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

1 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 scientific 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 differential operators are capable to capture the behaviour of multifaceted media having diffusion 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 definitions 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 11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34. 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 efficiency of the future scheme, we consider fifth-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 fluid 39 and also has N-soliton solutions 40.

In the present scenario, many important and nonlinear models are methodically and effectively analysed with the help of fractional calculus. There have been diverse definitions are suggested by many senior research scholars, for instance, Riemann, Liouville, Caputo and Fabrizio. However, these definitions 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 find the solution for diverse classes of nonlinear complex problems. But some issues were pointed out in CF derivative, like non-singular kernel and non-local, 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 defined with the aid of Mittag--Leffler functions 42. This fractional derivative buried all the above-cited issues and helps us to understand the natural phenomena systematically and effectively.

In the present framework, we consider the fractional Caudrey-Dodd-Gibbon (FCDG) equation of the form

where α is fractional-order and defined with AB fractional operator. The fractional-order is introduced in order to incorporate the memory effects 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 effective and more accurate methods in order to find 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 profitably and effectively 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 well-known transform techniques.

In the present investigation, we put an effort to find 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 efficacy, 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 offers 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 different areas of science. Since FCDG equation plays a significant role in portraying several nonlinear phenomena and also which are the generalizations of diverse complex phenomena, many authors find and analysed the solution using analytical as well as numerical schemes 54,55,56,57,58,59,60,61.

2 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 effective while analysing the nature and physical behaviour of the models 62,63,64,65. Here, we define the basic notion of Atangana-Baleanu derivatives and integrals 42.

3 Fundamental idea of the considered scheme

In this segment, we consider the arbitrary order differential equation in order to demonstrate the fundamental solution procedure of the projected algorithm

with the initial condition

where aABCDtαvx,t symbolise the AB derivative of v(x, t), f(x, t) signifies the source term, R and N respectively denotes the linear and nonlinear differential operator. On using the LT on Eq. (10), we have after simplification

The non-linear operator is defined as follows

Here, φ(x, t; q) is the real-valued function with respect to x, t and (q∈[0,1n]). Now, we define a homotopy as follows

where L is signifying LT,q∈[0,1n](n≥1) is an auxiliary parameter. For q = 0 and q=1n, the results are given below hold true

Thus, by intensifying q from 0 to 1n, the solution φ(x, t; q) varies from v₀(x, t) to v(x, t). By using the Taylor theorem near to q, we defining φ(x, t; q) in series form and then we get

where

The series (14) converges at q=1n for the proper chaise of v₀(x, t), n and ℏ. Then

Now, m-times differentiating Eq. (15) with q and later dividing by m! and then putting q = 0, we obtain

where the vectors are defined as

On applying inverse LT on Eq. (19), one can get

where

and

In Eq. (22), H_m signifies homotopy polynomial and presented as follows

and φ(x, t; q) = φ₀ + qφ₁ + q²φ₂ + ….

By the aid of Eqs. (21) and (22), one can get

Using the Eq. (25), one can get the series of v_m (x, t). Lastly, the series q-HATM solution is defined as

4 Solution for FCDG equation

In order to present the solution procedure and efficiency 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 different fractional order. By the help of Eq. (2), we have

with initial condition

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) = 1, is given as follows

where

On applying inverse LT on Eq. (31), it reduces to

On simplifying the above equation systematically by using u0x,t=1sμ2sech2μx we can evaluate the terms of the series solution

5 Existence of solutions for the future model

Here, we considered the fixed-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

6 Results and discussion

In this manuscript, we find the solution for CDG equation having arbitrary order using a novel scheme namely, q-HATM with the help of Mittag-Leffler law. In the present segment, we demonstrate the effect of fractional order in the obtained solution with distinct parameters offered by the future method. In Figures 1 to 3, the nature of q-HATM solution for different 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 (ℏ), the ℏ-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 ℏ, 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 effective to analyse the considered complex coupled fractional order equations.

Figure 1

(a) Surface of u(x, t), (b) 2D plot of u(x, t) at t = 10 at μ = 0.5, ℏ = −1, n = 1 and α = 0.5.

Figure 2

(a) Surface of u(x, t), (b) 2D plot of u(x, t) at t = 10 at μ = 0.5, ℏ = −1, n = 1 and α = 0.75.

Figure 3

(a) Surface of u(x, t), (b) 2D plot of u(x, t) at t = 10 at μ = 0.5, ℏ = −1, n = 1 and α = 1.

Figure 4

ℏ-curves for q-HATM solution with distinct α at x = 1 and t = 0.01 for (a) n = 1 and (b) n = 2.

7 Conclusion

In this study, the q-HATM is applied lucratively to find the solution for arbitrary order CDG equations. Since AB derivatives and integrals having fractional order are defined with the help of generalized Mittag-Leffler function as the non-singular and non-local kernel, the present investigation illuminates the effeteness of the considered derivative. The existence and uniqueness of the obtained solution are demonstrated with the fixed point hypothesis. The results obtained by the future scheme are more stimulating as compared to results available in the literature. Further, the projected algorithm finds 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 proficiently analysed by applying the concept of calculus with fractional order. The present investigation 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 effective and very accurate, and which can be applied to analyse the diverse classes of nonlinear problems arising in science and technology.