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In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form <br/>
<br/>
for where
Ω is a bounded domain in
*R ^{N}* with a piecewise smooth boundary
is a constant,
is the Riemann-Liouville fractional derivative of order
a of

*u*with respect to

*t*and is the Laplacian operator in the Euclidean

*N*-space

*R*subject to the condition

^{N}Fractional differential equations are generalizations of classical differential equations to an arbitrary non integer order and have gained considerable importance due to the fact that these equations are applied in real world problems arising in various branches of science and technology [

In this paper, we study the oscillatory behavior of solutions of nonlinear neutral fractional differential equations with forced term of the form

where

where

In what follows, we always assume without mentioning that

(A_{1})

(A_{2})

(A_{3})

(A_{4})

(A_{5})

(A_{6})

A function _{1}) ((E), (B_{2})) if it satisfies in the domain G and the boundary condition (B_{1}), (B_{2}). The solution of _{1}) or (E), (B_{2}) is said to be oscillatory in the domain _{1}) up to now. To develop the qualitative properties of fractional partial differential equations, it is very interesting to study the oscillatory behavior of (E) and (B_{1}). The purpose of this paper is to establish some new oscillation criteria for (E) by using a generalized Riccati technique and integral averaging technique. Our results are essentially new.

In this section, we give the definitions of fractional derivatives and integrals and some notations which are useful throughout this paper. There are several kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left sided definition on the half-axis

Definition 2.1. The Riemann-Liouville fractional partial derivative of order

provided the right hand side is point wise defined on

Definition 2.2. The Riemann-Liouville fractional integral of order

provided the right hand side is pointwise defined on

Definition 2.3. The Riemann-Liouville fractional derivative of order

provided the right hand side is pointwise defined on

Lemma 2.1. Let

Then

We introduce a class of function P. Let

The function

C_{1})

C_{2})

Lemma 3.1. If _{1}) for which

with

Proof. Let

Using Green’s formula and boundary condition (B_{1}) it follows that

and

Also from (A_{3}), (A_{5}), we obtain

and using and Jensen’s inequality we get

In view of (1), (7)-(10) and A_{6}, (6) yield

This completes the proof.

Lemma 3.2. Let _{1}) defined on

1)

2)

Proof. From Lemma 3.1, the function

for

Lemma 3.3. Let _{1}) defined on

Proof. From Case (I),

This completes the proof.

Lemma 3.4. Let _{1}) defined on

Proof. In this case the function

This completes the proof.

Theorem 3.1. Assume that

be continuous functions such that

Assume also that there exists a positive nondecreasing function

where

and

where

Then every solution _{1}) is oscillatory in

Proof. Suppose that _{1}), which has no zero in

Let

Case (I): For this case _{5}), (16) yields

Define the function

then

From

Let

substituting

Thus for all

Then, by (22) and (C_{2}), for

Then, by (14) and (C_{2}), we have

which contradicts (14).

Case (II): Assume that

Let

Integrating (26) from

condition (15) implies that the last inequality has no eventually positive solution, a contradiction. This completes the proof.

Corollary 3.1. Let conditions of Theorem 3.1 be hold. If the inequality (16) has no eventually positive solutions, then every solution _{1}) is oscillatory in

Corollary 3.2. Let assumption (14) in Theorem 3.1 be replaced by

and

Then every solution _{1}) is oscillatory in

Let

Corollary 3.3. Let assumption (14) in Theorem 3.1 be replaced by

for some integer_{1}) is oscillatory in

Next we establish conditions for the oscillation of all solutions of (E), (B_{1}) subject to the following con- ditions:

C_{3})

C_{4})

Theorem 3.2. In addition to conditions (C_{3}) and (C_{4}) assume _{1}) are oscillatory if

and

where

Proof. Suppose that _{1}), which has no zero in

Let

and

Let

Integrating the last inequality from

since

Letting

where

From Lemma 3.2 there are two possible cases for

Integrating the last inequality from

By (C_{4}) and Lemma 3.3, we have from (32)

Letting

For this case

and

From (34) and (35) we have

which contradicts (27).

Next we consider the case that

Consider

Let

here we have used (C_{4}), (37) and Lemma 3.4. Integrating the last inequality from

and so letting

which contradicts (28). This completes the proof.

Next we consider (E), (B_{1}) subject to the following conditions:

C_{5})

Theorem 3.3. In addition to conditions (C_{3}) and (C_{5}) assume that

and

Then every solution _{1}) is oscillatory in

Proof. Without loss of generality we may assume that _{1}). Therefore

If

which contradicts (38). For this case

We consider the fractional differential

Let

according as

Integrating and rearranging we obtain

and so letting

which contradicts (39). This completes the proof.

In this section we establish sufficient conditions for the oscillation of all solutions of (E), (B_{2}). For this we need the following:

The smallest eigen value

is positive and the corresponding eigen function

Theorem 4.1. Let all the conditions of Theorem 3.1 be hold. Then every solution of (E), (B_{2}) oscillates in

Proof. Suppose that _{2}), which has no zero in

We obtain for

Using Green’s formula and boundary condition (B_{2}) it follows that

and for

Also from (A_{3}), (A_{5}), we obtain

and using and Jensen’s inequality we get

Set

In view of (41)-(45) and (A_{6}), (40) yield

for

Using the above theorem, we derive the following Corollaries.

Corollary 4.1. If the inequality (46) has no eventually positive solutions, then every solution _{2}) is oscillatory in G.

Corollary 4.2. Let the conditions of Corollary 3.2 hold; then every solution _{2}) is oscillatory in G.

Corollary 4.3. Let the conditions of Corollary 3.3 hold; then every solution _{2}) is oscillatory in G.

Theorem 4.2. Let the conditions of Theorem 3.2 hold; then every solution _{2}) is oscillatory in G.

Theorem 4.3. Let the conditions of Theorem 3.3 hold; then every solution _{2}) is oscillatory in G.

The proof Theorems 4.2 and 4.3 are similar to that of Theorem 4.1 and ends details are omitted.

In this section we give some examples to illustrate our results established in Sections 3 and 4.

Example 1. Consider the fractional neutral partial differential equation

for

Example 1 is particular case of Equation (E). Here

and

It is easy to see that

Here n = 1, m = 1, so we have

Take

Here m = 1, n = 1 so we have

Consider

Choose

Thus all the conditions of Corollary 3.3 are satisfied. Hence every solution of (E_{1}), (47) oscillates in

Example 2. Consider the fractional neutral partial differential equation

for

Here

It is easy to see that

Take

Consider

Choose

Thus all the conditions of Corollary 3.3 are satisfied. Therefore every solution of (E_{2}), (48) oscillates in

The authors thank Prof. E. Thandapani for his support to complete the paper. Also the authors express their sincere thanks to the referee for valuable suggestions.

V.Sadhasivam,J.Kavitha, (2015) Forced Oscillation of Solutions of a Fractional Neutral Partial Functional Differential Equation. Applied Mathematics,06,1302-1317. doi: 10.4236/am.2015.68124