1.
Introduction
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
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
with initial condition
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.
2.
Some preliminary definitions
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
Definition 2.2. The Caputo-Hadamard derivative [20] of fractional order α of function g(t) is defined as follows
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
where γ represents differential operator and is given as γ=t1−ρddt.
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
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
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
3.
Elementary description of analytical methods
3.1. q-Homotopy analysis generalized transform method (q-HAGTM)
To illustrate basic working plan of the first implemented analytical scheme, consider a non-homogeneous nonlinear fractional differential equation
with initial condition v(x,0)=℘(x) for any x∈R, 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
Now, on utilizing the generalized LT of Katugampola derivative of fractional order in Caputo type, we get
On simplifying the Eq (3.3), we get
The nonlinear operator can be written in the following manner
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 n≥1. Now the homotopy can be developed in this way
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 ℏ 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
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
where,
If the initial guess v0(x,t), the convergence control parameter ℏ and asymptotic parameter n are described appropriately, then Eq (3.8) converges at q=1n. Then we get the following equation
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
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
Employing the inverse generalized LT operator on Eq (3.12), we attain the subsequent result
where χk is defined as
and we express the value of Rk(→vk−1) in an enhanced manner as
In Eq (3.15), Ak represents the homotopy polynomial [25] and is given as
and
Using Eq (3.15) in Eq (3.13), we get
Hence, by using Eq (3.18), the various components of vk(x,t) for k≥1 can be determined and we obtain q-HAGTM solution given by the subsequent equation as
3.2. Homotopy perturbation generalized transform method (HPGTM)
To demonstrate fundamental working plan of the next implemented analytical scheme, take a non-homogeneous nonlinear fractional differential equation
with initial condition v(x,0)=℘(x) for any x∈R, 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
Now, on utilizing the generalized LT of Katugampola derivative of fractional order in Caputo type, we get
On simplifying the Eq (3.22), we get
Now, operating the inverse generalized LT on Eq (3.23), we obtain the following equation
where, F(x,t) stands for the term arising from the prescribed initial condition and the source term. Next, we implement the HPM
The nonlinear terms can be decomposed as
using the He's polynomials Hk(v) that are given as
On using Eqs (3.25) and (3.26) in Eq (3.24), we get
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
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
4.
Analysis of uniqueness and convergence of the solution
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+ℏ)+ℏ[δ′3+4((A+B)δ1A+B2δ1)+3(δ2Aδ1+δ1Bδ2)+(δ3A+Bδ3)]T.
Proof. Here, the solution of FMDP Eq (1.2) is given as
where,
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
Now, on applying convolution theorem [22] for generalized LT, we obtain
Now, on implementing mean value theorem [26,27], we get
On simplifying Eq (4.5), we obtain the subsequent relation as
where, λ=(n+ℏ)+ℏ[δ′3+4((A+B)δ1A+B2δ1)+3(δ2Aδ1+δ1Bδ2)+(δ3A+Bδ3)]T.
It yields (1−λ)|v−v∗| and here 0<λ<1, hence |v−v∗|=0 which confers that v=v∗.
Therefore, we can say that the obtained solution is unique.
Theorem 4.2. (Convergence Theorem). Let F:B→B be a nonlinear mapping, where B is a Banach space, also assume that
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,u0∈B, converges to the fixed point of F and
Proof. Let (C[I],‖.‖) be the Banach space of all continuous functions on I associated with the norm, given as ‖f(t)‖=maxt∈I|f(t)|.
Now, to prove the convergence of this solution, we will show that {vj} is a Cauchy sequence in the Banach space.
Now, employing convolution theorem for generalized LT, we obtain
Now, applying mean value theorem, we obtain
then we have
Setting k=j+1, it gives
Using triangular inequality, we have
As 0<λ<1, so 1−λk−j−1<1, then we have
Since ‖v1−v0‖<∞, so as k→∞ then ‖vk−vj‖→0.
Hence, the sequence {vj} is convergent as it is a Cauchy sequence in C[I].
5.
Solution of fractional Degasperis-Procesi equation
5.1. Solution by applying q-HAGTM
The FMDP equation associated with Katugampola fractional derivative is given as
with initial condition
The exact solution [11] of standard modified Degasperis-Procesi equation obtained by substituting α=1 in Eq (5.1) is given as
Now, by employing generalized LT on Eq (5.1) and using initial approximation given by Eq (5.2), we obtain
Now, the nonlinear operator is given as follows
and the value of Rk(→vk−1) is given as
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
Hence, performing in the similar way, we can find rest of the components vk,k≥2, and approximate solution using qHAGTM is obtained.
Consequently, q-HAGTM solution is given as
5.2. Solution by applying HPGTM
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
Now, operating the inverse generalized LT on Eq (5.9), we obtain
Employing HPM, we get subsequent equation
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
Hence, performing in the similar way, we can find rest of the components vk,k≥2 and approximate solution using HPGTM is obtained.
Consequently, HPGTM solution is given as
6.
Numerical results
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 1–14. Figures 1–4 represent the behaviour of solution v(x,t) obtained by q-HAGTM and Figures 5–8 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.
7.
Conclusions
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.
Conflict of interest
The authors declare that there is no conflict of interests.