Forced Oscillation of Solutions of a Fractional Neutral Partial Functional Differential Equation

V. Sadhasivam, J. Kavitha^*

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal Dt. Tamil Nadu, India.

DOI: 10.4236/am.2015.68124                

Abstract

In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form

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^N subject to the condition

Keywords

Fractional, Neutral, Oscillation, Partial, Functional

for where is a bounded domain in with a piecewise smooth boundary is a constant, is the Riemann-Liouville fractional derivative of order of u with respect to t and is the Laplacian operator in the Euclidean N-space subject to the condition

Keywords:

1. Introduction

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 [1] -[5] . Neutral delay differential equations have applications in electric networks containing Lossless transmission lines and population dynamics [6] . Several papers concerning neutral parabolic differential equations have appeared recently (for example see [7] [8] ). The oscillatory theory of solutions of fractional differential equations has received a great deal of attention [9] - [15] . In the last few years, many authors studied the oscillation of a time-fractional partial differential equations [16] [17] . There are only few works has been done on oscillation of forced neutral fractional partial differential equations.

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

where is a bounded domain in with a piecewise smooth boundary is a constant, is the Riemann-Liouville fractional derivative of order of with respect to and is the Laplacian operator in the Euclidean N-space (ie). Equation (E) is supplemented with the boundary condition

(B₁)

where is the unit exterior normal vector to and is non negative continuous function on and

(B₂)

In what follows, we always assume without mentioning that

(A₁) such that

(A₂), and, are non negative constants,

(A₃) and

(A₄), and are nonnegative constants,;;

(A₅) are convex in, and for

(A₆) such that

A function is called a solution of (E), (B₁) ((E), (B₂)) if it satisfies in the domain G and the boundary condition (B₁), (B₂). The solution of of equations (E), (B₁) or (E), (B₂) is said to be oscillatory in the domain if for any positive number there exists a point such that holds. Particularly no work has been known with (E) and (B₁) 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 (B₁). 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.

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 several kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann-Liouville left sided definition on the half-axis. The following notations will be used for the convenience.

(1)

Definition 2.1. The Riemann-Liouville fractional partial derivative of order with respect to t of a function is given by

(2)

provided the right hand side is point wise defined on where is the gamma function.

Definition 2.2. The Riemann-Liouville fractional integral of order of a function on the half-axis is given by

(3)

provided the right hand side is pointwise defined on.

Definition 2.3. The Riemann-Liouville fractional derivative of order of a function on the half-axis is given by

(4)

provided the right hand side is pointwise defined on where is the ceiling function of.

Lemma 2.1. Let be the solution of (E) and

Then.

3. Oscillation of (E), (B₁)

We introduce a class of function P. Let

The function is said to belong to the class, if

C₁) for, for

C₂) has a continuous and non-positive partial derivative on with respect to s.

Lemma 3.1. If is a solution of (E), (B₁) for which in then the function is defined by (1) satisfy the fractional differential inequality

(5)

with and for

Proof. Let Integrating (E) with respect to over we have

(6)

Using Green’s formula and boundary condition (B₁) it follows that

(7)

and

(8)

Also from (A₃), (A₅), we obtain

(9)

and using and Jensen’s inequality we get

(10)

In view of (1), (7)-(10) and A₆, (6) yield

This completes the proof.

Lemma 3.2. Let be a positive solution of the (E), (B₁) defined on then the function where is defined by (1) satisfies one of the following con- ditions:

1)

2) for all

Proof. From Lemma 3.1, the function satisfies the inequality (5) and and

for From (5) and the hypothesis we have and

for Hence is monotonic and eventually of one sign. This completes the proof.

Lemma 3.3. Let be a positive solution of (E), (B₁) defined on and suppose Case (1) of Lemma 3.2 holds, then

(11)

Proof. From Case (I), is positive and increasing for, and by the definition of, we obtain and

for

This completes the proof.

Lemma 3.4. Let be a positive solution of (E), (B₁) defined on and suppose Case (2) of Lemma 3.2 holds, then

(12)

Proof. In this case the function is positive and nonincreasing for and therefore without loss of generality we may assume from the definition of and is also nonincreasing for. Hence which implies (12).

This completes the proof.

Theorem 3.1. Assume that for, where are positive constants. Let

be continuous functions such that and

(13)

Assume also that there exists a positive nondecreasing function such that

(14)

where

and

(15)

where and.

Then every solution of (E), (B₁) is oscillatory in.

Proof. Suppose that is a non oscillatory solution of (E), (B₁), which has no zero in for some. Without loss of generality we may assume that and in where is chosen so large that Lemmas 3.1 to 3.4 hold for From Lemma 3.1 the function defined by (1) satisfy the inequality

(16)

Let Then satisfies either Case (1) or Case (2) of Lemma 3.2.

Case (I): For this case and Using Lemma 3.3 and (A₅), (16) yields

(17)

Define the function by the generalized Riccati substititution

(18)

then

(19)

From for we have and consequently by (19) for, we obtain that

(20)

Let Then and so the last inequality becomes

(21)

substituting with multiplying both sides of (21) by and integrating from to for we have

Thus for all, we conclude that

(22)

Then, by (22) and (C₂), for we obtain

(23)

Then, by (14) and (C₂), we have

(24)

which contradicts (14).

Case (II): Assume that satisfies (11). Using hypothesis and Lemma 3.3, we have from (16)

(25)

Let Then and so the last inequality becomes

(26)

Integrating (26) from to we have

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 of (E), (B₁) is oscillatory in.

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

and

Then every solution of (E), (B₁) is oscillatory in.

Let for some integer. Then Theorem 3.1, implies the following the result.

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

for some integer. Then every solution of (E), (B₁) is oscillatory in.

Next we establish conditions for the oscillation of all solutions of (E), (B₁) subject to the following con- ditions:

C₃)

C₄) for and is a ratio of odd integers.

Theorem 3.2. In addition to conditions (C₃) and (C₄) assume for all. Then all the solutions of (E), (B₁) are oscillatory if

(27)

and

(28)

where

Proof. Suppose that is a non oscillatory solution of (E), (B₁), which has no zero in for some Without loss of generality we may assume that and in Then the function defined by (1) satisfies the inequality (16).

Let Then for From (16), we have

(29)

and for,

Let Then therefore the above inequality becomes

Integrating the last inequality from to, we have

(30)

since is bounded above. From (30) we obtain

Letting we obtain

(31)

where is defined by (28) and is an arbitrary large number.

From Lemma 3.2 there are two possible cases for. First we consider that for Let Then using this in (16) we have

Integrating the last inequality from to, we have

(32)

By (C₄) and Lemma 3.3, we have from (32)

(33)

Letting we have

(34)

For this case is increasing, so there exists a number such that for Thus there exists a such that

(35)

and since as

From (34) and (35) we have

(36)

which contradicts (27).

Next we consider the case that and for From (31), we have

(37)

Consider since is an odd ratio integer.

Let Then

here we have used (C₄), (37) and Lemma 3.4. Integrating the last inequality from to, we obtain

and so letting, we obtain

which contradicts (28). This completes the proof.

Next we consider (E), (B₁) subject to the following conditions:

C₅) for and is a ratio of odd positive integers.

Theorem 3.3. In addition to conditions (C₃) and (C₅) assume that

(38)

and

(39)

Then every solution of (E), (B₁) is oscillatory in.

Proof. Without loss of generality we may assume that and in is a solution of (E), (B₁). Therefore

If for we have from (34) and (36). For large we have and Therefore from (36), we obtain

which contradicts (38). For this case for from (33)

We consider the fractional differential where such that

Let Then

according as or and is decreasing. Since for where is a constant, there exist positive number such that

Integrating and rearranging we obtain

and so letting we have

which contradicts (39). This completes the proof.

4. Oscillation of (E), (B₂)

In this section we establish sufficient conditions for the oscillation of all solutions of (E), (B₂). For this we need the following:

The smallest eigen value of the Dirichlet problem

is positive and the corresponding eigen function is positive in.

Theorem 4.1. Let all the conditions of Theorem 3.1 be hold. Then every solution of (E), (B₂) oscillates in.

Proof. Suppose that is a non oscillatory solution of (E), (B₂), which has no zero in for some Without loss of generality, we may assume that and in Multiplying both sides of the Equation (E) by and integrating with respect to over.

We obtain for,

(40)

Using Green’s formula and boundary condition (B₂) it follows that

(41)

and for

(42)

Also from (A₃), (A₅), we obtain

(43)

and using and Jensen’s inequality we get

(44)

Set

(45)

In view of (41)-(45) and (A₆), (40) yield

(46)

for Rest of the proof is similar to that of Theorems 3.1 and hence the details are omitted.

Using the above theorem, we derive the following Corollaries.

Corollary 4.1. If the inequality (46) has no eventually positive solutions, then every solution of (E), (B₂) is oscillatory in G.

Corollary 4.2. Let the conditions of Corollary 3.2 hold; then every solution of (E), (B₂) is oscillatory in G.

Corollary 4.3. Let the conditions of Corollary 3.3 hold; then every solution of (E), (B₂) is oscillatory in G.

Theorem 4.2. Let the conditions of Theorem 3.2 hold; then every solution of (E), (B₂) is oscillatory in G.

Theorem 4.3. Let the conditions of Theorem 3.3 hold; then every solution of (E), (B₂) 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.

5. Examples

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

(E1)

for with the boundary condition

(47)

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 and we get

Thus all the conditions of Corollary 3.3 are satisfied. Hence every solution of (E₁), (47) oscillates in In fact is such a solution.

Example 2. Consider the fractional neutral partial differential equation

(E2)

for with the boundary condition

(48)

Here

and

It is easy to see that

Take

Consider

Choose and we get

Thus all the conditions of Corollary 3.3 are satisfied. Therefore every solution of (E₂), (48) oscillates in In fact is such a solution.

Acknowledgements

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.

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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