A reliable analytical technique for fractional Caudrey-Dodd-Gibbon equation with Mittag-Leffler kernel

Abstract The pivotal aim of the present work is to find the solution for fractional Caudrey-Dodd-Gibbon (CDG) equation using q-homotopy analysis transform method (q-HATM). The considered technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the projected fractional-order model. In order to illustrate and validate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The obtained results elucidate that, the considered algorithm is easy to implement, highly methodical as well as accurate and very effective to examine the nature of nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.


Introduction
Fractional calculus (FC) was originated in Newton's time, but lately, it fascinated the attention of many scholars. From the last thirty years, the most intriguing leaps in P. Veeresha scienti c and engineering applications have been found within the framework of FC. The concept of the fractional derivative has been industrialized due to the complexities associated with a heterogeneities phenomenon. The fractional di erential operators are capable to capture the behaviour of multifaceted media having di usion process. It has been a very essential tool and many problems can be illustrated more conveniently and more accurately with differential equations having arbitrary order. Due to the swift development of mathematical techniques with computer software's, many researchers started to work on generalised calculus to present their viewpoints while analysing many complex phenomena.
Numerous pioneering directions are prescribed for the diverse de nitions of fractional calculus by many senior researchers, and which prearranged the foundation [1][2][3][4][5][6]. Calculus with fractional order is associated to practical ventures and it extensively employed to nanotechnology [7], human diseases [8,9], chaos theory [10], and other areas . The numerical and analytical solution for these equations illustrating these models have an impartment role in portraying nature of nonlinear problems ascends in connected areas of science.
In order to demonstrate the e ciency of the future scheme, we consider fth-order nonlinear CGD equation of the form [35,36] The above equation is a class of KdV equation and further, it possesses distinct and diverse properties. The CGD equation is also familiar as Sawada-Kotera equation [37]. Due to the importance of the considered problem, it has been magnetized the attention of many researchers from diverse areas. In 1984, Weiss illustrated the Painleve' property for the Eq. (1) [38]. It has been proved that it has a strong physical background in uid [39] and also has Nsoliton solutions [40].
In the present scenario, many important and nonlinear models are methodically and e ectively analysed with the help of fractional calculus. There have been diverse de nitions are suggested by many senior research scholars, for instance, Riemann, Liouville, Caputo and Fabrizio. However, these de nitions have their own limitations. The Riemann-Liouville derivative is unable to explain the importance of the initial conditions; the Caputo derivative has overcome this shortcoming but is impotent to explain the singular kernel of the phenomena. Later, in 2015 Caputo and Fabrizio defeated the above obliges [41], and many researchers are considered this derivative in order to analyse and nd the solution for diverse classes of nonlinear complex problems. But some issues were pointed out in CF derivative, like non-singular kernel and nonlocal, these properties are very essential in describing the physical behaviour and nature of the nonlinear problems. In 2016, Atangana and Baleanu introduced and natured the novel fractional derivative, namely AB derivative. This novel derivative de ned with the aid of Mittag-Le er functions [42]. This fractional derivative buried all the above-cited issues and helps us to understand the natural phenomena systematically and e ectively.
In the present framework, we consider the fractional Caudrey-Dodd-Gibbon (FCDG) equation of the form where α is fractional-order and de ned with AB fractional operator. The fractional-order is introduced in order to incorporate the memory e ects and hereditary consequence in the phenomenon and these properties aid us to capture essential physical properties of the nonlinear problems.
Recently, many mathematicians and physicists developed very e ective and more accurate methods in order to nd and analyse the solution for complex and nonlinear problems arisen in science and engineering. In connection with this, the homotopy analysis method (HAM) proposed by Chinese mathematician Liao Shijun [43]. HAM has been pro tably and e ectively applied to study the behaviour of nonlinear problems without perturbation or linearization. But, for computational work, HAM requires huge time and computer memory. To overcome this, there is an essence of the amalgamation of a considered method with wellknown transform techniques.
In the present investigation, we put an e ort to nd and analysed the behaviour of the solution obtained for the FCDG equation by applying q-HATM. The future algorithm is the combination of q-HAM with LT [44]. Since q-HATM is an improved scheme of HAM; it does not require discretization, perturbation or linearization. Recently, due to its reliability and e cacy, the considered method is exceptionally applied by many researchers to understand physical behaviour diverse classes of complex problems [45][46][47][48][49][50][51][52][53]. The projected method o ers us more freedom to consider the diverse class of initial guess and the equation type complex as well as nonlinear problems; because of this, the complex NDEs can be directly solved. The novelty of the future method is it aids a modest algorithm to evaluate the solution and it natured by the homotopy and axillary parameters, which provides the rapid convergence in the obtained solution for a nonlinear portion of the given problem. Meanwhile, it has prodigious generality because it plausibly contains the results obtained by many algorithms like q-HAM, HPM, ADM and some other traditional techniques. The considered method can preserve great accuracy while decreasing the computational time and work in comparison with other methods. The considered nonlinear problem recently fascinated the attention of researchers from di erent areas of science. Since FCDG equation plays a signi cant role in portraying several nonlinear phenomena and also which are the generalizations of diverse complex phenomena, many authors nd and analysed the solution using analytical as well as numerical schemes [54][55][56][57][58][59][60][61].

Preliminaries
Recently, many authors considered these derivatives to analyse a diverse class of models in comparison with classical order as well as other fractional derivatives, and they prove that AB derivative is more e ective while analysing the nature and physical behaviour of the models [62][63][64][65]. Here, we de ne the basic notion of Atangana-Baleanu derivatives and integrals [42].
De nition 2. The AB derivative of fractional order for a De nition 3. The fractional AB integral related to the nonlocal kernel is de ned by The following Lipschitz conditions respectively hold true for both Riemann-Liouville and AB derivatives de ned in Eqs. (3) and (4) [42], has a unique solution and which is dened as [42]

Fundamental idea of the considered scheme
In this segment, we consider the arbitrary order di erential equation in order to demonstrate the fundamental solution procedure of the projected algorithm where ABC signi es the source term, R and N respectively denotes the linear and nonlinear di erential operator. On using the LT on Eq. (10), we have after simpli cation The non-linear operator is de ned as follows Here, φ(x, t; q) is the real-valued function with respect to x, t and q ∈ , n . Now, we de ne a homotopy as follows (14) where L is signifying LT, q ∈ , n (n ≥ ) is the embedding parameter and ̸ = is an auxiliary parameter. For q = and q = n , the results are given below hold true Thus, by intensifying q from to n , the solution φ(x, t; q) varies from v (x, t) to v (x, t). By using the Taylor theorem near to q, we de ning φ (x, t; q) in series form and then we get where The series (14) converges at q = n for the proper chaise of v (x, t) , n and . Then Now, m-times di erentiating Eq. (15) with q and later dividing by m! and then putting q = , we obtain where the vectors are de ned as On applying inverse LT on Eq. (19), one can get where and km = , m ≤ , n, m > .
In Eq. (22), Hm signi es homotopy polynomial and presented as follows and φ (x, t; q) = φ + qφ + q φ + . . . . By the aid of Eqs. (21) and (22), one can get Using the Eq. (25), one can get the series of vm (x, t). Lastly, the series q-HATM solution is de ned as

Solution for FCDG equation
In order to present the solution procedure and e ciency of the future scheme, in this segment, we consider FCDG equation of fractional order. Further by the help of obtained results, we made an attempt to capture the behaviour of q-HATM solution for di erent fractional order. By the help of Eq. (2), we have with initial condition u (x, ) = µ sech (µx).
Taking LT on Eq. (27) and then using the Eq. (28), we get The non-linear operator N is presented with the help of future algorithm as below The deformation equation of m-th order by the help of q-HATM at H(x, t) = , is given as follows where On applying inverse LT on Eq. (31), it reduces to On simplifying the above equation systematically by using u (x, t) = s µ sech (µx) we can evaluate the terms of the series solution

Existence of solutions for the future model
Here, we considered the xed-point theorem in order to demonstrate the existence of the solution for the considered model. Since the considered model cited in the system (27) is non-local as well as complex; there are no particular algorithms or methods exist to evaluate the exact solutions. However, under some particular conditions the existence of the solution assurances. Now, the system (27) is considered as follows: The foregoing system is transformed to the Volterra integral equation using the Theorem 2, and which as follows Theorem 3. The kernel G satis es the Lipschitz condition and contraction if the condition ≤ δ + δ a + b + ab + δ < holds.
Proof. In order to prove the required result, we consider the two functions u and u , then where δ is the di erential operator. Since u and u are bounded, we have u (x, t) ≤ a and u (x, t ) ≤ b.
Putting η = δ + δ a + b + ab + δ in the above inequality, then we have t ) . (38) This gives, the Lipschitz condition is obtained for G . Further, we can see that if ≤ δ + δ a + b + ab + δ < , then it implies the contraction. The recursive form of Eq. (36) de ned as follows The associated initial condition is The successive di erence between the terms is presented as follows Notice that By using Eq. (38) after applying the norm on the Eq. (41), one can get We prove the following theorem by using the above result. Theorem 4. The solution for the system (27) will exist and unique if we have speci c t then Proof. Let us consider the bounded function u (x, t) satisfying the Lipschitz condition. Then, by Eqs. (42) and (44), we have Therefore, the continuity as well as existence for the obtained solutions is proved. Subsequently, in order to show the Eq. (44) is a solution for the Eq. (29), we consider In order to obtain require a result, we consider Similarly, at t we can obtain As n approaches to ∞, we can see that form Eq. (50), Kn (x, t) tends to .
Next, it is a necessity to demonstrate uniqueness for the solution of the considered model. Suppose u * (x, t) be the other solution, then we have On applying norm, the Eq. (48) simpli es to On simpli cation From the above condition, it is clear that But < λ < , therefore limn, m→∞ Sn − Sm = . Hence, {Sn}is the Cauchy sequence. Similarly, we can demonstrate for the second case. This proves the required result. Theorem 6. For the series solution (26) of the Eq. (10), the maximum absolute error is presented as Proof: By the help of Eq. (56), we get This ends the proof.

Results and discussion
In this manuscript, we nd the solution for CDG equation having arbitrary order using a novel scheme namely, q-HATM with the help of Mittag-Le er law. In the present segment, we demonstrate the e ect of fractional order in the obtained solution with distinct parameters o ered by the future method. In Figures 1 to 3, the nature of q-HATM solution for di erent arbitrary order is presented in terms of 2D plots. From these plots, we can see that considered problem conspicuously depends on fractional order. In order to analyse the behaviour of obtained solution with respect to homotopy parameter ( ), the -curves are drowned for diverse µ and presented in Figure 4. In the plots, the horizontal line represents the convergence region of the q-HATM solution and these curves aid us to adjust and handle the convergence province of the solution. For an appropriate value of , the achieved solution quickly converges to the exact solution. Further, the small variation in the physical behaviour of the complex models stimulates the enormous new results to analyse and understand nature in a better and systematic manner. Moreover, from all the plots we can see that the considered method is more accurate and very e ective to analyse the considered complex coupled fractional order equations.

Conclusion
In this study, the q-HATM is applied lucratively to nd the solution for arbitrary order CDG equations. Since AB derivatives and integrals having fractional order are dened with the help of generalized Mittag-Le er function as the non-singular and non-local kernel, the present investigation illuminates the e eteness of the considered derivative. The existence and uniqueness of the obtained solution are demonstrated with the xed point hypothesis. The results obtained by the future scheme are more stimulating as compared to results available in the literature. Further, the projected algorithm nds the solution for the nonlinear problem without considering any discretization, perturbation or transformations. The present investigation illuminates, the considered nonlinear phenomena noticeably depend on the time history and the time instant and which can be pro ciently analysed by applying the concept of calculus with fractional order. The present in- vestigation helps the researchers to study the behaviour nonlinear problems gives very interesting and useful consequences. Lastly, we can conclude the projected method is extremely methodical, more e ective and very accurate, and which can be applied to analyse the diverse classes of nonlinear problems arising in science and technology. [5] Podlubny I., Fractional Di erential Equations, Academic Press, New York, 1999.