Research article Special Issues

Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

  • We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor ω considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.

    Citation: Sayed Saifullah, Amir Ali, Zareen A. Khan. Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel[J]. AIMS Mathematics, 2022, 7(4): 5275-5290. doi: 10.3934/math.2022293

    Related Papers:

    [1] Xiangmei Li, Kamran, Absar Ul Haq, Xiujun Zhang . Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature. AIMS Mathematics, 2020, 5(5): 5287-5308. doi: 10.3934/math.2020339
    [2] Canlin Gan, Weiwei Wang . Existence result for the critical Klein-Gordon-Maxwell system involving steep potential well. AIMS Mathematics, 2023, 8(11): 26665-26681. doi: 10.3934/math.20231364
    [3] Mustafa Inc, Hadi Rezazadeh, Javad Vahidi, Mostafa Eslami, Mehmet Ali Akinlar, Muhammad Nasir Ali, Yu-Ming Chu . New solitary wave solutions for the conformable Klein-Gordon equation with quantic nonlinearity. AIMS Mathematics, 2020, 5(6): 6972-6984. doi: 10.3934/math.2020447
    [4] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [5] Yaojun Ye, Lanlan Li . Global existence and blow-up of solutions for logarithmic Klein-Gordon equation. AIMS Mathematics, 2021, 6(7): 6898-6914. doi: 10.3934/math.2021404
    [6] Khalid Khan, Amir Ali, Manuel De la Sen, Muhammad Irfan . Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels. AIMS Mathematics, 2022, 7(2): 1580-1602. doi: 10.3934/math.2022092
    [7] Khalid K. Ali, Mohamed S. Mohamed, Weam G. Alharbi, M. Maneea . Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method. AIMS Mathematics, 2024, 9(9): 24654-24676. doi: 10.3934/math.20241201
    [8] Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari, Adel Almarashi . Lump-type kink wave phenomena of the space-time fractional phi-four equation. AIMS Mathematics, 2024, 9(12): 34372-34386. doi: 10.3934/math.20241637
    [9] Qiuying Li, Xiaoxiao Zheng, Zhenguo Wang . Orbital stability of periodic standing waves of the coupled Klein-Gordon-Zakharov equations. AIMS Mathematics, 2023, 8(4): 8560-8579. doi: 10.3934/math.2023430
    [10] Shabir Ahmad, Aman Ullah, Ali Akgül, Fahd Jarad . A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations. AIMS Mathematics, 2022, 7(5): 9389-9404. doi: 10.3934/math.2022521
  • We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor ω considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.



    Fractional-order calculus has received considerable attention in the engineering and physical sciences over the last few decades to model a number of diverse phenomena in robotic-technology, bio-engineering, control theory, viscoelasticity diffusion model, relaxation processes and signal processing [1,2]. The order of derivatives, as well as integrals in the fractional-order calculus, is arbitrary. Therefore, fractional-order NPDEs have developed a fundamental interest in generalising integer-order NPDEs to model complex systems in thermodynamics, engineering, fluid dynamics and optical physics [3].

    The enormous advantage of using fractional differential equations (FDEs) in modeling real-world problems is their global behavior together with preserving memory [4] which is not present in integer-order differential equations. It has also been noted that FDEs fastly converge to ordinary differential equations (ODEs) in a case when fractional-order is equal to one. Moreover, fractional calculus can clarify the basic features of various models and processes them more precisely than integer-order [5]. Several techniques have been applied to study analytical as well as numerical solutions of FNPDEs such as, Variational Iteration Method (VIM) [6], Laplace transforms Method [7], the double Laplace transform [8], invariant subspace method [9], Integral transform [10], Sumudu transform Method (STM) [11], natural transform [12] and Adomian decomposition method (ADM) [13].

    The Klein-Gordon equation (KGE) considered herein is a basic non-linear evolution equation that arises in relativistic quantum Mechanics. It was formulated by Erwin Schrödinger for the non-relativistic wave equation in quantum physics, while precisely studied by the famous physicists O. Klein and W. Gordon (as it is named after their work) in 1926 [14,15]. The KGE has an extensive variety of applications in classical field theory [16] as well as in quantum field theory [17]. It has also been extensively used in numerous areas of physical phenomena such as in solid-state physics, dispersive wave-phenomena, nonlinear optics, elementary particles behavior, dislocations propagation in crystals, and different class of soliton solutions [18]. Here, we investigate equation of the form [19]

    ωψtω2ψx2+pψ+qg(ψ)=r(x,t),  1<ω2, (1.1)

    together with

    ψ(x,0)=F(x),ψt(x,0)=G(x),

    where ψ=ψ(x,t), g(ψ) and r(x,t) represent nonlinear term and external function respectively.

    The nonlinear differential equations involve numerous fractional differential operators, such as, Caputo, Hilfer, Riemann-Liouville (R-L), Atangana-Baleanu in Caputo's sense, and Caputo-Fabrizio, [20]. The above fractional operators are very useful in FC due to the complexities of fractional-PDEs/ODEs because standard operators cannot handle some equations to obtain explicit solutions. The Caputo fractional derivative is the basic idea of fractional derivatives. All the fractional derivatives will reduce in Caputo or Riemann-Liouville fractional derivatives after some parametric replacement. One can assume that the fractional derivative could provide a power-law of the local behavior of non-differentiable functions. The Caputo fractional derivative was introduced by Michele Caputo in 1967 [21] to study initial/boundary value problems in many areas of real-world phenomena. The Caputo's derivative has many advantages as it is the most important tool for dealing with integer order models in a fractional sense with suitable subsidiary conditions [23]. Most of the problems have been handled precisely using Caputo operator [24].

    The integer order KGE has broadly studied by using a variety of methods [25]. Time-fractional Klein Gordon equations with Caputo's fractional operator have also been extensively studied using a variety of numerical and analytical techniques [26]. Here, we apply double Laplace transform with decomposition method to study the general solutions of the governing model with power law. Some particular examples are also studied numerically with some physical analysis. For preliminaries and some basic definitions of Caputo's derivative, see [31] and the reference therein.

    Definition 1. Let us suppose ψ(x,t) lies in xtplane, the double Laplace transform (DLT) of ψ(x,t) is defined by [32]

    LxLt[ψ(x,t)]=0erx0estψdxdt,

    where, r,s(C).

    Definition 2. Application of DLT on fractional-order operator in Caputo's sense gives

    LxLt{CDωxψ(x,t)}=rω¯ψ(r,s)n1k=0rω1kLt{kψ(0,t)xk},

    and

    LxLt{CDβtψ(x,t)}=sβ¯ψ(p,s)m1k=0sβ1kLx{kψ(x,0)tk},

    where, m=[β]+1 and n=[ω]+1. Hence, we infer that

    LxLtψ(x)v(t)=¯ψ(p)¯v(s)=Lxψ(x)Ltv(t).

    The inverse DLT L1xL1t{¯ψ}=ψ, is represented by

    L1xL1t{¯ψ(x,t)}=12πic+iciestd+idiepx¯ψ(p,s)dpds,

    where Re(p)c and Re(s)d, and c,dR to be chosen appropriately.

    It is often more challenging to find the closed form of series solution to a nonlinear FDEs due to their complexity. Therefore question arises about the existence of the solution to such FDEs. For this, we utilize the applications of fixed point theory to study whether the solution of our considered system exists. So far, in the literature there exists no such theory for the existence of our considered system. We use here for the first time the βl-Geraghty type contraction to show that there exists a solution to the considered model. So we progress as follows

    CaDωtψψx+pψ+qg(ψ)=r,1<ω2, (2.1)

    with

    ψ(x,0)=F(x),ψt(x,0)=G(x). (2.2)

    The above equation can also be expressed in the form

    CaDωtψ=H(x,t,ψ),1<ω2, (2.3)

    where

    H(x,t,ψ)=ψxpψqg(ψ)+r. (2.4)

    For the existence of the above model, we use the following notions.

    Let Ω be the family of continuous and increasing functions defined as l:[0,)[0,) satisfying

    l(qx)ql(x)qx,q>1,

    and the elements of Θ are non-decreasing functions, such that

    ε:[0,)[0,1ρ21),whereρ11.

    Definition 3. Suppose that (M,d) be a complete b-metric space: Let T:MM also consider that F:M×M[0,) with F(m,n)l(ρ31d(Tm,Tn))ε(ld(m,n))l(d(m,n)), for m,nM,whereρ11,εΘandlΩ. Then T is known a generalized Fl-Gergaghty type contraction mapping.

    Definition 4. Consider T:MM, where M is non-empty and F:M×M[0,), where β(m,n)1β(Mm,Mn)m,nM, then T is called β-admissible mapping.

    First we show that there exists the fixed point for the considered model Eq (2.3), for this we apply the following theorem.

    Theorem 1. [33] Let T:MM be a generalized Fl-Gergaghty type contraction such that

    (1) T is β-admissible.

    (2) There exists υ0Mwithβ(υ0,Tυ0)1.

    (3) {υn}M,limnυn=υ, where υM and β(υn,υn+1)1β(υn,υ)1,

    then a fixed point for T. Let M=C(π,R) and d:M×M[0,), where π=[0,1]×[0,1] given by

    d(u,v)=||(uv)2||=supm[0,M]t[0,T](uv)2,

    thus (M,d) be a complete b-metric space. The following theorem shows the existence of solution of the considered model Eq (2.3).

    Theorem 2. Suppose that J:R2R such that

    (1) |H(x,t,ψ(x,t))H(x,t,ϕ(x,t))|α+133ε(l|uv|2)l(|uv|2), for x[0,X],t[0,T], and u,vM with J(u,v)0.

    (2) there exists u1M with J(u1,Tu1)0, where T:MM is defined by Tuj=u0u0+u1ut+IωtH(x,t,ψ(x,t)).

    (3) for u,vM,J(u,v)0J(Tun,Tv).

    (4) {un}M, unu where uM and mathcalJ(un,un+1)mathcalJ(un,u)0, for nN.Then there exists a solution of the model Eq (2.3).

    Proof. Applying the fractional integral to Eq (2.3), we obtain

    ψ(x,t)=C0ψ(x,0)+C1ψt(x,0)+IωtH(x,t,ψ(x,t))=Tψ(x,t).

    Here we prove that T has a fixed point using the above technique, thus

    |Tψ(x,t)Tϕ(x,t)|2=|IωtH(x,t,ψ(x,t))IωtH(x,t,ϕ(x,t))|2Iωt{|H(x,t,ψ(x,t))H(x,t,ϕ(x,t))|}2={1ω|H(x,t,ψ(x,t))H(x,t,ϕ(x,t))|}2{ωω33ωTωt0(ts)ω1ε(l(|uv|2)l(|uv|2))|}2{ωTωt0(ts)ω1ε(l(supx[0,X]t[0,T]|uv|2)l(supx[0,X]t[0,T]|uv|2))|}2133ε(l(d(u,v))l(d(u,v))).

    Hence for u,vC(π with J(u,v)0 we have 27||(TuTv)2||ε(l(d(u,v))l(d(u,v))). Now F:C([0,X]×[0,T],R)×C([0,X]×[0,T],R)[0,) by

    β(u,v)={1ifJ(u,v)0,0else,

    and

    β(u,v)l(27d(Tu,Tv))27d(Tu,Tv)ε(ld(u,v)ld(u,v)).

    Thus, T is an βl-contraction. Now to show that T is β-admissible, we have from condition (iii)

    β(u,v)1J(u,v)0J(Tu,Tv)0β(Tu,Tv)1.

    For u,vC(π,R), from condition (ii) we have uC(π,R). Such that β(u0,Tu0)1. Similarly from (iv) and Theorem 1, there exists uC(π,R), such that u=Tu. Therefore we proved that the model Eq (2.3) has a solution.

    Here, we study the above technique, which is a composition of DLT with the decomposition method. This method can be applied to find the general series solutions for various PDEs/ODEs. This is an efficient technique to study the analytical solutions of several nonlinear systems [34]. Let us consider the general non-linear system

    Lψ+Rψ+Nψ=r(x,t). (3.1)

    Here, L and R is linear and nonlinear operators, r(x,t) is some particular external function and N is nonlinearity in the system. The convergence analysis of the considered technique can be seen in [35].

    General solution of proposed model in Caputo's sense

    Using the technique defined above and expressing Eq (1.1) in the form

    CDωtψ(x,t)2ψx2+pψ+qg(ψ)=r(x,t),1<ω2, (3.2)

    with

    ψ(x,0)=F(x),ψt(x,t)=G(x). (3.3)

    Applying DLT to above equation, we obtain

    LxLt{CDωtψ}LxLt{2x2ψ}+pLxLt{ψ}+qLxLt{gψ}=LxLt{r(x,t)}. (3.4)

    Applying DLT on fractional order, gives

    LxLt{ψ}=1sLx{ψ(x,0)}+1s2Lx{ψt(x,0)}+1sωLxLt{2x2ψ}+p1sωLxLt{ψ}+q1sωLxLt{g(ψ)}+LxLt{r(x,t)}. (3.5)

    Similarly, applying Laplace transform on Eq (3.3), gives

    Lx{ψ(x,0)}=¯F(p),Lx{ψt(x,0)}=t¯G(p). (3.6)

    Now consider

    ψ=n=0ψn, (3.7)

    where the non-linear term can be degraded as

    g(ψ)=i=0An, (3.8)

    where An, is given by [36]

    An=1n!dndλn[nk=0λkg(ψk)]λ=0. (3.9)

    Finally, applying inverse DLT to Eq (3.2), using Eq (3.6) and Eq (3.9), gives

    ψ0=L1xL1t[1s¯F(p,0)]+tL1xL1t[1s2¯G(p,0)]=ψ(x,0),ψ1=L1xL1t[1sωLxLt{ψ0xx}]pL1xL1t[1sωLxLt{ψ0}]qL1xL1t[1sωLxLt{A0}]+[1sωLxLt{r(x,t)}],ψ2=L1xL1t[1sωLxLt{ψ1xx}]pL1xL1t[1sωLxLt{ψ1}]qL1xL1t[1sωLxLt{A1}],ψ3=L1xL1t[1sωLxLt{ψ2xx}]pL1xL1t[1sωLxLt{ψ2}]qL1xL1t[1sωLxLt{A2}].

    In a similar manner, other terms can be computed. Final result can be obtained as

    ψ(x,t)=n=0ψn(x,t). (3.10)

    which is the general solution of Eq (3.2) in series form by using the proposed method as discussed above.

    Here, we present numerical examples on the TFKG equation in Caputo's sense given as Eq (3.2) and discuss the behaviour of each example. We apply the aforesaid technique discussed in Section 3, to obtain the approximate solution of the problems.

    Example 1. Consider the nonlinear TFKG equation

    Dωtψ2ψx2+34ψ32ψ3=0,1<ω2,p=34,q=32, (4.1)
    g(ψ)=ψ3,r(x,t)=0, (4.2)

    with

    ψ(x,0)=sech(x),ψt(x,0)=12sech(x)tanh(x). (4.3)

    For α=2, the exact solution of Eq (4.1) can be obtained in the form [19]

    ψ(x,t)=sech(x+t2). (4.4)

    Consider TFKG Eq (4.1) in Caputo's sense

    CDωtψ2ψx2+34ψ32ψ3=0,1<ω2. (4.5)

    Applying MDLDM scheme discussed in Section 3, we obtain

    ψ0=sech(x)+t2sech(x)tanh(x),ψ1=tωΓ(ω+1)[1432sech2(x)]sech(x)+tω+1Γ(ω+2)[11874sech2(x)]sech(x)tanh(x)2!tω+2Γ(ω+3)[98sech3(x)tanh2(x)]+3!tω+3Γ(ω+4)[316sech3(x)tanh3(x)].

    In a similar manner, other terms can be computed. Final result can be obtained as

    ψ(x,t)=n=0ψn(x,t). (4.6)

    Discussion

    The error analysis between series solution Eq (4.6) and the exact solution Eq (4.4) is shown in Table 1, while the surface behaviour is shown in Figure 1 reveals that Eq (4.1) is depends on time (t). It should be noted that when the time (t) is small enough, there is less extent of error exist amongst the approximate and exact solutions obtained by the MDLDM method. Figure 2 [left panel] shows the absolute of wave solution Eq (4.6) with deviations in (α) with t=0.65 in comparison with exact solution Eq (4.4). Notice that numerical result, Eq (4.6) exactly matches to the exact solution Eq (4.4). This shows that the governing equation admits a soliton solution. Figure 2 [right panel] represents Eq (4.6) reveals that the amplitude of the solitary potentials blow-up as t increases. The 3D profiles for Eq (4.6) is shown in Figure 3 versus x for t=0.65. It reveals the progression of localized mode in the governing system. The solution obtained as Eq (4.6) versus x for t=1(dashedline),0.8(solidcurve),0.6(dottedcurve) Figure 4, when ω=2and1.7 are also depicted. Clearly, one can see the wave amplitude enhancement with variations in t that concludes that coefficient (ω) considerably increases the wave amplitudes.

    Table 1.  Comparison between the exact solution (4.4) with approximate solution obtained in the form (4.6).
    (x, t) Exact ψ Exactψ (x, t) Exact ψ Exactψ
    (-6, 0.6) 0.0067 0.0067 2.4083 ×105 (-4, 0.6) 0.0494 0.0492 1.7609×104
    (-2, 0.6) 0.3536 0.3529 6.2194×104 (0, 0.6) 0.9566 0.9550 1.6000×103
    (2, 0.6) 0.1985 0.1992 6.7883×104 (4, 0.6) 0.0271 0.0273 1.5198×104
    (6, 0.6) 0.0037 0.0037 2.0728×105 (-6, 0.2) 0.0055 0.0055 8.4718 ×107
    (-4, 0.2) 0.0405 0.0405 6.2013×106 (-2, 0.2) 0.2926 0.2926 2.4219×105
    (0, 0.2) 0.9950 0.9950 2.0749×105 (2, 0.2) 0.2413 0.2413 2.4874×105
    (4, 0.2) 0.0331 0.0331 5.9043×106 (6, 0.2) 0.0045 0.0045 8.0587×107
    (-6, 0.05) 0.0051 0.0051 1.2989×108 (-4, 0.05) 0.0375 0.0375 9.5114×108
    (-2, 0.05) 0.2723 0.2723 3.8310×107 (0, 0.05) 0.9997 0.9997 8.1360×108
    (2, 0.05) 0.2595 0.2595 3.8564×107 (4, 0.05) 0.0357 0.0357 9.3954×108
    (6, 0.05) 0.0048 0.0048 1.2828×108

     | Show Table
    DownLoad: CSV
    Figure 1.  The surface plot of the error analysis given in Table 1.
    Figure 2.  The left plot portrays comparison between Eq (4.4) and Eq (4.6) for various values of ω, while the right panel portrays solution profiles of ψ(x,t) vs t for various values of ω.
    Figure 3.  The surface plot of Eq (4.6) for the parameters used in the Figure 2's left panel.
    Figure 4.  The solution profiles of Eq (4.6) for desperate values of ω with desperate values of t.

    Example 2. Consider the time-fractional nonlinear KGE in the form

    Dωtϕ2ϕx2+qϕ3=0,1<ω2,p=0q=1, (4.7)
    y(ϕ)=ϕ3,r(x,t)=0, (4.8)

    with

    ϕ(x,0)=Rtan(λx),ϕt(x,0)=Rηλsec2(λx), (4.9)

    where

    R=ρκandλ=ρ2(σ+η2).

    The parameters ρ, κ, σ, and ηR. It should be noted that, for α=2, an exact solution of Eq (4.7) can be obtained in the form [37]

    ϕ(x,t)=Rtan[λ(x+ηt)]. (4.10)

    Writing Eq (4.7) in Caputo's sense gives

    CDωtϕ2ϕx2+ϕ3=0,1<ω2. (4.11)

    The series solution of Eq (4.11) with conditions (4.9) gives

    ϕ0=Rtan(λx)+tRηλsec2(λx),ϕ1=(tωΓ(ω+1))[2Rλ2sec2(λx)tan(λx)R2tan3(λx)]+Rηλ(tω+1Γ(ω+2))[4λ2sec2(λx)tan2(λx)+2λ2sec4(λx)3R2tan2(λx)sec2(λx)]3R3λ2η2(2!tω+2Γ(ω+3))[sec4(λx)tan(λx)]R3λ3η3(3!tω+3Γ(ω+4))[sech6(λx)].

    In a similar manner, other terms can be computed. The final result can be written in the form

    ϕ=n=0ϕn. (4.12)

    Discussion

    The parameters as κ=1,σ=8.5,η=0.05, and ρ=1 are considered for numerical illustration. The error amongst the approximate and exact solutions of Eq (4.7) is shown in Table 2 and its corresponding surface plot is presented in Figure 5. The numerical solution, Eq (4.12) and exact solution Eq (4.10) is depicted in Figure 6 [left panel], for t=7 and for different values of time-fractional coefficient (ω). It is noted that TFKG Eq (4.12) may admits the excitation in the system. This amount enrichment in ω overturned the wave amplitude as it interrupt the dispersion/nonlinearity effects. To see the effect of a temporal variable (t) on the wave solution, Eq (4.12) is displayed in Figure 6 [right panel] which shows that ϕ(x,t) increases with time. Further, the 3D profiles for Eq (4.12) is shown versus x with t=7 in Figure 7 for (ω=2), which represents the physical behaviour of Eq (4.12). It shows the advancement of localized mode excitations in the governing equation. The solution of Eq (4.12) versus x with t=7 (dashed curve), 6 (solid green curve), 5 (dotted curve), in Figure 8, with ω=2and1.5 respectively is depicted. Clearly, the wave amplitude increases with deviations in t. It infers that the fractional order (ω) significantly increases the wave amplitudes.

    Table 2.  Comparison between approximate solution obtained in the form (4.12) with exact solution (4.7).
    (x, t) Exact ϕ Exactϕ (x, t) Exact ϕ Exactϕ
    (-10, 0.02) 0.9845 0.9845 4×107 (-8, 0.02) 0.9596 0.9596 4×106
    (-6, 0.02) 0.8968 0.8965 3×105 (-4, 0.02) 0.7488 0.7486 2×105
    (-2, 0.02) 0.4504 0.4503 1×105 (0, 0.02) 0 0 0
    (2, 0.02) 0.4504 0.4503 1×105 (4, 0.02) 0.7488 0.7486 2×105
    (6, 0.02) 0.8968 0.8965 3×105 (8, 0.02) 0.9596 0.9596 4×106
    (10, 0.02) 0.9845 0.9845 4×107 (-10, 0.05) 0.9845 0.9845 1.8751×105
    (-8, 0.05) 0.9595 0.9596 4.8543×105 (-6, 0.05) 0.8966 0.8968 1.2196×104
    (-4, 0.05) 0.7486 0.7488 2.8545 ×104 (-2, 0.05) 0.4499 0.4499 5.8734×104
    (0, 0.05) 0.0012 0.0012 1.1300×106 (2, 0.05) 0.4508 0.4505 3.7952×104
    (4, 0.05) 0.7491 0.7488 2.4730×104 (6, 0.05) 0.8969 0.8968 1.1554×104
    (8, 0.05) 0.9596 0.9596 4.7525×105 (10, 0.05) 0.9845 0.9845 1.8587×105

     | Show Table
    DownLoad: CSV
    Figure 5.  The surface plot for Table 2.
    Figure 6.  Comparison between Eq (4.10) and Eq (4.12) for different values of ω [left panel]. The solution profiles of ψ(x,t) against time (t) interval with various values of ω [right panel].
    Figure 7.  The surface plot for the parameters used for the left panel of Figure 6.
    Figure 8.  The solution profiles of Eq (4.12) for different ω with different values of time(t).

    We have studied the TFKG equation using double Laplace transforms with the decomposition method. The general solution of proposed system is obtained as a class of general series solution. It is relevant to note that following only two iterations, fairly precise results are obtained that converges to the exact solution of the governing equation. The proposed method offers perfect numerical results without any alteration and complicated numerical methods for the governing equation in the fractional case. The numerical results obtained for particular examples are compared with the exact solutions at the classical order. The result profiles with physical interpretations for different fraction orders were revealed explicitly.

    Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R8). Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    It is declared that all the authors have no conflict of interest regarding this manuscript.



    [1] I. Podlubny, Fractional differential equations, Academic, New York, 1999.
    [2] Z. A. Khan, H. Ahmad, Qualitative properties of solutions of fractional differential and difference equations arising in physical models, Fractals, 29 (2021), 1–10. http://dx.doi.org/10.1142/S0218348X21400247 doi: 10.1142/S0218348X21400247
    [3] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
    [4] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
    [5] R. Toledo-Hernandez, V. Rico-Ramirez, A. Gustavo Iglesias-Silva, Urmila M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chem. Eng. Sci., 117 (2014), 217–228. https://doi.org/10.1016/j.ces.2014.06.034 doi: 10.1016/j.ces.2014.06.034
    [6] Y. Zhang, X. J. Yang, An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics, Appl. Math. Mod., 40 (2016), 1793–1799. https://doi.org/10.1016/j.apm.2015.08.017 doi: 10.1016/j.apm.2015.08.017
    [7] L. Kexue, P. Jigen, Laplace transform and fractional differential equations, Appl. Math. Lett., 24 (2011), 2019–2023. https://doi.org/10.1016/j.aml.2011.05.035 doi: 10.1016/j.aml.2011.05.035
    [8] S. Saifullah, A. Ali, M. Irfan, K. Shah, Time-fractional Klein-Gordon equation with solitary/shock waves solutions, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/6858592
    [9] R. Gazizov, A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl., 66 (2013), 576–584. https://doi.org/10.1016/j.camwa.2013.05.006 doi: 10.1016/j.camwa.2013.05.006
    [10] D. G. Duffy, Transform methods for solving partial differential equations, CRC Press, 2004. https://doi.org/10.1201/9781420035148
    [11] S. T. Demiray, H. Bulut, F. Bin, M. Belgacem, Sumudu transform method for analytical solutions of fractional type ordinary differential equations, Math. Probl. Eng., 2015 (2015). https://doi.org/10.1155/2015/131690 doi: 10.1155/2015/131690
    [12] M. Valizadeh, Y. Mahmoudi, F. Dastmalchi Saei, Application of natural transform method to fractional pantograph delay differential equations, J. Math., 2019 (2019). https://doi.org/10.1155/2019/3913840
    [13] D. J. Evans, H. Bulut, A new approach to the gas dynamics equation: An application of the decomposition method, Int. J. Comput. Math., 79 (2002), 752–761. https://doi.org/10.1080/00207160211297 doi: 10.1080/00207160211297
    [14] W. Gordon, Der comptoneffekt nach der schrödingerschen theorie, Zeitschrift für Physik, 40 (1926). https://doi.org/10.1007/BF01390840 doi: 10.1007/BF01390840
    [15] O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Zeitschrift für Physik, 37 (1926). https://doi.org/10.1007/BF01397481
    [16] D. Bambusi, S. Cuccagna, On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential, Am. J. Math., 133 (2011), 1421–1468. https://doi.org/10.1353/ajm.2011.0034
    [17] W. Bao, X. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189–229. https://doi.org/10.1007/s00211-011-0411-2 doi: 10.1007/s00211-011-0411-2
    [18] N. A. Khan, F. Riaz, A. Ara, A note on soliton solutions of Klein-Gordon-Zakharov equation by variational approach, Nonlinear Eng., 5 (2016), 135–139. https://doi.org/10.1515/nleng-2016-0001 doi: 10.1515/nleng-2016-0001
    [19] B. Bülbül, M. Sezer, A new approach to numerical solution of nonlinear Klein-Gordon equation, Math. Probl. Eng., 2013 (2013). https://doi.org/10.1155/2013/869749
    [20] A. Atangana, J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Meth. Part. D. E., 34 (2018), 1502–1523. https://doi.org/10.1002/num.22195 doi: 10.1002/num.22195
    [21] M. Caputo, Linear model of dissipation whose Q is almost frequency independent–-II, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
    [22] Z. A. Khan, R. Gul, K. Shah, On impulsive boundary value problem with Riemann-Liouville fractional order derivative, J. Funct. Space., 2021 (2021), 1–11. https://doi.org/10.1155/2021/8331731 doi: 10.1155/2021/8331731
    [23] Z. A. Khan, I. Ahmad, K. Shah, Applications of fixed point theory to investigate a system of fractional order differential equations, J. Func. Space., 2021 (2021), 1–7. https://doi.org/10.1155/2021/1399764 doi: 10.1155/2021/1399764
    [24] M. Çiçek, C. Yakar, M. B. Gücen, Practical stability in terms of two measures for fractional order dynamic systems in Caputo's sense with initial time difference, J. Franklin I., 2 (2014), 732–742. https://doi.org/10.1016/j.jfranklin.2013.10.009 doi: 10.1016/j.jfranklin.2013.10.009
    [25] J. Saelao, N. Yokchoo, The solution of Klein-Gordon equation by using modified Adomian decomposition method, Math. Comput. Simulat., 171 (2020), 94–102. https://doi.org/10.1016/j.matcom.2019.10.010 doi: 10.1016/j.matcom.2019.10.010
    [26] M. M. Khader, N. H. Swetlam, A. M. S. Mahdy, The chebyshev collection method for solving fractional order Klein-Gordon equation, Wseas Trans. Math., 13 (2014), 31–38.
    [27] A. Wazwaz, Abdul-Majid, The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation, Appl. Math. Comput., 167 (2005), 1179–1195. https://doi.org/10.1016/j.amc.2004.08.006 doi: 10.1016/j.amc.2004.08.006
    [28] X. Li, B. Y. Guo, A Legendre pseudospectral method for solving nonlinear Klein-Gordon equation, J. Comp. Math., 15 (1997), 105–126.
    [29] H. Li, X. H. Meng, B. Tian, Bilinear form and soliton solutions for the coupled nonlinear Kleain-Gordon equations, Inter. J. Mod. Phys. B, 26 (2012). https://doi.org/10.1142/S0217979212500579 doi: 10.1142/S0217979212500579
    [30] A. M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Commun. Nonlinear. Sci., 13 (2008), 889–901. https://doi.org/10.1016/j.cnsns.2006.08.005 doi: 10.1016/j.cnsns.2006.08.005
    [31] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 204 (2006).
    [32] I. N. Sneddon, The use of integral transforms, Tata McGraw Hill Edition, 1974.
    [33] H. Afshari, H. Aydi, E. Karapinar, Existence of fixed points of set-valued mappings in b-metric spaces, E. Asian Math. J., 32 (2016), 319–332. https://doi.org/10.7858/eamj.2016.024 doi: 10.7858/eamj.2016.024
    [34] F. Rahman, A. Ali, S. Saifullah, Analysis of time-fractional ϕ4-equation with singular and non-singular kernels, J. Appl. Comput. Math., 7 (2021), 192. https://doi.org/10.1007/s40819-021-01128-w doi: 10.1007/s40819-021-01128-w
    [35] K. Khan, Z. Khan, A. Ali, M. Irfan, Investigation of Hirota equation: Modified double Laplace decomposition method, Phys. Scripta, 96 (2021). https://doi.org/10.1088/1402-4896/ac0d33 doi: 10.1088/1402-4896/ac0d33
    [36] G. Adomian, Modification of the decomposition approach to heat equation, J. Math. Anal. Appl., 124 (1987), 290–291. https://doi.org/10.1016/0022-247X(87)90040-0 doi: 10.1016/0022-247X(87)90040-0
    [37] D. Huang, G. Zou, L. W. Zhang, Numerical approximation of nonlinear Klein-Gordon equation using an element-free approach, Math. Probl. Eng., 2015 (2015). https://doi.org/10.1155/2015/548905 doi: 10.1155/2015/548905
  • This article has been cited by:

    1. Sajjad Ali, Aman Ullah, Shabir Ahmad, Kamsing Nonlaopon, Ali Akgül, Analysis of Kink Behaviour of KdV-mKdV Equation under Caputo Fractional Operator with Non-Singular Kernel, 2022, 14, 2073-8994, 2316, 10.3390/sym14112316
    2. Aman Ullah, Shabir Ahmad, Mustafa Inc, Fractal fractional analysis of modified KdV equation under three different kernels, 2022, 24680133, 10.1016/j.joes.2022.04.025
    3. Husna Zafar, Amir Ali, Khalid Khan, Muhammad Noveel Sadiq, Analytical Solution of Time Fractional Kawahara and Modified Kawahara Equations by Homotopy Analysis Method, 2022, 8, 2349-5103, 10.1007/s40819-022-01296-3
    4. Shabir Ahmad, Mansour F. Yassen, Mohammad Mahtab Alam, Soliman Alkhati, Fahd Jarad, Muhammad Bilal Riaz, A numerical study of dengue internal transmission model with fractional piecewise derivative, 2022, 39, 22113797, 105798, 10.1016/j.rinp.2022.105798
    5. Abdul Jamal, Aman Ullah, Shabir Ahmad, Shahzad Sarwar, Ali Shokri, A survey of (2+1)-dimensional KdV–mKdV equation using nonlocal Caputo fractal–fractional operator, 2023, 46, 22113797, 106294, 10.1016/j.rinp.2023.106294
    6. Rubayyi T. Alqahtani, Shabir Ahmad, Ali Akgül, On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels, 2022, 10, 2227-7390, 1102, 10.3390/math10071102
    7. Badr Saad T. Alkahtani, Analytical Study of the Complexities in a Three Species Food Web Model with Modified Caputo–Fabrizio Operator, 2023, 7, 2504-3110, 105, 10.3390/fractalfract7020105
    8. Asif Khan, Amir Ali, Shabir Ahmad, Sayed Saifullah, Kamsing Nonlaopon, Ali Akgül, Nonlinear Schrödinger equation under non-singular fractional operators: A computational study, 2022, 43, 22113797, 106062, 10.1016/j.rinp.2022.106062
    9. Changjin Xu, Zixin Liu, Yicheng Pang, Ali Akgül, Dumitru Baleanu, Dynamics of HIV-TB coinfection model using classical and Caputo piecewise operator: A dynamic approach with real data from South-East Asia, European and American regions, 2022, 165, 09600779, 112879, 10.1016/j.chaos.2022.112879
    10. Shaimaa A. M. Abdelmohsen, Mansour F. Yassen, Shabir Ahmad, Ashraf M. M. Abdelbacki, Javed Khan, Theoretical and numerical study of the rumours spreading model in the framework of piecewise derivative, 2022, 137, 2190-5444, 10.1140/epjp/s13360-022-02921-2
    11. Khalid Khan, Obaid Algahtani, Muhammad Irfan, Amir Ali, Electron-acoustic solitary potential in nonextensive streaming plasma, 2022, 12, 2045-2322, 10.1038/s41598-022-19206-4
    12. Badr S. Alkahtani, Mathematical Modeling of COVID-19 Transmission Using a Fractional Order Derivative, 2022, 7, 2504-3110, 46, 10.3390/fractalfract7010046
    13. JAVED KHAN, MATI UR RAHMAN, MUHAMMAD BILAL RIAZ, JAN AWREJCEWICZ, A NUMERICAL STUDY ON THE DYNAMICS OF DENGUE DISEASE MODEL WITH FRACTIONAL PIECEWISE DERIVATIVE, 2022, 30, 0218-348X, 10.1142/S0218348X22402605
    14. Arshad Hussain, Sayed Saifullah, Amir Ali, Theory and applications of integral transform: analytical and numerical study of nonlinear partial differential equations, 2022, 97, 0031-8949, 084004, 10.1088/1402-4896/ac7d7b
    15. Shabir Ahmad, Aman Ullah, Ali Akgül, Fahd Jarad, A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations, 2022, 7, 2473-6988, 9389, 10.3934/math.2022521
    16. Zareen A. Khan, Javed Khan, Sayed Saifullah, Amir Ali, Hemant Kumar Nashine, Dynamics of Hidden Attractors in Four-Dimensional Dynamical Systems with Power Law, 2022, 2022, 2314-8888, 1, 10.1155/2022/3675076
    17. Mohammad Partohaghighi, Zahrasadat Mirtalebi, Ali Akgül, Muhammad Bilal Riaz, Fractal–fractional Klein–Gordon equation: A numerical study, 2022, 42, 22113797, 105970, 10.1016/j.rinp.2022.105970
    18. H M Ahmed, R M Hafez, W M Abd-Elhameed, A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices, 2024, 99, 0031-8949, 045250, 10.1088/1402-4896/ad3482
    19. Zhoujin Cui, Tao Lu, Bo Chen, Ammar Alsinai, Traveling Wave Solutions of the Conformable Fractional Klein–Gordon Equation With Power Law Nonlinearity, 2024, 2024, 2314-4629, 10.1155/2024/8367957
    20. Muhammad SİNAN, Kamal SHAH, Thabet ABDELJAWAD, Ali AKGUL, Analysis of Nonlinear Mathematical Model of COVID-19 via Fractional-Order Piecewise Derivative, 2023, 5, 2687-4539, 27, 10.51537/chaos.1210461
    21. Manoj Singh, Approximation of the Time-Fractional Klein-Gordon Equation using the Integral and Projected Differential Transform Methods, 2023, 8, 2455-7749, 672, 10.33889/IJMEMS.2023.8.4.039
    22. Khan Shehzada, Aman Ullah, Sayed Saifullah, Ali Akgül, Fractional generalized perturbed KdV equation with a power Law kernel: A computational study, 2023, 12, 26667207, 100298, 10.1016/j.rico.2023.100298
    23. Itishree Sahu, Saumya Ranjan Jena, An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics, 2024, 10, 26668181, 100744, 10.1016/j.padiff.2024.100744
    24. Meina Liu, Lin Lu, Yujing Chen, Chao Zhang, New periodic solutions and solitary wave solutions for the time-fractional differential equations, 2024, 99, 0031-8949, 115239, 10.1088/1402-4896/ad8317
    25. Ihsan Ullah, Aman Ullah, Shabir Ahmad, Hijaz Ahmad, Taher A. Nofal, A survey of KdV-CDG equations via nonsingular fractional operators, 2023, 8, 2473-6988, 18964, 10.3934/math.2023966
    26. Shabir Ahmad, Sayed Saifullah, Analysis of the seventh-order Caputo fractional KdV equation: applications to the Sawada–Kotera–Ito and Lax equations, 2023, 75, 0253-6102, 085002, 10.1088/1572-9494/acded7
    27. Inayat Ullah, Aman Ullah, Shabir Ahmad, Ali Akgül, Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels, 2023, 9, 2349-5103, 10.1007/s40819-023-01498-3
    28. Mallanagoud Mulimani, S. Kumbinarasaiah, A numerical study on the nonlinear fractional Klein–Gordon equation, 2023, 2731-6734, 10.1007/s43994-023-00091-0
    29. Ghulam Saddique, Salman Zeb, Amir Ali, Solitary Wave Solution of Korteweg–De Vries Equation by Double Laplace Transform with Decomposition Method, 2025, 11, 2349-5103, 10.1007/s40819-024-01813-6
    30. Najeeb Alam Khan, Sahar Altaf, Nadeem Alam Khan, Muhammad Ayaz, Haar wavelet Arctic puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator, 2025, 26668181, 101114, 10.1016/j.padiff.2025.101114
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2298) PDF downloads(118) Cited by(30)

Figures and Tables

Figures(8)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog