Methods | hmax | NoDv |
[15] | 0.126 | 16 |
[20] | 0.577 | 75 |
[21] | 0.675 | 45 |
[22] | 0.728 | 45 |
[23] | 0.752 | 84 |
Theorem 3.1 | 1.141 | 71 |
Theoretical maximal value | 1.463 | − |
In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter ℏ, the results presented underscored the remarkable efficiency and accuracy of this approach.
Citation: Y. Massoun, C. Cesarano, A. K Alomari, A. Said. Numerical study of fractional phi-4 equation[J]. AIMS Mathematics, 2024, 9(4): 8630-8640. doi: 10.3934/math.2024418
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In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter ℏ, the results presented underscored the remarkable efficiency and accuracy of this approach.
Over the last two decades, many researches used LKF method to get stability results for time-delay systems [1,2]. The LKF method has two important technical steps to reduce the conservatism of the stability conditions. The one is how to construct an appropriate LKF, and the other is how to estimate the derivative of the given LKF. For the first one, several types of LKF are introduced, such as integral delay partitioning method based on LKF [3], the simple LKF [4,5], delay partitioning based LKF [6], polynomial-type LKF [7], the augmented LKF [8,9,10]. The augmented LKF provides more freedom than the simple LKF in the stability criteria because of introducing several extra matrices. The delay partitioning based LKF method can obtain less conservative results due to introduce several extra matrices and state vectors. For the second step, several integral inequalities have been widely used, such as Jensen inequality [11,12,13,14], Wirtinger inequality [15,16], free-matrix-based integral inequality [17], Bessel-Legendre inequalities [18] and the further improvement of Jensen inequality [19,20,21,22,23,24,25]. The further improvement of Jensen inequality [22] is less conservative than other inequalities. However, The interaction between the delay partitioning method and the further improvement of Jensen inequality [23] was not considered fully, which may increase conservatism. Thus, there exists room for further improvement.
This paper further researches the stability of distributed time-delay systems and aims to obtain upper bounds of time-delay. A new LKF is introduced via the delay partitioning method. Then, a less conservative stability criterion is obtained by using the further improvement of Jensen inequality [22]. Finally, an example is provided to show the advantage of our stability criterion. The contributions of our paper are as follows:
∙ The integral inequality in [23] is more general than previous integral inequality. For r=0,1,2,3, the integral inequality in [23] includes those in [12,15,21,22] as special cases, respectively.
∙ An augmented LKF which contains general multiple integral terms is introduced to reduce the conservatism via a generalized delay partitioning approach. For example, the ∫tt−1mhx(s)ds, ∫tt−1mh∫tu1x(s)dsdu1, ⋯, ∫tt−1mh∫tu1⋯∫tuN−1x(s)dsduN−1⋯du1 are added as state vectors in the LKF, which may reduce the conservatism.
∙ In this paper, a new LKF is introduced based on the delay interval [0,h] is divided into m segments equally. From the LKF, we can conclude that the relationship among x(s), x(s−1mh) and x(s−m−1mh) is considered fully, which may yield less conservative results.
Notation: Throughout this paper, Rm denotes m-dimensional Euclidean space, A⊤ denotes the transpose of the A matrix, 0 denotes a zero matrix with appropriate dimensions.
Consider the following time-delay system:
˙x(t)=Ax(t)+B1x(t−h)+B2∫tt−hx(s)ds, | (2.1) |
x(t)=Φ(t),t∈[−h,0], | (2.2) |
where x(t)∈Rn is the state vector, A,B1,B2∈Rn×n are constant matrices. h>0 is a constant time-delay and Φ(t) is initial condition.
Lemma2.1. [23] For any matrix R>0 and a differentiable function x(s),s∈[a,b], the following inequality holds:
∫ba˙xT(s)R˙x(s)ds≥r∑n=0ρnb−aΦn(a,b)TRΦn(a,b), | (2.3) |
where
ρn=(n∑k=0cn,kn+k+1)−1, |
cn,k={1,k=n,n≥0,−n−1∑t=kf(n,t)ct,k,k=0,1,⋯n−1,n≥1, |
Φl(a,b)={x(b)−x(a),l=0,l∑k=0cl,kx(b)−cl,0x(a)−l∑k=1cl,kk!(b−a)kφk(a,b)x(t),l≥1, |
f(l,t)=t∑j=0ct,jl+j+1/t∑j=0ct,jt+j+1, |
φk(a,b)x(t)={∫bax(s)ds,k=1,∫ba∫bs1⋯∫bsk−1x(sk)dsk⋯ds2dss1,k>1. |
Remark2.1. The integral inequality in Lemma 2.1 is more general than previous integral inequality. For r=0,1,2,3, the integral inequality (2.3) includes those in [12,15,21,22] as special cases, respectively.
Theorem3.1. For given integers m>0,N>0, scalar h>0, system (2.1) is asymptotically stable, if there exist matrices P>0, Q>0, Ri>0,i=1,2,⋯,m, such that
Ψ=ξT1Pξ2+ξT2Pξ1+ξT3Qξ3−ξT4Qξ4+m∑i=1(hm)2ATdRiAd−m∑i=1r∑n=0ρnωn(t−imh,t−i−1mh)Ri×ωn(t−imh,t−i−1mh)<0, | (3.1) |
where
ξ1=[eT1ˉET0ˉET1ˉET2⋯ˉETN]T, |
ξ2=[ATdET0ET1ET2⋯ETN]T, |
ξ3=[eT1eT2⋯eTm]T, |
ξ4=[eT2eT3⋯eTm+1]T, |
ˉE0=hm[eT2eT3⋯eTm+1]T, |
ˉEi=hm[eTim+2eTim+3⋯eTim+m+1]T,i=1,2,⋯,N, |
Ei=hm[eT1−eTim+2eT2−eTim+3⋯eTm−eTm(i+1)+1]T,i=0,1,2,⋯,N, |
Ad=Ae1+B1em+1+B2m∑i=0em+1+i, |
ωn(t−imh,t−i−1mh)={ei−ei+i,n=0,n∑k=0cn,kei−cn,0ei+1−n∑k=1cn,kk!e(k−1)m+k+1,n≥1, |
ei=[0n×(i−1)nIn×n0n×(Nm+1−i)]T,i=1,2,⋯,Nm+1. |
Proof. Let an integer m>0, [0,h] can be decomposed into m segments equally, i.e., [0,h]=⋃mi=1[i−1mh,imh]. The system (2.1) is transformed into
˙x(t)=Ax(t)+B1x(t−h)+B2m∑i=1∫t−i−1mht−imhx(s)ds. | (3.2) |
Then, a new LKF is introduced as follows:
V(xt)=ηT(t)Pη(t)+∫tt−hmγT(s)Qγ(s)ds+m∑i=1hm∫−i−1mh−imh∫tt+v˙xT(s)Ri˙x(s)dsdv, | (3.3) |
where
η(t)=[xT(t)γT1(t)γT2(t)⋯γTN(t)]T, |
γ1(t)=[∫tt−1mhx(s)ds∫t−1mht−2mhx(s)ds⋮∫t−m−1mht−hx(s)ds],γ2(t)=mh[∫tt−1mh∫tu1x(s)dsdu1∫t−1mht−2mh∫t−1mhu1x(s)dsdu1⋮∫t−m−1mht−h∫t−m−1mhu1x(s)dsdu1],⋯, |
γN(t)=(mh)N−1×[∫tt−1mh∫tu1⋯∫tuN−1x(s)dsduN−1⋯du1∫t−1mht−2mh∫t−1mhu1⋯∫t−1mhuN−1x(s)dsduN−1⋯du1⋮∫t−m−1mht−h∫t−m−1mhu1⋯∫t−m−1mhuN−1x(s)dsduN−1⋯du1], |
γ(s)=[x(s)x(s−1mh)⋮x(s−m−1mh)]. |
The derivative of V(xt) is given by
˙V(xt)=2ηT(t)P˙η(t)+γT(t)Qγ(t)−γT(t−hm)Qx(t−hm)+m∑i=1(hm)2˙xT(t)Ri˙x(t)−m∑i=1hm∫t−i−1mht−imh˙xT(s)Ri˙x(s)ds. |
Then, one can obtain
˙V(xt)=ϕT(t){ξT1Pξ2+ξT2Pξ1+ξT3Qξ3−ξT4Qξ4+m∑i=1(hm)2ATdRiAd}ϕ(t)−m∑i=1hm∫t−i−1mht−imh˙xT(s)Ri˙x(s)ds, | (3.4) |
ϕ(t)=[xT(t)γT0(t)γT1(t)⋯γTN(t)]T, |
γ0(t)=[xT(t−1mh)xT(t−2mh)⋯xT(t−h)]T. |
By Lemma 2.1, one can obtain
−hm∫t−i−1mht−imh˙xT(s)Ri˙x(s)ds≤−r∑l=0ρlωl(t−imh,t−i−1mh)Ri×ωl(t−imh,t−i−1mh). | (3.5) |
Thus, we have ˙V(xt)≤ϕT(t)Ψϕ(t) by (3.4) and (3.5). We complete the proof.
Remark3.1. An augmented LKF which contains general multiple integral terms is introduced to reduce the conservatism via a generalized delay partitioning approach. For example, the ∫tt−1mhx(s)ds, ∫tt−1mh∫tu1x(s)dsdu1, ⋯, ∫tt−1mh∫tu1⋯∫tuN−1x(s)dsduN−1⋯du1 are added as state vectors in the LKF, which may reduce the conservatism.
Remark3.2. For r=0,1,2,3, the integral inequality (3.5) includes those in [12,15,21,22] as special cases, respectively. This may yield less conservative results. It is worth noting that the number of variables in our result is less than that in [23].
Remark3.3. Let B2=0, the system (2.1) can reduces to system (1) with N=1 in [23]. For m=1, the LKF in this paper can reduces to LKF in [23]. So the LKF in our paper is more general than that in [23].
This section gives a numerical example to test merits of our criterion.
Example 4.1. Consider system (2.1) with m=2,N=3 and
A=[01−100−1],B1=[0.00.10.10.2],B2=[0000]. |
Table 1 lists upper bounds of h by our methods and other methods in [15,20,21,22,23]. Table 1 shows that our method is more effective than those in [15,20,21,22,23]. It is worth noting that the number of variables in our result is less than that in [23]. Furthermore, let h=1.141 and x(0)=[0.2,−0.2]T, the state responses of system (1) are given in Figure 1. Figure 1 shows the system (2.1) is stable.
Methods | hmax | NoDv |
[15] | 0.126 | 16 |
[20] | 0.577 | 75 |
[21] | 0.675 | 45 |
[22] | 0.728 | 45 |
[23] | 0.752 | 84 |
Theorem 3.1 | 1.141 | 71 |
Theoretical maximal value | 1.463 | − |
In this paper, a new LKF is introduced via the delay partitioning method. Then, a less conservative stability criterion is obtained by using the further improvement of Jensen inequality. Finally, an example is provided to show the advantage of our stability criterion.
This work was supported by Basic Research Program of Guizhou Province (Qian Ke He JiChu[2021]YiBan 005); New Academic Talents and Innovation Program of Guizhou Province (Qian Ke He Pingtai Rencai[2017]5727-19); Project of Youth Science and Technology Talents of Guizhou Province (Qian Jiao He KY Zi[2020]095).
The authors declare that there are no conflicts of interest.
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