Research article Special Issues

Computational analysis of fractional modified Degasperis-Procesi equation with Caputo-Katugampola derivative

  • Main aim of the current study is to examine the outcomes of nonlinear partial modified Degasperis-Procesi equation of arbitrary order by using two analytical methods. Both methods are based on homotopy and a novel adjustment with generalized Laplace transform operator. Nonlinear terms are handled by using He's polynomials. The fractional order modified Degasperis-Procesi (FMDP) equation, is capable to describe the nonlinear aspects of dispersive waves. The Katugampola derivative of fractional order in the caputo type is employed to model this problem. The numerical results and graphical representation demonstrate the efficiency and accuracy of applied techniques.

    Citation: Jagdev Singh, Arpita Gupta. Computational analysis of fractional modified Degasperis-Procesi equation with Caputo-Katugampola derivative[J]. AIMS Mathematics, 2023, 8(1): 194-212. doi: 10.3934/math.2023009

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  • Main aim of the current study is to examine the outcomes of nonlinear partial modified Degasperis-Procesi equation of arbitrary order by using two analytical methods. Both methods are based on homotopy and a novel adjustment with generalized Laplace transform operator. Nonlinear terms are handled by using He's polynomials. The fractional order modified Degasperis-Procesi (FMDP) equation, is capable to describe the nonlinear aspects of dispersive waves. The Katugampola derivative of fractional order in the caputo type is employed to model this problem. The numerical results and graphical representation demonstrate the efficiency and accuracy of applied techniques.



    The nonlinear partial Degasperis-Procesi equation is a very important differential equation, that arises in the modeling of dispersive water wave propagation. In mathematical physics, the modified Degasperis-Procesi differential equation is written as

    vtvxxt+4v2vx=3vxvxx+vvxxx. (1.1)

    The Degasperis-Procesi [1] equation was discovered by, 'Antonio Degasperis' and 'Michela Procesi', while researching asymptotically integrable partial differential equations. In addition, this third order nonlinear modified dispersive Degasperis-Procesi equation, is also considered for the modeling of shallow water dynamics. Because of these properties, this equation is centre of attraction for many researchers.

    Due to its local nature, model (1.1) can not describe the entire memory effect of the system. Thus, in order to involve whole memory of the system, we modify the model (1.1) by changing ordinary time derivative to the Katugampola fractional derivative in the Caputo sense.

    In this research work, we are considering the nonlinear time fractional modified Degasperis-Procesi (FMDP) equation that models the unidirectional propagation of two-dimensional shallow water waves over a flat plate. Hence, FMDP equation associated with the Caputo-Katugampola fractional derivative is given as

    KCaDα,ρtv(x,t)vxxt+4v2vx=3vxvxx+vvxxx, (1.2)

    with initial condition

    v0(x,t)=v(x,0)=(x)=158sech2(x2). (1.3)

    In last few decades, many researchers and mathematicians have taken a great interest in the study of fractional calculus and its application areas, like physical sciences, chemistry, engineering, life sciences, etc. The main reason behind their interest in fractional calculus is that, these fractional order models give more accurate results in comparison to the integer order models. Many definitions of fractional calculus, fractional order derivatives (integrals) and their various properties are available [2,3,4,5,6]. Usually, it is tough to obtain the exact solutions of fractional order differential equations. So, several numerical and analytical methods are available to obtain the approximate solution of a differential equation of fractional order. Many analytical and numerical methods are given to obtain the more efiicient and approximate results of FMDP equation. Some of them are: homotopy perturbation technique applied by Zhang et al. [7] to get the solitary wave solution of modified Degasperis-Procesi and Camassa-Holm equations, variational homotopy perturbation method (VHPM) given by Yousif et al. [8] is a coupling of variational iteration method and the homotopy perturbation approach. Gupta et al. [9] obtained the approximate analytical solution of modified fractional Degasperis-Procesi equation by using the homotopy perturbation technique. Abourabia et al. [10] gave the analytical solutions of Camassa-Holm and Degasperis-Procesi equations by employing three different methods, which are the Cole-Hopf method, the Schwarzian derivatives method and the factorization method, the q-homotopy analysis sumudu transform method (q-HASTM) applied by Dubey et al. [11] to examine the results of fractional modified Degasperis-Procesi equation. Singh et al. [12] employed homotopy analysis transform method (HATM) to determine the solution of fractional fish farm model. An efficient computational approach, namely q-homotopy analysis transform method (q-HATM), implemented by Singh et al. [13] to analyze the local fractional Poisson equation. The homotopy perturbation sumudu transform method (HPSTM) is used by Goswami et al. [14] to obtain the solution of time-fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system.

    In this research work, we are studying the FMDP equation by employing two techniques. One is q-homotopy analysis generalized transform method (q-HAGTM), which is a graceful coupling of q-homotopy analysis method [15], generalized Laplace transform (LT) [16] and homotopy polynomials, and the another is homotopy perturbation generalized transform method (HPGTM). The HPGTM is a mixture of the homotopy perturbation method [17], generalized LT and He's polynomials [18]. Reason behind the application of these techniques is the potential of combining two powerful computational methods for analysing nonlinear fractional order equations.

    The current article is organized as follows: Section 2 contains some preliminary definitions. In section 3, the elementary procedure of both analytic methods (q-HAGTM and HPGTM) is mentioned. In section 4, FMDP equation is analyzed by using q-HAGTM and HPGTM. Section 5 is devoted to numerical results. Lastly, section 6 presents the conclusion of this research work.

    Here, we present some definitions and results related to fractional operators and generalized LT [4,16,19,20,21,22,23,24].

    Definition 2.1. The Caputo derivative [4] of non-integer order α of function g(t) is given as follows

    (CaDαtg)(t)=1Γ(1α)ta(tw)αg(w)dw, where 0<α1. (2.1)

    Definition 2.2. The Caputo-Hadamard derivative [20] of fractional order α of function g(t) is defined as follows

    (CHaDαtg)(t)=1Γ(1α)ta(logtw)αδg(w)dw, (2.2)

    where δ represents differential operator and is defined as δ=tddt.

    Definition 2.3. The Katugampola derivative of fractional order α of function g(t) in Caputo type [22] is given as

    (KCaDα,ρtg)(t)=1Γ(1α)ta(tρwρρ)αγg(w)duw1ρ, (2.3)

    where γ represents differential operator and is given as γ=t1ρddt.

    As, we can observe that if ρ=1, then the Katugampola fractional derivative given by Eq (2.3) reduces to the Caputo derivative of fractional order α and if ρ approaches to 0, then the fractional derivative given by Eq (2.3) reduces to the Caputo-Hadamard derivative of fractional order α.

    Definition 2.4. Let g,h:[a,)R be two real-valued functions in such a way that h(t) is continuous and h(t)>0 on [a,), then generalized LT [16] of function g(t) is defined as

    Lh{g(t)}(s)=aes(h(t)h(a))g(t)h(t)dt, (2.4)

    here, s is used as generalized LT parameter.

    On putting h(t)=t and a=0 in Eq (2.4), the generalized LT reduces to the classical LT and if we put h(t)=tρρ and a=0 then the generalized LT becomes the ρ-LT and is defined as

    Ltρρ{g(t)}(s)=0estρρg(t)dtt1ρ. (2.5)

    In this paper, we are using generalized LT given by Eq (2.5).

    Definition 2.5. Generalized LT of Katugampola derivative of fractional order in Caputo type [22] is given as follows

    Ltρρ{(KCaDα,ρtg)(t)}(s)=sαLtρρ{g(t)}(s)sα1g(0). (2.6)

    To illustrate basic working plan of the first implemented analytical scheme, consider a non-homogeneous nonlinear fractional differential equation

    KCaDα,ρtv(x,t)+Rv(x,t)+Nv(x,t)=ϕ(x,t),m1<αm,mN, (3.1)

    with initial condition v(x,0)=(x) for any xR, here, v(x,t) is a function of x and t, KCaDα,ρt is the Katugampola fractional derivative of order α, R is bounded linear operator of x and t. The general nonlinear operator is presented by N, which is Lipschitz continuous and ϕ(x,t) is a source term.

    Using generalized LT on Eq (3.1), we get

    Ltρρ[KCaDα,ρtv(x,t)]+Ltρρ[Rv(x,t)+Nv(x,t)]=Ltρρ[ϕ(x,t)]. (3.2)

    Now, on utilizing the generalized LT of Katugampola derivative of fractional order in Caputo type, we get

    sαLtρρ[v(x,t)](s)sα1v(x,0)+Ltρρ[Rv(x,t)+Nv(x,t)]Ltρρ[ϕ(x,t)]=0. (3.3)

    On simplifying the Eq (3.3), we get

    Ltρρ[v(x,t)](s)s1v(x,0)+sα[Ltρρ[Rv(x,t)+N(x,t)]]sαLtρρ[ϕ(x,t)]=0. (3.4)

    The nonlinear operator can be written in the following manner

    N[ψ(x,t;q)]=Ltρρ[ψ(x,t;q)]s1ψ(x,0;q)+sα[Ltρρ[Rψ(x,t;q)+Nψ(x,t;q)]Ltρρ[ϕ(x,t)]]. (3.5)

    In Eq (3.5) ψ(x,t;q) represents a function of x,t and q, also q is an embedding parameter s.t. q[0,1n], where n1. Now the homotopy can be developed in this way

    (1nq)Ltρρ[ψ(x,t;q)v0(x,t)]=qN[ψ(x,t;q)], (3.6)

    where, Ltρρ represents the generalized LT operator, v0(x,t) is an initial approximation of v(x,t), ψ(x,t;q) is an unknown function and is a nonzero auxiliary parameter. Moreover, it may be clarified that, by substituting the values of embedding parameter q=0 as well as q=1n, it gives

    ψ(x,t;0)=v0(x,t),ψ(x,t;1n)=v(x,t), (3.7)

    respectively. Thus, we can note that as the value of q varies from 0 to 1n, the solution of ψ(x,t;q) changes from initial approximation v0(x,t) to the solution v(x,t). The Taylor's series extension of function ψ(x,t;q) is given as follows

    ψ(x,t;q)=v0(x,t)+k=1vk(x,t)qk, (3.8)

    where,

    vk(x,t)=1k!kqk{ψ(x,t;q)}|q=0. (3.9)

    If the initial guess v0(x,t), the convergence control parameter and asymptotic parameter n are described appropriately, then Eq (3.8) converges at q=1n. Then we get the following equation

    v(x,t)=v0(x,t)+k=1vk(x,t)(1n)k. (3.10)

    Result given by Eq (3.10) must be one of the solutions of studied nonlinear fractional differential equation. With the aid of Eqs (3.10) and (3.6), the governing equation can be obtained as

    vk={v1(x,t),v2(x,t),v3(x,t),...,vk(x,t)}. (3.11)

    On differentiating Eq (3.6) k-times w.r.t. q and then dividing by k!, after that putting q=0, it gives the subsequent equation

    Ltρρ[vk(x,t)χkvk1(x,t)]=Rk(vk1). (3.12)

    Employing the inverse generalized LT operator on Eq (3.12), we attain the subsequent result

    vk(x,t)=χkvk1(x,t)+L1tρρ[Rk(vk1)], (3.13)

    where χk is defined as

    χk={0,k1,n,k>1, (3.14)

    and we express the value of Rk(vk1) in an enhanced manner as

    Rk(vk1)=Ltρρ[vk1(x,t)](1χkn)[s1v(x,0)+sαLtρρ[ϕ(x,t)]]+sαLtρρ[Rvk1+Ak1]. (3.15)

    In Eq (3.15), Ak represents the homotopy polynomial [25] and is given as

    Ak=1Γ(k)[kqkNψ(x,t;q)]q=0, (3.16)

    and

    ψ(x,t;q)=ψ0+qψ1+q2ψ2+... (3.17)

    Using Eq (3.15) in Eq (3.13), we get

    vk(x,t)=(χk+)vk1(x,t)(1χkn)L1tρρ[s1v(x,0)+sαLtρρ[ϕ(x,t)]]+L1tρρ[sαLtρρ[Rvk1+Ak1]]. (3.18)

    Hence, by using Eq (3.18), the various components of vk(x,t) for k1 can be determined and we obtain q-HAGTM solution given by the subsequent equation as

    v(x,t)=k=0vk(x,t)(1n)k. (3.19)

    To demonstrate fundamental working plan of the next implemented analytical scheme, take a non-homogeneous nonlinear fractional differential equation

    KCaDα,ρtv(x,t)+Rv(x,t)+Nv(x,t)=ϕ(x,t),m1<αm,mN, (3.20)

    with initial condition v(x,0)=(x) for any xR, here, v(x,t) is a function of x and t, KCaDα,ρt is the Katugampola fractional derivative of order α, R is bounded linear operator of x and t. The general nonlinear operator is presented by N, which is Lipschitz continuous and ϕ(x,t) is a source term.

    Employing generalized LT on Eq (3.20), we get

    Ltρρ[KCaDα,ρtv(x,t)]+Ltρρ[Rv(x,t)+Nv(x,t)]=Ltρρ[ϕ(x,t)]. (3.21)

    Now, on utilizing the generalized LT of Katugampola derivative of fractional order in Caputo type, we get

    sαLtρρ[v(x,t)](s)sα1v(x,0)+Ltρρ[Rv(x,t)+Nv(x,t)]Ltρρ[ϕ(x,t)]=0. (3.22)

    On simplifying the Eq (3.22), we get

    Ltρρ[v(x,t)](s)=(x)s+1sαLtρρ[ϕ(x,t)]1sα[Ltρρ[Rv(x,t)+Nv(x,t)]]. (3.23)

    Now, operating the inverse generalized LT on Eq (3.23), we obtain the following equation

    v(x,t)=F(x,t)L1tρρ[1sα{Ltρρ[Rv(x,t)+Nv(x,t)]}], (3.24)

    where, F(x,t) stands for the term arising from the prescribed initial condition and the source term. Next, we implement the HPM

    v(x,t)=k=0pkvk(x,t). (3.25)

    The nonlinear terms can be decomposed as

    Nv(x,t)=k=0pkHk(v), (3.26)

    using the He's polynomials Hk(v) that are given as

    Hk(v0,v1,,...,vk)=1k!kpk[N(kj=0pjvj)]p=0,k=0,1,2,3,... (3.27)

    On using Eqs (3.25) and (3.26) in Eq (3.24), we get

    k=0pkvk(x,t)=F(x,t)p[L1tρρ{1sαLtρρ(Rk=0pkvk(x,t)+k=0pkHk(v))}], (3.28)

    that is a combination of generalized LT and HPM utilizing He's polynomials. Next, on equating the coefficients of like powers of p, we obtain the subsequent approximations

    p0:v0(x,t)=F(x,t), (3.29)
    p1:v1(x,t)=L1tρρ[1sαLtρρ[Rv0(x,t)+H0(v)]], (3.30)
    p2:v2(x,t)=L1tρρ[1sαLtρρ[Rv1(x,t)+H1(v)]], (3.31)
    p3:v3(x,t)=L1tρρ[1sαLtρρ[Rv2(x,t)+H2(v)]]. (3.32)

    Hence, we can find remaining components vk(x,t) completely by proceeding in the same way, and we get the series solution. Finally, the approximate solution of the problem using this technique is presented as

    v(x,t)=limKKk=0vk(x,t). (3.33)

    In this part, we check the uniqueness and convergence of the obtained solution.

    Theorem 4.1. (Uniqueness Theorem). The solution of FMDP Eq (1.2) is unique, while 0<λ<1, where, λ=(n+)+[δ3+4((A+B)δ1A+B2δ1)+3(δ2Aδ1+δ1Bδ2)+(δ3A+Bδ3)]T.

    Proof. Here, the solution of FMDP Eq (1.2) is given as

    v(x,t)=k=0vk(x,t)(1n)k, (4.1)

    where,

    vk(x,t)=(χk+)vk1(x,t)(1χkn)v0(x,t)L1tρρ[1sαLtρρ(v(k1)xxt4Ak1+3Bk1+Ck1)]. (4.2)

    Let, v and v be two different solutions of FMDP Eq (1.2) s.t. |v|A,|v|B, then using the Eq (4.2), we have

    |vv|=|(n+)(vv)L1tρρ[1sαLtρρ{(vxxtvxxt)4(v2vxv2vx)+3(vxvxxvxvxx)+(vvxxxvvxxx)}]|. (4.3)

    Now, on applying convolution theorem [22] for generalized LT, we obtain

    |vv|(n+)|vv|+t0(|vxxtvxxt|+4|v2vxv2vx|+3|vxvxxvxvxx|+|vvxxxvvxxx|)1Γ(α)[(tρwρ)ρ]α1wρ1dw.
    |vv|(n+)|vv|+t0(|3x2t(vv)|+4|vx(vv)(v+v)+v2x(vv)|+3|x(vv)2vx2+vx2x2(vv)|+|(vv)3vx3+v3x3(vv)|)1Γ(α)[(tρwρ)ρ]α1wρ1dw.
    |vv|(n+)|vv|+t0(δ3|(vv)|+4((A+B)δ1A+B2δ1)|(vv)|+3(δ2Aδ1+δ1Bδ2)|(vv)|+(δ3A+Bδ3)|(vv)|)1Γ(α)[(tρwρ)ρ]α1wρ1dw. (4.4)

    Now, on implementing mean value theorem [26,27], we get

    |vv|(n+)|vv|+(δ3|(vv)|+4((A+B)δ1A+B2δ1)|(vv)|+3(δ2Aδ1+δ1Bδ2)|(vv)|+(δ3A+Bδ3)|(vv)|)T. (4.5)

    On simplifying Eq (4.5), we obtain the subsequent relation as

    |vv|λ|vv|, (4.6)

    where, λ=(n+)+[δ3+4((A+B)δ1A+B2δ1)+3(δ2Aδ1+δ1Bδ2)+(δ3A+Bδ3)]T.

    It yields (1λ)|vv| and here 0<λ<1, hence |vv|=0 which confers that v=v.

    Therefore, we can say that the obtained solution is unique.

    Theorem 4.2. (Convergence Theorem). Let F:BB be a nonlinear mapping, where B is a Banach space, also assume that

    F(v)F(u)|vu|,v,uB. (4.7)

    Then by the fixed point theory [26,27] of Banach space, we know that F has a fixed point. Also, the sequence formed by using q-HAGTM solution having an arbitrary solution of v0,u0B, converges to the fixed point of F and

    vkvjλj1λv1v0,v,uB. (4.8)

    Proof. Let (C[I],.) be the Banach space of all continuous functions on I associated with the norm, given as f(t)=maxtI|f(t)|.

    Now, to prove the convergence of this solution, we will show that {vj} is a Cauchy sequence in the Banach space.

    vkvj=maxtI|(vkvj)|,
    vkvj=maxtI|(n+)(v(k1)v(j1))L1tρρ[1sαLtρρ{(v(k1)xxtv(j1)xxt)4(v2(k1)v(k1)xv2(j1)v(j1)x)+3(v(k1)xv(k1)xxv(j1)xv(j1)xx)+(v(k1)v(k1)xxxv(j1)v(j1)xxx)}]|.
    vkvjmaxtI[(n+)|v(k1)v(j1)|+L1tρρ{1sαLtρρ(|v(k1)xxtv(j1)xxt|+4|v2(k1)v(k1)xv2(j1)v(j1)x|+3|v(k1)xv(k1)xxv(j1)xv(j1)xx|+|v(k1)v(k1)xxxv(j1)v(j1)xxx|)}].

    Now, employing convolution theorem for generalized LT, we obtain

    vkvjmaxtI[(n+)|v(k1)v(j1)|+t0(|v(k1)xxtv(j1)xxt|+4|v2(k1)v(k1)xv2(j1)v(j1)x|+3|v(k1)xv(k1)xxv(j1)xv(j1)xx|+|v(k1)v(k1)xxxv(j1)v(j1)xxx|)1Γ(α)[(tρwρ)ρ]α1wρ1dw].
    vkvjmaxtI[(n+)|v(k1)v(j1)|+t0(δ3|v(k1)v(j1)|+4((A+B)δ1A+B2δ1)|v(k1)v(j1)|+3(δ2Aδ1+δ1Bδ2)|v(k1)v(j1)|+(δ3A+Bδ3)|v(k1)v(j1)|)1Γ(α)[(tρwρ)ρ]α1wρ1dw].

    Now, applying mean value theorem, we obtain

    vkvjmaxtI[(n+)|v(k1)v(j1)|+(δ3|v(k1)v(j1)|+4((A+B)δ1A+B2δ1)|v(k1)v(j1)|+3(δ2Aδ1+δ1Bδ2)|v(k1)v(j1)|+(δ3A+Bδ3)|v(k1)v(j1)|)T],

    then we have

    vkvjλvk1vj1.

    Setting k=j+1, it gives

    vj+1vjλvjvj1λ2vj1vj2...λjv1v0.

    Using triangular inequality, we have

    vkvjvj+1vj+vj+2vj+1+...+vkvk1[λj+λj+1+...+λk1]v1v0λj[1+λ+λ2+...+λkj1]v1v0λj(1λkj11λ)v1v0.

    As 0<λ<1, so 1λkj1<1, then we have

    vkvjλj1λv1v0.

    Since v1v0<, so as k then vkvj0.

    Hence, the sequence {vj} is convergent as it is a Cauchy sequence in C[I].

    The FMDP equation associated with Katugampola fractional derivative is given as

    KCaDα,ρtv(x,t)vxxt+4v2vx=3vxvxx+vvxxx, (5.1)

    with initial condition

    v0(x,t)=v(x,0)=(x)=158sech2(x2). (5.2)

    The exact solution [11] of standard modified Degasperis-Procesi equation obtained by substituting α=1 in Eq (5.1) is given as

    v(x,t)=158sech2(x25t4). (5.3)

    Now, by employing generalized LT on Eq (5.1) and using initial approximation given by Eq (5.2), we obtain

    Ltρρ[v(x,t)]s1(x)sαLtρρ[vxxt]+4sαLtρρ[v2vx]3sαLtρρ[vxvxx]sαLtρρ[vvxxx]=0. (5.4)

    Now, the nonlinear operator is given as follows

    N[ψ(x,t;q)]=Ltρρ[ψ(x,t;q)]1s(x)sαLtρρ[ψxxt(x,t;q)]+4sαLtρρ[ψ2(x,t;q)ψx(x,t;q)]3sαLtρρ[ψx(x,t;q)ψxx(x,t;q)]sαLtρρ[ψ(x,t;q)ψxxx(x,t;q)], (5.5)

    and the value of Rk(vk1) is given as

    Rk(vk1(x,t))=Ltρρ[vk1(x,t)](1χkn)[1s(x)]sαLtρρ[v(k1)xxt4A(k1)+3B(k1)+C(k1)]. (5.6)

    Now, making use of the initial approximation (x)=158sech2(x2) and iterative formula given by Eq (3.13), we attain the subsequent iterative terms of the approximate solution

    v1(x,t)=[42(x)(x)3(x)(x)(x)(x)]1Γ(α+1)(tρρ)α. (5.7)

    Hence, performing in the similar way, we can find rest of the components vk,k2, and approximate solution using qHAGTM is obtained.

    Consequently, q-HAGTM solution is given as

    v(x,t)=limKKk=0vk(x,t)(1n)k. (5.8)

    In this part, we are finding the approximate solution of FMDP Eq (5.1) with an initial guess given by Eq (5.2) using HPGTM.

    Employing generalized LT on both sides of Eq (5.1) and utilizing initial guess (5.2), we get

    Ltρρ[v(x,t)]=1s(x)+1sαLtρρ[vxxt4v2vx+3vxvxx+vvxxx]. (5.9)

    Now, operating the inverse generalized LT on Eq (5.9), we obtain

    v(x,t)=(x)+L1tρρ[1sαLtρρ[vxxt4v2vx+3vxvxx+vvxxx]]. (5.10)

    Employing HPM, we get subsequent equation

    k=0pkvk(x,t)=(x)+p[L1tρρ{1sαLtρρ(k=0pk(vk)xxt4k=0pkAk(v)+3k=0pkBk(v)+k=0pkCk(v))}], (5.11)

    where Ak(v),Bk(v) and Ck(v) are He's polynomials, which represent the nonlinear terms.

    On equating the coefficients of like powers of p, we get

    p0:v0(x,t)=(x), (5.12)
    p1:v1(x,t)=L1tρρ[1sαLtρρ{(v0)xxt4A0(v)+3B0(v)+C0(v)}]=[42(x)(x)3(x)(x)(x)(x)]1Γ(α+1)(tρρ)α.

    Hence, performing in the similar way, we can find rest of the components vk,k2 and approximate solution using HPGTM is obtained.

    Consequently, HPGTM solution is given as

    v(x,t)=limKKk=0vk(x,t). (5.13)

    In this part, we analyze the solutions of FMDP equation obtained by q-HAGTM and HPGTM. The numerical simulation of the discussed problem is performed for numerous values of the time variable t, space variable x and fractional order α. The given table shows the comparative study of solutions attained by implemented techniques versus exact solution. Table 1 shows that approximate solutions obtained by implemented techniques are quite close to their exact solution. The outcomes of this numerical simulation are presented in the form of Figures 114. Figures 14 represent the behaviour of solution v(x,t) obtained by q-HAGTM and Figures 58 represent the behaviour of solution v(x,t) obtained by HPGTM with respect to x,t and for distinct values of α. Figure 9 is plotted for the exact solution of the classical modified Degasperis-Procesi equation. Figure 10 (for q-HAGTM) and Figure 11 (for HPGTM) represent the response of v(x,t) w.r.t. time variable for various values of α. Figure 12 (for q-HAGTM) and Figure 13 (for HPGTM) depict the aspect of v(x,t) w.r.t. space variable. Figure 14 expresses the n-curves for q-HAGTM solution at various values of α.

    Table 1.  Comparative study of exact solution and obtained solutions for v(x,t) when α=1,=1 and n=1.
    (x,t) Exact Solution Approximate Approximate Absolute Error
    (q-HAGTM) (HPGTM)
    (8, 0.05) -0.002848800949 -0.002515546006 -0.002515546006 0.000333254943
    (9, 0.05) -0.001048518941 -0.0009255163340 -0.0009255163340 0.0001230026070
    (10, 0.05) -0.0003857967557 -0.0003404917396 -.0003404917396 0.0000453050161
    (8, 0.1) -0.003227787763 -0.002516809482 -0.002516809482 0.000710978281
    (9, 0.1) -0.001188083378 -0.0009256875444 -0.0009256875444 0.0002623958336
    (10, 0.1) -0.0004371590085 -0.0003405149212 -0.0003405149212 0.0000966440873
    (8, 0.2) -0.004143548046 -0.002519336432 -0.002519336432 0.001624211614
    (9, 0.2) -0.001525391967 -0.0009260299653 -0.0009260299653 0.0005993620017
    (10, 0.2) -0.0005613046914 -0.0003405612846 -0.0003405612846 0.0002207434068

     | Show Table
    DownLoad: CSV
    Figure 1.  The surface of v(x,t) for q-HAGTM solution w.r.t. x and t for α=1.
    Figure 2.  The surface of v(x,t) for q-HAGTM solution w.r.t. x and t for α=0.75.
    Figure 3.  The surface of v(x,t) for q-HAGTM solution w.r.t. x and t for α=0.50.
    Figure 4.  The surface of v(x,t) for q-HAGTM solution w.r.t. x and t for α=0.25.
    Figure 5.  The surface of v(x,t) for HPGTM solution w.r.t. x and t for α=1.
    Figure 6.  The surface of v(x,t) for HPGTM solution w.r.t. x and t for α=0.75.
    Figure 7.  The surface of v(x,t) for HPGTM solution w.r.t. x and t for α=0.50.
    Figure 8.  The surface of v(x,t) for HPGTM solution w.r.t. x and t for α=0.25.
    Figure 9.  The surface of Exact solution v(x,t) w.r.t. x and t.
    Figure 10.  Response of q-HAGTM solution v(x,t) w.r.t. t for distinct values of α.
    Figure 11.  Response of HPGTM solution v(x,t) w.r.t. t for distinct values of α.
    Figure 12.  Nature of v(x,t) w.r.t. x at t=0.005.
    Figure 13.  Nature of v(x,t) w.r.t. x at t=0.005.
    Figure 14.  n-curves for distinct values of α at x=1, t=0.05 and =1.

    In this current work, we successfully implemented two techniques, namely q-HAGTM and HPGTM, to analyze the approximate series solution of FMDP equation. Graphical representation of the obtained results indicates that the implemented techniques are powerful and efficient for solving FMDP equation. The comparative study of approximate solutions with exact solution shows the accuracy and applicability of the applied techniques. Hence, we can conclude that the applied methods are efficient to solve such types of problems arising in physical sciences.

    The authors declare that there is no conflict of interests.



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