Oscillatory Behaviour of the Nonlinear Damped Fractional Partial Dynamic Equation

Abstract

In this paper, a partial dynamic equation of fractional order is considered with Neumann and Dirichlet boundary conditions, and we studied the oscillation properties of the fractional partial dynamic equation on time scales. Riccati transformation technique is used to establish oscillation criteria for the fractional partial dynamic equation. The obtained results are verified with examples.

1. Introduction

In recent years, the importance of fractional-order calculus has been much motivated by researchers rather than the integer order because of increasing applications in signal processing, neural networks, and electrical and mechanical engineering. Definitions for fractional derivatives in continuous and discrete cases using various operators and functions have been given by many such as Hadamard, Euler, Riemann, and Grunwald, but fundamental properties of derivatives were not satisfied by these derivatives, and with these derivatives, solving the differential equations of fractional order is not easy. In [1], the definition of the conformable fractional derivative has been given, and it has satisfied some fundamental properties of derivatives. The properties and application of the conformable derivative have been presented in [2, 3] and the references cited therein. Meanwhile, the oscillation properties of solutions are an important qualitative tool to study the solutions of dynamical systems. The oscillation of different types of both integer-order and fractional-order differential, difference, and dynamic equations has been investigated by many authors [4–20]. The time-scale calculus was introduced by Stefan Hilger to unify the theory of continuous and discrete cases. The time-scale analysis for multivariable cases is discussed in [21–23], and the fractional time-scale calculus is studied in [24]. To the best of our knowledge, we observe that the oscillation of solutions of the fractional partial dynamic equation on the time scale was not considered so far.

This motivates the authors to establish oscillation results of the following fractional partial dynamic equation with the damped term:where with the Neumann boundary conditionand the Dirichlet boundary conditionwhere is the conformable fractional partial dynamic operator with respect to the time variable , is a bounded domain in with piecewise smooth boundary , is the unit exterior normal vector to and , and is the Laplacian operator. , and are real-valued and rd-continuous functions on , is conformable fractional differentiable with continuous, and with . The continuous function satisfies for .

A time scale is a nonempty closed subset of the real numbers which is unbounded above. The time-scale interval of the form is denoted by for , .

A nontrivial solution which satisfies (1) on along with boundary condition (2) (or (3)) is called oscillatory if it has neither eventually positive solution nor eventually negative solution . Equation (1) is oscillatory if all the solutions of it are oscillatory.

2. Preliminaries

The following definitions can be found in [25], where there is a detailed introduction to time-scale calculus. A time scale is a nonempty closed subset of the real numbers . We will use intervals of the form for . For a point , we have the following definitions: the forward jump operator is defined as . The backward jump operator is defined as . The graininess is defined as . A point is said to be right dense if . A function is said to be rd-continuous if it is continuous at each right-dense point and there exists a finite left limit of at all left-dense points. The set of rd-continuous functions is denoted by .

To define derivatives, we introduce

Definition 1 (see [24]). Let , and . For , we define to be the number (provided it exists) with the property that, given any , there is a -neighborhood of such that is called the conformable fractional derivative of of order at , and the conformable fractional derivative at 0 is .

Lemma 1. Let and , and assume that is continuous at where with . Suppose that, for all , there is a -neighborhood of such thatwhere denotes the partial derivative of with respect to . Then,

Proof. The proof is analogous to the proof of Theorem 1.117 in [25]. □

Definition 2. Let . Then, the class is a collection of functions such that for , for , and has a nonpositive continuous -partial fractional derivative .

A function is said to be regressive provided for each . Let be the set of functions that are rd-continuous and regressive. Also, we define . For and , the generalized exponential function is defined by

For convenience, we use the following:

3. Oscillation of (1) with Boundary Condition (2)

Lemma 2. If the fractional dynamic inequalitieshave no and , respectively, then every solution of (1) and (2) is oscillatory.

Proof. Suppose that is an of (1) and (2). Then, by the definition of , there exists such that for , which gives for . Now, taking integration to (1) in connection with over , we getApplying Green’s formula and (2), we getBy using the condition on and applying Jensen’s inequality, we obtainFrom (12)–(14), we haveUsing that is continuous in connection with on the closed and bounded set , we have independent of in Lemma 1. Therefore, the conditions of Lemma 1 are satisfied. Thus, . Hence, from (15), we obtain (10). A similar argument is used for the of (1)and (2) to obtain (11). □

Lemma 3. Assume that and . If (10) has an , then there exists sufficiently large such that and on .

Proof. Given is an of (10), there is sufficiently large so that on . Now, we haveSince , we get , and is decreasing on . Therefore, we get that is the or . Suppose that on for sufficiently large . Then,Letting and using the condition that , we get which is a contradiction to , and hence, . □

The following lemma holds if we proceed as in the above lemma with an .

Lemma 4. Assume that and . If (11) has an , then there exists sufficiently large such that and on .

Theorem 1. Assume that , , and exists with for some , . If there exists on such thatthen (1) and (2) are oscillatory.

Proof. Suppose that the solution of (1) and (2) is the . Define a generalized Riccati function asBy Lemma 3, it is lucid that and , respectively. Hence, for . Then,By taking -integration from to , we obtainwhich is a contradiction to (18). Similarly, we get a contradiction while assuming that is the of (1) and (2). □

Theorem 2. Assume that , , and for some , . If there exists on such thatthen (1) and (2) are oscillatory.

Proof. Suppose that the solution of (1) and (2) is the . Define a generalized Riccati function asBy Lemma 3, it is lucid that and , respectively. Hence, for . Then,By taking -integration from to , we obtainwhich is a contradiction to (22). Similarly, we get a contradiction while assuming that is the of (1) and (2). □

Remark 1. In Theorem 1, (18) can be replaced byfor .
By Theorem 1, we haveMultiplying the above inequality by and taking -integration from to , we getUsing integration by parts, we obtainBy using the definition of the function and rearranging the terms in the above inequality, we havewhich contradicts (26). Likewise, we obtain a contradiction when we take is the of (1) and (2).
Similarly, (22) can be replaced by the following in Theorem 2:for .

4. Oscillation of (1) with Boundary Condition (3)

In this section, we use the eigenvalues of the Laplacian, . The principal eigenvalue is positive, and the corresponding eigenfunction is also positive in the interior of ; see [26], Theorem 2, page 356. Furthermore, we normalize this eigenvector so that .

Lemma 5. If is an of (1) and (3), then there exists such that and

Also, if is an , then there exists such that and

Proof. Since is an of (1) satisfying (3), there exists such that . in the interior of , which implies that . Multiply (1) by , and integrate in connection with over ; then,On the right-hand side of (34), we havewhich implies thatApplying Jensen’s inequality and condition on to the third term of the above inequality and using Lemma 1 and continuity of in the first term of the above inequality, we obtainA similar proof will be used for the to obtain (33). □

The following lemmas and theorems consider the solution of (1) and (3), and the proofs of the following are similar to those of lemmas and theorems proved for the solution of (1) and (2). Thus, we omit the proofs.

Lemma 6. Assume that and . If (32) has an , then there is sufficiently large such that and on .

Lemma 7. Assume that and . If (33) has an , then there is sufficiently large such that and on .

Theorem 3. Assume that , , and exists with for some , . If there exists on such thatthen (1) and (3) are oscillatory.

Theorem 4. Assume that , , and for some , . If there exists on such thatthen (1) and (3) are oscillatory.

Remark 2. In Theorem 3, (38) can be replaced bySimilarly, (39) can be replaced by the following in Theorem 4:

5. Examples

The equations considered below are an illustration to validate our established results.

Example 1. Considerwith . Here, ; ; ; ; ; ; ; and . It is lucid that , , , ,Let . Then, every condition of Theorem 2 is fulfilled. Hence, (42) with the given boundary condition is oscillatory.

Example 2. Considerwith . Here, ; ; ; ; ; ; ; ; and .
Therefore, we get , , , , and which implies . Also,Let . Then, all the conditions of Theorem 4 are fulfilled. Hence, (44) with the given boundary condition is oscillatory.

6. Conclusion

In this article, we established the oscillation criteria for the fractional partial dynamic equation on time scales. The obtained results are improved in the sense that which provide sufficient criteria for the oscillation of the considered equation with two boundary conditions. Finally, numerical examples are also presented to validate the theoretical results of this study. Additionally, more complex systems, including the fractional dynamic equation with time delay, are interesting research topics and will be examined in our future works.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The third author was supported by the Department of Science and Technology, New Delhi, India, under the FIST Programme (SR/FST/MS1-115/2016). This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (no. 2021R1F1A1048937) to the last author.