# Oscillatory Behaviour of the Nonlinear Damped Fractional Partial Dynamic Equation

Authors: R. Ramesh; S. Harikrishnan; P. Prakash; Yong-Ki Ma
Journal: Advances in Mathematical Physics
 (2022)

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## 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.


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## Body

## 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:(1)ℓηχΔαηξ,ηΔαη+ℏηχΔαηξ,η+υξ,ηfχξ,η=τη∇ξ2χξ,η,where ξ,η∈Ω×η0,∞Θ with the Neumann boundary condition(2)∂χξ,η∂ν=0,ξ,η∈∂Ω×η0,∞Θ,and the Dirichlet boundary condition(3)χξ,η=0,ξ,η∈∂Ω×η0,∞Θ,where Δαη is the conformable fractional partial dynamic operator with respect to the time variable η, Ω is a bounded domain in ℝn with piecewise smooth boundary ∂Ω, ν is the unit exterior normal vector to ∂Ω and α∈0,1, and ∇ξ2=∑i=1n∂ξiξi2 is the Laplacian operator. ℓη,ℏη, and τη are real-valued and rd-continuous functions on 0,∞Θ, ℓη>0 is conformable fractional differentiable with ℓηΔαη continuous, and υξ,η∈CΩ¯×η0,∞Θ,ℝ with υη=minξ∈Ωυξ,η. The continuous function f satisfies χfχ>0 for χ≠0.A time scaleΘ is a nonempty closed subset of the real numbers ℝ which is unbounded above. The time-scale interval of the form η0,∞Θ is denoted by η0,∞Θ=η0,∞∩Θ for η0∈Θ, η0>0.A nontrivial solutionχξ,η which satisfies (1) on Ω×η0,∞Θ along with boundary condition (2) (or (3)) is called oscillatory if it has neither eventually positive solution EPS nor eventually negative solution ENS. 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 Θ is a nonempty closed subset of the real numbers ℝ. We will use intervals of the form t0,∞Θ=t0,∞∩Θ for t0∈Θ. For a point t∈Θ, we have the following definitions: the forward jump operator is defined as σt=infs∈Θ,s>t. The backward jump operator is defined as ρt=sups∈Θ,s<t. The graininess is defined as μt=σt−t. A point t∈Θ is said to be right dense if σt=t. A function g:Θ⟶ℝ is said to be rd-continuous if it is continuous at each right-dense point and there exists a finite left limit of g at all left-dense points. The set of rd-continuous functions is denoted by CrdΘ,ℝ.To define derivatives, we introduce(4)Θk=Θ\ρsupΘ,supΘifsupΘ<∞,ΘifsupΘ<∞.Definition 1 (see [24]).
Letf:Θ⟶ℝ;η∈Θk, and α∈0,1. For η>0, we define fΔαη to be the number (provided it exists) with the property that, given any ϵ>0, there is a δ-neighborhood Nη∈Θ of η,δ>0 such that(5)fση−fκη1−α−fΔαηση−κ≤εση−κfor allκ∈Nη,fΔαη is called the conformable fractional derivative of f of order α at η, and the conformable fractional derivative at 0 is fΔα0=limη⟶0+fΔαη.Lemma 1.
Leta∈Θk and b∈Θ, and assume that f:Θ×Θk⟶ℝ is continuous at η,η where η∈Θk with a<η. Suppose that, for all ϵ>0, there is a δ-neighborhood U of ηi such that(6)fσiη−fiκηηi1−α−fΔαηiησiη−κ≤εσiη−κfor allκ∈U,where fΔαηi denotes the partial derivative of f with respect to ηi. Then,(7)gη=∫aηfη,κΔκ⇒gΔαηiη=∫aηfΔαηiη,κΔκ+fσiη,η,hη=∫ηbfη,κΔκ⇒hΔαηiη=∫ηbfΔαηiη,κΔκ−fσiη,η.Proof.
The proof is analogous to the proof of Theorem 1.117 in [25]. □Definition 2.
LetB=η,κ|η≥κ≥η0. Then, the class Λ is a collection of functions Λ∈CrdB,ℝ such that Λη,η=0 for η≥η0, Λη,κ>0 for η>κ≥η0, and Λ has a nonpositive continuous α-partial fractional derivative ΛΔακη,κ.A functiong:Θ⟶ℝ is said to be regressive provided 1+μtgt≠0 for each t∈Θk. Let ℛ be the set of functions that are rd-continuous and regressive. Also, we define ℛ+=g∈ℛ:1+μtgt>0,t∈Θ. For γ∈ℛ and s,t∈Θ, the generalized exponential function is defined by(8)eγt,s=exp∫stζμτγτΔτ,ζμz=1μlog1+μz,μ≠0,z,μ=0.For convenience, we use the following:(9)ϒη=∫Ωχξ,ηdξ,Rη=−ηα−1ℏηℓη.


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

Lemma 2.
If the fractional dynamic inequalities(10)ℓηϒΔαηΔα+ℏηϒΔαη+υηfϒη≤0,η≥η1,(11)ℓηϒΔαηΔα+ℏηϒΔαη+υηfϒη≥0,η≥η1,have no EPS and ENS, respectively, then every solution χξ,η of (1) and (2) is oscillatory.Proof.
Suppose thatχξ,η is an EPS of (1) and (2). Then, by the definition of EPS, there exists η1≥η0 such that χξ,η>0 for η≥η1, which gives ϒη>0 for η≥η1. Now, taking integration to (1) in connection with ξ over Ω, we get(12)∫ΩℓηχΔαηξ,ηΔαηdξ+∫ΩℏηχΔαηξ,ηdξ+∫Ωυξ,ηfχξ,ηdξ=∫Ωτη∇ξ2χξ,ηdξ.
Applying Green’s formula and (2), we get(13)∫Ω∇ξ2χξ,ηdξ=∫∂Ω∂χξ,η∂νdS=0.
By using the condition onυ and applying Jensen’s inequality, we obtain(14)∫Ωυξ,ηfχξ,ηdξ≥υη∫Ωfχξ,ηdξ≥υηfϒη.
From (12)–(14), we have(15)∫ΩℓηχΔαηξ,ηΔαηdξ+ℏη∫ΩχΔαηξ,ηdξ+υηfϒη≤0.
Using thatℓηχΔαηξ,ηΔαη is continuous in connection with ξ on the closed and bounded set G¯⊆ℛn, we have δ independent of η in Lemma 1. Therefore, the conditions of Lemma 1 are satisfied. Thus, ∫Gχξ,ηdξΔαη=∫GχΔαηξ,ηdξ. Hence, from (15), we obtain (10). A similar argument is used for the ENSχξ,η of (1)and (2) to obtain (11). □Lemma 3.
Assume that∫η0∞Δακ/ℓκ=∞ and Rη∈ℛ+. If (10) has an EPSϒη, then there exists sufficiently large η1∈η0,∞Θ such that ℓηϒΔαη/eRηη,η0Δα<0 and ΥΔαη>0 on η,∞Θ.Proof.
Givenϒη is an EPS of (10), there is sufficiently large η2 so that ϒη>0 on η,∞Θ. Now, we have(16)ℓηϒΔαηeRηη,η0Δα=eRηη,η0ℓηϒΔαηΔα−ℓηϒΔαηeRηη,η0ΔαeRηη,η0eRηση,η0=eRηη,η0ℓηϒΔαηΔα+ℏηϒΔαηeRηη,η0eRηη,η0eRηση,η0=−υηfϒηeRηση,η0<0.
SinceRη∈ℝ+, we get eRηση,η0>0, and ℓηϒΔαη/eRηη,η0 is decreasing on η2,∞Θ. Therefore, we get that ϒΔαη is the EPS or ENS. Suppose that ϒΔαη<0 on η2,∞Θ for sufficiently large η2>η1. Then,(17)ϒη−ϒη3=∫η2ηℓκϒΔακℓκΔακ≤ℓη2ϒΔαη2∫η2ηΔακℓκ.
Lettingη⟶∞ and using the condition that ∫η∞Δακ/ℓκ=∞, we get limη⟶∞ϒη=−∞ which is a contradiction to ϒη>0, and hence, ϒΔαη>0. □The following lemma holds if we proceed as in the above lemma with anENSϒη.Lemma 4.
Assume that∫η0∞Δακ/ℓκ=∞ and Rη∈ℛ+. If (11) has an ENSϒη, then there exists sufficiently large η1∈η0,∞Θ such that ℓηϒΔαη/eRηη,η0Δα>0 and ϒΔαη<0 on η,∞Θ.Theorem 1.
Assume that∫η0∞Δακ/ℓκ=∞, Rη∈ℛ+, and f′ exists with f′ξ≥N for some N>0, ∀ξ≠0. If there exists zη>0 on H,∞Θ such that(18)limsupη⟶∞∫HηυκzκeRκσκ,η0−ℓκzΔακ24NzκeRηκ,η0Δακ=∞,then (1) and (2) are oscillatory.Proof.
Suppose that the solutionχξ,η of (1) and (2) is the EPS. Define a generalized Riccati function as(19)wη=zηℓηϒΔαηfϒηeRηη,η0.
By Lemma3, it is lucid that ϒ>0 and ϒΔα≥0, respectively. Hence, wη≥0 for η≥η1. Then,(20)wΔαη=zηfϒηeRηη,η0ℓηϒΔαηΔα−eRηη,η0ΔαℓηϒΔαηeRηη,η0eRηση,η0+ℓσηϒΔασηeRηση,η0fϒηzΔαη−zηfϒηΔαfϒηfϒση≤−υηzηeRηση,η0+zΔαηwσηzση−NzηeRηη,η0ℓηwσηzση2≤−υηzηeRηση,η0+ℓηzΔαη24NzηeRηη,η0.
By takingα-integration from η1 to η, we obtain(21)limsupη⟶∞∫η1ηυκzκeRκσκ,η0−ℓκzΔακ24NzκeRηκ,η0Δακ≤wη1−wη≤wη1≤∞,which is a contradiction to (18). Similarly, we get a contradiction while assuming that χξ,η is the ENS of (1) and (2). □Theorem 2.
Assume that∫η0∞Δακ/ℓκ=∞, Rη∈ℛ+, and fξ≥Ex for some E>0, ∀ξ≠0. If there exists zη>0 on H,∞Θ such that(22)limsupη⟶∞∫HηEυκzκeRκσκ,η0−ℓκzΔακ24zκeRηκ,η0Δακ=∞,then (1) and (2) are oscillatory.Proof.
Suppose that the solutionχξ,η of (1) and (2) is the EPS. Define a generalized Riccati function as(23)wη=zηℓηϒΔαηϒηeRηη,η0.
By Lemma3, it is lucid that ϒ>0 and ϒΔα≥0, respectively. Hence, wη≥0 for η≥η1. Then,(24)wΔαη=zηϒηeRηη,η0ℓηϒΔαηΔα−eRηη,η0ΔαℓηϒΔαηeRηη,η0eRηση,η0+ℓσηϒΔασηeRηση,η0ϒηzΔαη−zηϒηΔαϒηϒση≤−EυηzηeRηση,η0+zΔαηwσηzση−zηeRηη,η0ℓηwσηzση2≤−EυηzηeRηση,η0+ℓηzΔαη24zηeRηη,η0.
By takingα-integration from η1 to η, we obtain(25)limsupη⟶∞∫η1ηEυκzκeRκσκ,η0−ℓκzΔακ24zκeRηκ,η0Δακ≤wη1−wη≤wη1≤∞,which is a contradiction to (22). Similarly, we get a contradiction while assuming that χξ,η is the ENS of (1) and (2). □Remark 1.
In Theorem1, (18) can be replaced by(26)limsupη⟶∞1Λη,H∫HηΛη,κυκzκeRκσκ,η0−ℓκzΔακ24NzκeRηκ,η0Δακ=∞,for Λ∈Λ.
By Theorem1, we have(27)wΔαη≤−υηzηeRηση,η0+ℓηzΔαη24NzηeRηη,η0.
Multiplying the above inequality byΛη,κ and taking α-integration from H to η, we get(28)∫HηΛη,κwΔακΔακ≤∫HηΛη,κ−υκzκeRκσκ,η0+ℓκzΔακ24NzκeRηκ,η0Δακ.
Using integration by parts, we obtain(29)Λη,κwηHη−∫HηΛΔαη,κwσκΔακ≤∫HηΛη,κ−υκzκeRκσκ,η0+ℓκzΔακ24NzκeRηκ,η0Δακ.
By using the definition of the functionΛ and rearranging the terms in the above inequality, we have(30)1Λη,H∫HηΛη,κυκzκeRκσκ,η0−ℓκzΔακ24NzκeRηκ,η0Δακ≤wH≤∞,which contradicts (26). Likewise, we obtain a contradiction when we take χξ,η is the ENS of (1) and (2).
Similarly, (22) can be replaced by the following in Theorem 2:(31)limsupη⟶∞1Λη,H∫HηΛη,κEυκzκeRκσκ,η0−ℓκzΔακ24zκeRηκ,η0Δακ=∞,for Λ∈Λ.


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

In this section, we use the eigenvalues of the Laplacian,−∇2ϕξ=λϕξinG,ϕξ=0on∂G. The principal eigenvalue λ1 is positive, and the corresponding eigenfunction ϕ1 is also positive in the interior of G; see [26], Theorem 2, page 356. Furthermore, we normalize this eigenvector so that ∫Gϕ1=1.Lemma 5.
Ifχξ,η is an EPS of (1) and (3), then there exists η1≥η0 such that ϒ1η=∫Gχξ,ηϕ1ξdξ>0 and(32)ℓηϒ1ΔαηΔα+ℏηϒ1Δαη+υηfϒ1η≤0,∀η≥η1.Also, ifχξ,η is an ENS, then there exists η1≥η0 such that ϒ1η<0 and(33)ℓηϒ1ΔαηΔα+ℏηϒ1Δαη+υηfϒ1η≥0,∀η≥η1.Proof.
Sinceχξ,η is an EPS of (1) satisfying (3), there exists η1≥η0 such that χξ,η≥0. ϕ1ξ>0 in the interior of Ω, which implies that ϒ1η=∫Ωχξ,ηϕ1ξdξ>0. Multiply (1) by ϕ1ξ, and integrate in connection with ξ over Ω; then,(34)∫ΩℓηχΔαηξ,ηΔαηϕ1ξdξ+∫ΩℏηχΔαηξ,ηϕ1ξdξ+∫Ωυξ,ηfχξ,ηϕ1ξdξ=∫Ωτη∇ξ2χξ,ηϕ1ξdξ.
On the right-hand side of (34), we have(35)∫Ωτη∇ξ2χξ,ηϕ1ξdξ=−λτη∫Ωχξ,ηϕ1ξdξ=−λτηϒ1η≤0,which implies that(36)∫ΩℓηχΔαηξ,ηΔαηϕ1ξdξ+∫ΩℏηχΔαηξ,ηϕ1ξdξ+∫Ωυξ,ηfχξ,ηϕ1ξdξ≤0.
Applying Jensen’s inequality and condition onυ to the third term of the above inequality and using Lemma 1 and continuity of ℓηχΔαηξ,ηΔαη in the first term of the above inequality, we obtain(37)ℓηϒ1ΔαηΔα+ℏηϒ1Δαη+υηfϒ1η≤0.
A similar proof will be used for theENS to obtain (33). □The following lemmas and theorems consider the solutionϒ1η of (1) and (3), and the proofs of the following are similar to those of lemmas and theorems proved for the solution ϒη of (1) and (2). Thus, we omit the proofs.Lemma 6.
Assume that∫η0∞Δακ/ℓκ=∞ and Rη∈ℛ+. If (32) has an EPSϒη, then there is sufficiently large η1∈η0,∞Θ such that ℓηϒ1Δαη/eRηη,η0Δα<0 and ϒ1Δαη>0 on η1,∞Θ.Lemma 7.
Assume that∫η0∞Δακ/ℓκ=∞ and Rη∈ℛ+. If (33) has an ENSϒ1η, then there is sufficiently large η1∈η0,∞Θ such that ℓηϒ1Δαη/eRηη,η0Δα>0 and ϒ1Δαη<0 on η1,∞Θ.Theorem 3.
Assume that∫η0∞Δακ/ℓκ=∞, Rη∈ℛ+, and f′ exists with f′η≥N for some N≥0, ∀ξ≠0. If there exists ψη>0 on H,∞Θ such that(38)limsupη⟶∞∫HηυκψκeRκσκ,η0−ℓκψΔακ24NψκeRηκ,η0Δακ=∞,then (1) and (3) are oscillatory.Theorem 4.
Assume that∫η0∞Δακ/ℓκ=∞, Rη∈ℛ+, and fξ≥Ex for some E>0, ∀ξ≠0. If there exists ψη>0 on H,∞Θ such that(39)limsupη⟶∞∫HηEυκψκeRκσκ,η0−ℓκψΔακ24ψκeRηκ,η0Δακ=∞,then (1) and (3) are oscillatory.Remark 2.
In Theorem3, (38) can be replaced by(40)limsupη⟶∞1Λη,H∫HηΛη,κυκψκeRκσκ,η0−ℓκψΔακ24NψκeRηκ,η0Δακ=∞,forΛ∈Λ.
Similarly, (39) can be replaced by the following in Theorem 4:(41)limsupη⟶∞1Λη,H∫HηΛη,κEυκψκeRκσκ,η0−ℓκψΔακ24NψκeRηκ,η0Δακ=∞,forΛ∈Λ.


## 5. Examples

The equations considered below are an illustration to validate our established results.Example 1.
Consider(42)η−1χΔ0.5ηξ,ηΔ0.5η+η−3χΔ0.5ηξ,η+η−1χξ,η1+χξ,η=η∇ξ2χξ,η,with χx0,η=χxπ,η=0. Here, Θ=ℝ; Ω=0,π; α=0.5; ℓη=η−1; ℏη=η−3; fχξ,η=χξ,η1+χξ,η; υη=η−1; and η0=1. It is lucid that fχ≥χ⇒E=1, ση=η, μη=0, Rη=−η−5/2,(43)∫η0∞Δακℓκ=∫η0∞κ−1/2dκκ−1=∫η0∞κ1/2dκ=∞,1≥eRηη,η0=exp∫η0ηRκdκ≥1+∫1ηRκdκ=1−∫1ηκ−5/2dκ=1+23η−3/2−1≥13.
Letzη=η1/2. Then, every condition of Theorem 2 is fulfilled. Hence, (42) with the given boundary condition is oscillatory.Example 2.
Consider(44)η−1/2χΔ0.5ηξ,ηΔ0.5η+η−5/2χΔ0.5ηξ,η+η−1χξ,ηE+eχξ,η=η∇ξ2χξ,η,with χ0,η=χπ,η=0. Here, η∈2,∞ℤ; Θ=ℤ; Ω=0,π; α=0.5; ℓη=η−1/2; ℏη=η−5/2; υη=η−1; η0=2; and fχξ,η=χξ,ηγE+eχξ,η.
Therefore, we getfχ≥Eχ, ση=η+1, μη=1, Rη=−η−5/2, and 1+μηRη=1−η−5/2≥1−1/2>0 which implies Rη∈ℛ+. Also,(45)∫η0∞Δακℓκ=∫2∞κα−1Δκℓκ=∑κ=2∞κα−1κ−1/2=∑κ=2∞1=∞,1>eRηη,η0=exp∫η0ηRκΔκ≥1+∫2ηRκΔκ=1−∑κ=2η−1κ−5/2≥1−∫1η−1κ−5/2dκ=1+23η−1−3/2−1>13.
Letψη=η. 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.



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