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In this article, we will establish sufficient conditions for the interval oscillation of fractional partial differential equations of the form It is based on the information only on a sequence of subintervals of the time space rather than whole half line. We consider <i>f</i> to be monotonous and non monotonous. By using a generalized Riccati technique, integral averaging method, Philos type kernals and new interval oscillation criteria are established. We also present some examples to illustrate our main results.

Fractional differential equations are now recognized as an excellent source of knowledge in modelling dynamical processes in self similar and porous structures, electrical networks, probability and statistics, visco elasticity, electro chemistry of corrosion, electro dynamics of complex medium, polymer rheology, industrial robotics, economics, biotechnology, etc. For the theory and applications of fractional differential equations, we refer the monographs and journals in the literature [

In 1965, Wong and Burton [

In 1970, Burton and Grimer [

In 2009, Nandakumaran and Panigrahi [

In this article, we wish to study the interval oscillatory behavior of non linear fractional partial differential equations with damping term of the form

where

the Euclidean N-space

boundary condition

where γ denotes the unit exterior normal vector to

In what follows, we always assume without mentioning that

By a solution of

_{1}) and (E), (B_{2}) upto now. Motivativated by [_{1}) and (E), (B_{2}). Our results are essentially new.

Definition 1.1. A function

where

In this section, we will see the definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left sided definition on the half axis

For

Definition 2.1 [

provided the right hand side is pointwise defined on

Definition 2.2 [

provided the right hand side is pointwise defined on

Definition 2.3 [

provided the right hand side is pointwise defined on

Lemma 2.1 Let y be solution of

Then

In this section, we assume that

Theorem 3.1 If the fractional differential inequality

has no eventually positive solution, then every solution of

Proof. Suppose to the contrary that there is a non oscillatory solution

Using Green’s formula and boundary condition

By Jensen’s inequality and

By using

In view of (1), (6)-(8), (5) yield

Take

Therefore

Remark 3.1 Let

Then

Theorem (3.1) can be stated as, if the differential inequality

has no eventually positive solution then every solution of (E) and (B_{1}) is oscillatory in

Theorem 3.2 Suppose that the conditions (A_{1}) - (A_{5}) hold. Assume that for any

If there exist

where

Then every solution of

Proof. Suppose to the contrary that

Then for

By using

By assumption, if

therefore inequality (12) becomes

Let

Then

That is

Let

and integrating it over

Letting

On the other hand, substituting

Letting

Now we claim that every non trivial solution of differential inequality (9) has atleast one zero in

Suppose the contrary. By remark, without loss of generality, we may assume that there is a solution of (9) such that

which contradicts the assumption (11). Thus the claim holds.

We consider a sequence

at least one zero in

bitrary large zero. This contradicts the fact that

Theorem 3.3 Assume that the conditions (A_{1}) - (A_{5}) hold. Assume that there exist

and

where

Proof. For any

In (18) take

Dividing Equations (19) and (20) by

Then it follows by theorem 3.2 that every solution of

Consider the special case

Thus for

Theorem 3.4 Suppose that conditions (A_{1}) - (A_{5}) hold. If for each

where

Proof. Let

For any

From (21) we have

since

Hence every solution of

Let

Corollary 3.1 Assume that the conditions (A_{1}) - (A_{5}) hold. Assume for each

and

Then every solution of

Theorem 3.5 Suppose that the conditions (A_{1}) - (A_{5}) hold. If for each

and

Then every solution of

Proof. Clearly

Note that

and

Consider

Similarly we can prove other inequality

Next we consider

Theorem 3.6 Assume that the conditions (A_{1}) - (A_{5}) hold. If for each

and

Then every solution of

Proof. From (17)

Similarly we can prove that

If we choose

Corollary 3.2 Suppose that the conditions (A_{1}) - (A_{5}) hold. Assume for each

and

Then every solution of

Corollary 3.3 Suppose that the conditions (A_{1}) - (A_{5}) hold. Assume for each

and

Then every solution of

We now consider non monotonous situation

Theorem 4.1 Suppose that the conditions (A_{1}) - (A_{4}) and (A_{6}) hold. Assume that for any

If there exist

where

Then every solution of

Proof. Suppose to the contrary that

Then for

By using

By assumption, if

terval

terval

Therefore inequality (26) becomes

Let

Then

where

that is

The remaining part of the proof is the same as that of theorem 3.2 in section 3, and hence omitted.

Corollary 4.1 Suppose that the conditions (A_{1}) - (A_{4}) and (A_{6}) hold. Assume for each

and

Then every solution of

In this section, we establish sufficient conditions for the oscillation of all solutions of

The smallest eigen value

is positive and the corresponding eigen function

Theorem 5.1 Let all the conditions of Theorem 3.2 be hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Proof. Suppose to the contrary that there is a non oscillatory solution

Using Green’s formula and boundary condition

By using Jensen’s inequality and

Set

Therefore,

By using

In view of (31), (29)-(30), (32), (28) yield

Take

Rest of the proof is similar to that of Theorem 3.2 and hence the details are omitted.

Remark 5.1 If the differential inequality

has no eventually positive solution then every solution of

Theorem 5.2 Let the conditions of Theorem 3.3 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Theorem 5.3 Let the conditions of Theorem 3.4 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Corollary 5.1 Let the conditions of Corollary 3.1 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Theorem 5.4 Let the conditions of Theorem 3.5 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Theorem 5.5 Let the conditions of Theorem 3.6 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Corollary 5.2 Let the conditions of Corollary 3.2 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Corollary 5.3 Let the conditions of Corollary 3.3 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

Theorem 5.6 Let all the conditions of Theorem 4.1 be hold. Then every solution of (E), (B_{2}) is oscillatory in G.

Corollary 5.4 Let the conditions of Corollary 4.1 hold. Then every solution of (E) and (B_{2}) is oscillatory in G.

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

Example 6.1 Consider the fractional partial differential equation

for

Here

where

and

It is easy to see that

we take _{1}) - (A_{5}) hold. We may observe that

Using the property,

Consider

and

Thus all conditions of Corollary 3.1 are satisfied. Hence every solution of (E_{1}), (33) oscillates in_{1}) and (33).

Example 6.2 Consider the fractional partial differential equation

for

Here

where

and

It is easy to see that _{1}) - (A_{4}) and (A_{6}) hold. We may observe that

Consider

and

Thus, all the conditions of Corollary 4.1 are satisfied. Therefore, every solution of

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.

VadivelSadhasivam,JayapalKavitha, (2016) Interval Oscillation Criteria for Fractional Partial Differential Equations with Damping Term. Applied Mathematics,07,272-291. doi: 10.4236/am.2016.73025