Research article

Oscillation results for a fractional partial differential system with damping and forcing terms

  • Received: 11 September 2022 Revised: 11 November 2022 Accepted: 24 November 2022 Published: 02 December 2022
  • MSC : 34C10, 34K11, 35B05, 35R11

  • In this paper, we study the forced oscillation of solutions of a fractional partial differential system with damping terms by using the Riemann-Liouville derivative and integral. We obtained some new oscillation results by using the integral averaging technique. The obtained results are illustrated by using some examples.

    Citation: A. Palanisamy, J. Alzabut, V. Muthulakshmi, S. S. Santra, K. Nonlaopon. Oscillation results for a fractional partial differential system with damping and forcing terms[J]. AIMS Mathematics, 2023, 8(2): 4261-4279. doi: 10.3934/math.2023212

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  • In this paper, we study the forced oscillation of solutions of a fractional partial differential system with damping terms by using the Riemann-Liouville derivative and integral. We obtained some new oscillation results by using the integral averaging technique. The obtained results are illustrated by using some examples.



    The theory of fractional calculus has played an important role in engineering and natural sciences. Currently, the concept of fractional calculus has been effectively used in many social, physical, signal, image processing, biological and engineering problems. Further, it has been realized that a fractional system provides a more accurate interpretation than the integer-order system in many real modeling problems. For more details, one can refer to [1,2,3,4,5,6,7,8,9,10].

    Oscillation phenomena take part in different models of real world applications; see for instance the papers [11,12,13,14,15,16,17] and the papers cited therein. More precisely, we refer the reader to the papers [18,19] on bio-mathematical models where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. Recently and although it is rare, the study on the oscillation of fractional partial differential equations has attracted many researchers. In [20,21,22,23], the researchers have established the requirements of the oscillation for certain kinds of fractional partial differential equations.

    In [24], Luo et al. studied the oscillatory behavior of the fractional partial differential equation of the form

    D1+α+,tu(y,t)+p(t)Dα+,tu(y,t)+q(y,t)t0(t)αu(y,)d=a(t)Δu(y,t)+mi=1ai(t)Δu(y,tτi),(y,t)Q×R+=H

    subject to either of the following boundary conditions

    u(y,t)ν+β(y,t)u(y,t)=0,(y,t)Q×R+,u(y,t)=0,(y,t)Q×R+.

    They have obtained some sufficient conditions for the oscillation of all solutions of this kind of fractional partial differential equations by using the integral averaging technique and Riccati transformations.

    On other hand in [25], Xu and Meng considered a fractional partial differential equation of the form

    Dα+,t(r(t)Dα+,tu(y,t))+p(t)Dα+,tu(y,t)+q(y,t)f(u(y,t))=a(t)Δu(y,t)+mi=1bi(t)Δu(y,tτi),(y,t)Q×R+=H

    with the Robin boundary condition

    u(y,t)N+g(y,t)u(y,t)=0,(y,t)Q×R+,

    they obtained some oscillation criteria using the integral averaging technique and Riccati transformations.

    Prakash et al. [26] considered the oscillation of the fractional differential equation

    t(r(t)Dα+,tu(y,t))+q(y,t)f(t0(tv)αu(y,v)dv)=a(t)Δu(y,t),(y,t)Q×R+

    with the Neumann boundary condition

    u(y,t)N=0,(y,t)Q×R+,

    they obtained some oscillation criteria by using the integral averaging technique and Riccati transformations.

    Furthermore in [27], Ma et al. considered the forced oscillation of the fractional partial differential equation with damping term of the form

    t(r(t)Dα+,tu(y,t))+p(t)Dα+,tu(y,t)+q(y,t)f(u(y,t))=a(t)Δu(y,t)+˜g(y,t),(y,t)Q×R+

    with the boundary condition

    u(y,t)N+β(y,t)u(y,t)=0,(y,t)Q×R+,

    they obtained some oscillation criteria by using the integral averaging technique.

    From the above mentioned literature, one can notice that the Riccati transformation method has been incorporated into the proof of the oscillation results. Unlike previous results, however, we study in this paper the forced oscillation of the fractional partial differential equation with the damping term of the form

    t(a(t)t(r(t)g(Dα+,tu(y,t))))+p(t)t(r(t)g(Dα+,tu(y,t)))=b(t)Δu(y,t)+mi=1ai(t)Δu(y,tτi)q(y,t)t0(t)αu(y,)d+f(y,t),(y,t)Q×R+=H (1.1)

    via the application of the integral averaging technique only. Equation (1.1) is presented under a high degree of generality providing a general platform for many particular cases. Here, Dα+,tu(y,t) is the Riemann-Liouville fractional partial derivative of order α of u,α(0,1), Δ is the Laplacian in Rn, i.e.,

    Δu(y,t)=nr=12u(y,t)y2r,

    Q is a bounded domain of Rn with the piecewise smooth boundary Q and R+:=(0,).

    Further, we assume the Robin and Dirichlet boundary conditions

    u(y,t)N+γ(y,t)u(y,t)=0,(y,t)Q×R+ (1.2)

    and

    u(y,t)=0,(y,t)Q×R+, (1.3)

    where N is the unit outward normal to Q and γ(y,t)>0 is a continuous function on Q×R+. The following conditions are assumed throughout:

    (H1) a(t)C1([t0,);R+) and r(t)C2([t0,);R+);

    (H2) g(t)C2(R;R) is an increasing function and there exists a positive constant k such that yg(y)=k>0, yg(y)0 for y0;

    (H3) p(t)C([t0,);R) and A(t)=tt0p(ζ)a(ζ)dζ;

    (H4) b(t),ai(t)C(R+;R+) and τi are non-negative constants, iIm={1,2,,m};

    (H5) q(y,t)C(H;R+) and q(t)=minyQq(y,t);

    (H6) f(y,t)C(ˉH;R).

    By a solution of the problems (1.1) and (1.2) (or (1.1)–(1.3)), we mean a function u(y,t)C2+α(ˉQ×[0,)), which satisfies (1.1) on H and the boundary condition (1.2) (or (1.3)).

    A solution u(y,t) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is non-oscillatory.

    The rest of the paper is organized as follows. Some basic definitions and known lemmas are included in Section 2. In Sections 3 and 4, we study the oscillations of (1.1) and (1.2), and (1.1) and (1.3), respectively. Section 5 deals with some applications for the sake of showing the feasibility and effectiveness of our results. Lastly, we add a conclusion in Section 6.

    Before we start the main work, we present some basic lemmas and definitions which are applied in what follows.

    Definition 1. [4] The Riemann-Liouville fractional integral of order α>0 of a function y:R+R on the half-axis R+ is defined by

    (Iα+y)(t):=1Γ(α)t0(tϑ)α1y(ϑ)dϑ,t>0

    provided the right-hand side is pointwise defined on R+, where Γ is the gamma function.

    Definition 2. [4] The Riemann-Liouville fractional derivative of order α>0 of a function y:R+R on the half-axis R+ is defined by

    (Dα+y)(t):=dαdtα(Iαα+y)(t),t>0

    provided the right-hand side is pointwise defined on R+, where α is the ceiling function of α.

    Definition 3. [4] The Riemann-Liouville fractional partial derivative of order 0<α<1 with respect to t of a function u(y,t) is defined by

    (Dα+,tu)(y,t):=1Γ(1α)tt0(tϑ)αu(y,ϑ)dϑ

    provided the right-hand side is pointwise defined on R+.

    Lemma 1. [4] Let y be a solution of (1.1) and

    L(t):=t0(tϑ)αy(ϑ)dϑ

    for α(0,1) and t>0. Then

    L(t)=Γ(1α)(Dα+y)(t).

    Lemma 2. [4] Let α0,mN and D=ddt. If the fractional derivatives (Dαa+y)(t) and (Dα+ma+y)(t) exist, then

    (DmDαa+y)(t)=(Dα+ma+y)(t).

    Lemma 3. [4] If α(0,1), then

    (Iαa+Dαa+y)(t)=y(t)y1α(a)Γ(α)(ta)α1,

    where y1α(t)=(I1αa+y)(t).

    Lemma 4. [5] The smallest eigenvalue β0 of the Dirichlet problem

    Δω(y)+βω(y)=0 in Qω(y)=0 on Q

    is positive and the corresponding eigenfunction ϕ(y) is positive in Q.

    In this section, we establish the oscillation criteria for (1.1) and (1.2).

    Theorem 1. If (H1)(H6) are valid, limtI1α+U(0)=C0 and if

    lim inftt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d<0 (3.1)

    and

    lim suptt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d>0 (3.2)

    for some constants C0,C1 and C2 with F(t)=Qf(y,t)dy, then all solutions of (1.1) and (1.2) are oscillatory.

    Proof. If u(y,t) is a non-oscillatory solution of (1.1) and (1.2) then there exists a t00 such that u(y,t)>0 (or u(y,t)<0),tt0.

    Case 1. Let u(y,t)>0 for tt0. Integrating (1.1) over Q, we get

    ddt(a(t)ddt(r(t)g(Dα+U(t))))+p(t)ddt(r(t)g(Dα+U(t)))=b(t)QΔu(y,t)dy+mi=1ai(t)QΔu(y,tτi)dyQ(q(y,t)t0(t)αu(y,)d)dy+Qf(y,t)dy, (3.3)

    where U(t)=Qu(y,t)dy with U(t)>0. By (1.2) and Green's formula, we have

    QΔu(y,t)dy=QuNdζ=Qγ(y,t)u(y,t)dζ<0 (3.4)

    and

    QΔu(y,tτi)dy<0. (3.5)

    Also, by (H5), one can get

    Q(q(y,t)t0(t)αu(y,)d)dyq(t)t0(t)α(Qu(y,)dy)d=q(t)L(t), (3.6)

    where L(t)=t0(t)αU()d. Because of the inequalities (3.4)–(3.6), (3.3) becomes

    ddt(a(t)ddt(r(t)g(Dα+U(t))))+p(t)ddt(r(t)g(Dα+U(t)))q(t)L(t)+F(t)F(t).

    Thus, we get

    (eA(t)a(t)(r(t)g(Dα+U(t))))=eA(t)((a(t)(r(t)g(Dα+U(t))))+p(t)(r(t)g(Dα+U(t))))eA(t)F(t).

    Integrating the above inequality over [t0,t], one can get

    eA(t)a(t)(r(t)g(Dα+U(t)))tt0eA(ζ)F(ζ)dζ+C1,

    where

    C1=eA(t0)a(t0)(r(t0)g(Dα+U(t0))).

    Again integrating the above inequality over [t0,t], we get

    r(t)g(Dα+U(t))C2+tt0C1eA(τ)a(τ)dτ+tt01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ,

    where C2=r(t0)g(Dα+U(t0)). Then using (H5), we obtain

    Dα+U(t)kC2r(t)+1r(t)tt0C1eA(τ)a(τ)dτ+1r(t)tt01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ.

    Applying the Riemann-Liouville fractional integral operator of order α to the above inequality and using Lemma 3, we obtain

    U(t)I1α0U(0)Γ(α)tα1kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d.

    Then

    lim inftU(t)lim inftC0Γ(α)tα1+lim inft{kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d}.

    Therefore, by our hypothesis, as given by (3.1), we get lim inftU(t)0. This leads to a contradiction to U(t)>0.

    Case 2. Let u(y,t)<0 for tt0. Just as in Case 1, we can obtain that (3.3) holds and U(t)<0. By (1.2) and Green's formula, we get

    QΔu(y,t)dy=QuNdζ=Qγ(y,t)u(y,t)dζ>0 (3.7)

    and

    QΔu(y,tτi)dy>0. (3.8)

    Also, by (H5), we have

    Q(q(y,t)t0(t)αu(y,)d)dyq(t)t0(t)α(Qu(y,)dy)d=q(t)L(t). (3.9)

    Because of the inequalities (3.7)–(3.9), (3.3) becomes

    ddt(a(t)ddt(r(t)g(Dα+U(t))))+p(t)ddt(r(t)g(Dα+U(t)))q(t)L(t)+F(t)F(t), (3.10)

    that is,

    (eA(t)a(t)(r(t)g(Dα+U(t))))=eA(t)((a(t)(r(t)g(Dα+U(t))))+p(t)(r(t)g(Dα+U(t))))eA(t)F(t).

    Integrating the above inequality over [t0,t], we have

    eA(t)a(t)(r(t)g(Dα+U(t)))tt0eA(ζ)F(ζ)dζ+C1,

    where

    C1=eA(t0)a(t0)(r(t0)g(Dα+U(t0))).

    Again integrating the above inequality over [t0,t], we obtain

    r(t)g(Dα+U(t))C2+tt0C1eA(τ)a(τ)dτ+tt01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ,

    where C2=r(t0)g(Dα+U(t0)). Then using (H5), we obtain

    Dα+U(t)kC2r(t)+1r(t)tt0C1eA(τ)a(τ)dτ+1r(t)tt01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ.

    Applying the Riemann-Liouville fractional integral operator of order α to the above inequality and using Lemma 3, we obtain

    U(t)I1α0U(0)Γ(α)tα1kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d.

    Then

    lim suptU(t)lim suptC0Γ(α)tα1+lim supt{kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d}.

    Therefore, by our hypothesis given by (3.2), we get lim suptU(t)0. This leads to a contradiction to U(t)<0.

    In this section, we establish the oscillation criteria for (1.1) and (1.3).

    Theorem 2. If (H1)(H5) are valid, limtI1α+U1(0)=A1 and if

    lim inftt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d<0 (4.1)

    and

    lim suptt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d>0 (4.2)

    for some constants A1,C1 and C2 with

    F1(t)=Qf(y,t)ϕ(y)dyandU1(t)=Qu(y,t)ϕ(y)dy,

    then all solutions of (1.1) and (1.3) are oscillatory.

    Proof. If u(y,t) is a non-oscillatory solution of (1.1) and (1.3) then there exists a t00 such that u(y,t)>0 (or u(y,t)<0) for tt0.

    Case 1. Let u(y,t)>0 for tt0. Multiplying (1.1) by ϕ(y) and then integrating over Q, we get

    Qt(a(t)t(r(t)g(Dα+,tu(y,t))))ϕ(y)dy+Qp(t)t(r(t)g(Dα+,tu(y,t)))ϕ(y)dy=Qb(t)Δu(y,t)ϕ(y)dy+Qmi=1ai(t)Δu(y,tτi)ϕ(y)dyQ(q(y,t)t0(t)αu(y,)d)ϕ(y)dy+Qf(y,t)ϕ(y)dy. (4.3)

    By Lemma 4 and Green's formula, we have

    QΔu(y,t)ϕ(y)dy=Qu(y,t)Δϕ(y)dy=β0Qu(y,t)ϕ(y)dy<0 (4.4)

    and

    QΔu(y,tτi)ϕ(y)dy<0. (4.5)

    Also, by (H5), we get

    Q(q(y,t)t0(t)αu(y,)ϕ(y)d)dyq(t)t0(t)α(Qu(y,)ϕ(y)dy)d=q(t)L1(t), (4.6)

    where

    L1(t)=t0(t)αU1()d>0.

    Because of the inequalities (4.4)–(4.6), (4.3) becomes

    ddt(a(t)ddt(r(t)g(Dα+U1(t))))+p(t)ddt(r(t)g(Dα+U1(t)))q(t)L1(t)+F1(t)F1(t),

    that is,

    (eA(t)a(t)(r(t)g(Dα+U1(t))))=eA(t)[(a(t)(r(t)g(Dα+U1(t))))+p(t)(r(t)g(Dα+U1(t)))]eA(t)F1(t).

    Integrating the above inequality over [t0,t], we have

    eA(t)a(t)(r(t)g(Dα+U1(t)))tt0eA(ζ)F1(ζ)dζ+C1,

    where

    C1=eA(t0)a(t0)(r(t0)g(Dα+U1(t0))).

    Again integrating the above inequality over [t0,t], we have

    r(t)g(Dα+U1(t))C2+tt0C1eA(τ)a(τ)dτ+tt01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ,

    where C2=r(t0)g(Dα+U1(t0)). Then using (H5), we obtain

    Dα+U1(t)kC2r(t)+1r(t)tt0C1eA(τ)a(τ)dτ+1r(t)tt01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ.

    Applying the Riemann-Liouville fractional integral operator of order α to the above inequality and using Lemma 3, we obtain

    U1(t)I1α0U1(0)Γ(α)tα1kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d.

    Then

    lim inftU1(t)lim inftA1Γ(α)tα1+lim inft{kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d}.

    Therefore, by our hypothesis given by (4.1), we get lim inftU1(t)0. This leads to a contradiction to U1(t)>0.

    Case 2. Let u(y,t)<0 for tt0. Multiplying (1.1) by ϕ(y) and then integrating over Q, one can get (4.3). Using Green's formula, we have

    QΔu(y,t)ϕ(y)dy=Qu(y,t)Δϕ(y)dy=β0Qu(y,t)ϕ(y)dy>0 (4.7)

    and

    QΔu(y,tτi)ϕ(y)dy>0. (4.8)

    Also, by (H5), we have

    Q(q(y,t)t0(t)αu(y,)d)ϕ(y)dyq(t)t0(t)α(Qu(y,)ϕ(y)dy)d=q(t)L1(t), (4.9)

    where

    L1(t)=t0(t)αU1()d<0.

    Because of the inequalities (4.7)–(4.9), (4.3) becomes

    ddt(a(t)ddt(r(t)g(Dα+U1(t))))+p(t)ddt(r(t)g(Dα+U1(t)))q(t)L1(t)+F1(t)F1(t), (4.10)

    that is,

    (eA(t)a(t)(r(t)g(Dα+U1(t))))=eA(t)((a(t)(r(t)g(Dα+U1(t))))+p(t)(r(t)g(Dα+U1(t))))eA(t)F1(t).

    Integrating the above inequality over [t0,t], we get

    eA(t)a(t)(r(t)g(Dα+U1(t)))tt0eA(ζ)F1(ζ)dζ+C1,

    where

    C1=eA(t0)a(t0)(r(t0)g(Dα+U1(t0))).

    Again integrating the above inequality over [t0,t], we get

    r(t)g(Dα+U1(t))C2+tt0C1eA(τ)a(τ)dτ+tt01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ,

    where C2=r(t0)g(Dα+U1(t0)). Then using (H5), we obtain

    Dα+U1(t)kC2r(t)+1r(t)tt0C1eA(τ)a(τ)dτ+1r(t)tt01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ.

    Applying the Riemann-Liouville fractional integral operator of order α to the above inequality and using Lemma 3, we obtain

    U1(t)I1α0U1(0)Γ(α)tα1kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d.

    Then

    lim suptU1(t)lim suptA1Γ(α)tα1+lim supt{kΓ(α)t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F1(ζ)dζ)dτ]d}.

    Therefore, by our hypothesis given by (4.2), we get lim suptU1(t)0. This leads to a contradiction to U1(t)<0.

    In this section, we give two examples to illustrate our main results.

    Example 1. Let us consider the fractional partial differential system

    D52+,tu(y,t)=1πΔu(y,t)+2tΔu(y,t1)(y2+1t2)t0(t)12u(y,)d+e2tcos(t)sin(y),(y,t)(0,π)×R+ (5.1)

    with the condition

    uy(0,t)=uy(π,t)=0. (5.2)

    In the above, a(t)=1,r(t)=1,g(t)=t,α=1/2,p(t)=0,b(t)=1/π,m=1,a1(t)=2t,τ1=1, q(y,t)=(y2+1t2),f(y,t)=e2tcos(t)sin(y),Q=(0,π),q(t)=miny(0,π)q(y,t)=1/t2 and t0=0.

    Since F(t)=π0e2tcos(t)sin(y)dy=2e2tcos(t) and A(t)=0, we have

    t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=C2t0(t)1/2d+C1t0(t)1/2(0dτ)d+25t0(t)1/2(20e2τcos(τ)dτ+0e2τsin(τ)dτ20dτ)d=(C26/25)t0(t)1/2d+(C14/5)t0(t)1/2d+6/25t0e2(t)1/2cos()d+8/25t0e2(t)1/2sin()d.

    Fixing y2=t, then

    t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=2(C26/25)t+4/3(C14/5)t3/2+4/25e2t(3cos(t)t0e2y2cos(y2)dy+3sin(t)t0e2y2sin(y2)dy+4sin(t)t0e2y2cos(y2)dy4cos(t)t0e2y2sin(y2)dy). (5.3)

    Pointing out that

    |e2y2cos(y2)|e2y2,|e2y2sin(y2)|e2y2 and limtt0e2y2dy=2π4,

    we can conclude that

    limtt0e2y2cos(y2)dy and limtt0e2y2sin(y2)dy

    are convergent. Thus, we have that

    limt[cos(t)(3t0e2y2cos(y2)dy4t0e2y2sin(y2)dy)+sint(3t0e2y2sin(y2)dy+4t0e2y2cos(y2)dy)]

    is convergent. Fixing

    limtt0e2y2cos(y2)dy=P,limtt0e2y2sin(y2)dy=Q

    and considering the sequence

    tn=3π2+2nπarctan(3P4Q3Q+4P),

    we get

    limn{cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)}=(3P4Q)2+(3Q+4P)2sin(3π2+2nπarctan(3P4Q3Q+4P)+arctan(3P4Q3Q+4P))=5P2+Q2sin(3π2+2nπ)=5P2+Q2.

    Since limntn=, from (5.3), we have

    lim inftt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=lim infn{2(C26/25)tn+4/3(e1C14/5)t3/2n+4/25e2tn[cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)]}=<0.

    Similarly, fixing

    tn=π2+2nπarctan(3P4Q3Q+4P),

    we get

    limn[cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)]=5P2+Q2.

    Thus, from (5.3), we can get

    lim suptt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=lim supn{2(C26/25)tn+4/3(e1C14/5)t3/2n+4/25e2tn[cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)]}=>0.

    Therefore, by referring to Theorem 1, the solutions of (5.1) and (5.2) are oscillatory.

    Example 2. Let us consider the fractional partial differential system

    11875πD52+,tu(y,t)=1105t52Δu(y,t)+(1675×102πt316e2tcos(t)5πt3)t0(t)12u(y,)d+e2tcos(t)cos(10y),(y,t)(0,π)×(0,1.5) (5.4)

    with the condition

    u(0,t)=u(π,t)=0. (5.5)

    In the above, a(t)=1, r(t)=11875π, g(t)=t, α=1/2, p(t)=0, b(t)=12×105t52, m=1, a1(t)=12×105t52, τ1=0, q(y,t)= 1675×102πt3+16e2tcos(t)5πt3,f(y,t)= e2tcos(t)sin(y),Q=(0,π),q(t)=miny(0,π)q(y,t)= 1675×102πt3+16e2tcos(t)5πt3 and t0=0. It is obvious that β0=1 and ϕ(y)=sin(y). Since F1(t)=π0e2tcos(t)cos(10y)sin(y)dy= 299e2tcos(t) and A(t)=0, we have

    t0(t)α1r()(C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ)d=3750π99t0(t)α1(t0(τt0e2ζcos(ζ)dζ)dτ)d=50π11t0(t)1/2d+500π33t0(t)1/2d50π33[3t0e2(t)1/2cos()d+4t0e2(t)1/2sin()d].

    Fixing y2=t, then

    t0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=102πt11+2×10399t3/2102π33e2t{3cos(t)t0e2y2cos(y2)dy+3sin(t)t0e2y2sin(y2)dy+4sin(t)t0e2y2cos(y2)dy4cos(t)t0e2y2sin(y2)dy}. (5.6)

    Pointing out that

    |e2y2cos(y2)|e2y2,|e2y2sin(y2)|e2y2 and limtt0e2y2dy=2π4,

    we can conclude that

    limtt0e2y2cos(y2)dy and limtt0e2y2sin(y2)dy

    are convergent. Thus, we have that

    limt[cos(t)(3t0e2y2cos(y2)dy4t0e2y2sin(y2)dy)+sint(3t0e2y2sin(y2)dy+4t0e2y2cos(y2)dy)]

    is convergent. Fixing

    limtt0e2y2cos(y2)dy=P and limtt0e2y2sin(y2)dy=Q

    and considering the sequence

    tn=3π2+2nπarctan(3P4Q3Q+4P),

    we get

    limn{cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)}=(3P4Q)2+(3Q+4P)2sin(3π2+2nπarctan(3P4Q3Q+4P)+arctan(3P4Q3Q+4P))=5P2+Q2sin(3π2+2nπ)=5P2+Q2.

    Since limntn=, from (5.6), we have

    lim suptt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=lim supn{102πtn11tn+2×10399t3/2n102π33e2tn[cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)]}=>0.

    Similarly, fixing

    tn=π2+2nπarctan(3P4Q3Q+4P),

    we get

    limn{cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)}=5P2+Q2.

    Thus, from (5.6), we get

    lim inftt0(t)α1r()[C2+t0C1eA(τ)a(τ)dτ+t01eA(τ)a(τ)(τt0eA(ζ)F(ζ)dζ)dτ]d=lim infn{102πtn11tn+2×10399t3/2n102π33e2tn[cos(tn)(3tn0e2y2cos(y2)dy4tn0e2y2sin(y2)dy)+sin(tn)(3tn0e2y2sin(y2)dy+4tn0e2y2cos(y2)dy)]}=<0.

    Therefore, by referring to Theorem 2, the solutions of (5.4) and (5.5) are oscillatory. In fact, u(y,t)=t5/2cos(10y) is a solution of (5.4) and (5.5) and its oscillatory behavior is demonstrated in Figure 1.

    Figure 1.  Oscillatory behavior of u(y,t)=t5/2cos(10y).

    In this paper, we have obtained some new oscillation results for the fractional partial differential equation with damping and forcing terms under Robin and Dirichlet boundary conditions. The main results are proved by using only the integral averaging technique and without implementing the Riccati approach. Further, the obtained results are justified by some examples which can not be commented upon by using the previous results. Our results have been obtained for the general equation which may cover other particular cases.

    A. Palanisamy was supported by the University Grants Commission (UGC-Ref. No.: 958/(CSIR-UGC NET JUNE 2018)), New Delhi, India and the third author was partially supported by the DST-FIST Scheme (Grant No. SR/FST/MST-115/2016), New Delhi, India. J. Alzabut is thankful to Prince Sultan University and Ostim Technical University for their endless support. The authors are grateful to the reviewers for their precious help in improving this manuscript.

    The authors declare that they have no conflicts of interest.



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