proof Theorem 3.2, there are two cases for the sign of is eventually positive is similar to that of Theorem 3.2 and hence is omitted. Next,

Integrating both sides of the last inequality from
The proof is now complete.
Assume that V is an nonoscillatory solution of (8). Without loss of generality assume that V is an eventually positive solution of (8). Proceeding as in the proof of Theorem 3.2, there are two cases for the sign is eventually positive is similar to that of Theorem 3.3, and hence is omitted. The proof when is eventually negative is similar to that of Theorem 3.4, and thus is omitted. The proof is now complete.

Proof. Suppose that
is a nonoscillatory solution of ) (E and ) ( 2 B . Without loss of genearality, we may assume that Multiplying both sides of the Equation (E) by and then integrating with respect to x over .
 We obtain for Using Green's formula and boundary condition ) ( By using and Jensen's inequality, and ) ( Therefore, In view of (26)-(29), (25) yields Rest of the proof is similar to that of Theorems 3.2 and 3.3, and hence the details are omitted.

EXAMPLES
In this section we give some examples to illustrate the results established in Sections 3 .

Example 1.
Consider the fractional partial differential equation , with the boundary condition It is easy to see that and (14) hold. Therefore

Example 2.
Consider the fractional partial differential equation

Conclusion:
We have studied the oscillatory behavior for a class of fractional parabolic partial differential equation Riccatti transformation technique and the integral averaging method. We illustrated our main results by providing suitable examples. We believe that there is a wide scope for further study on this topic.