Some new solutions of the Caudrey\u2013Dodd\u2013Gibbon (CDG) equation using the conformable derivative

Naveed Ahmed,Sadaf Bibi,Imran Faisal,Umar Khan,Muhammad Rafiq,Syed Tauseef Mohyud-Din

doi:10.1186/s13662-019-2030-7

Abstract

New exact solutions of the space–time conformable Caudrey–Dodd–Gibbon (CDG) equation have been derived by implementing the conformable derivative. The generalized Riccati equation mapping method is applied to figure out twenty-seven forms of exact solutions, which are soliton, rational, and periodic ones. Also, for some suitable values of parameters, the exact solutions are found, namely dark, bell type, periodic, soliton, singular soliton, and several others, by using the conformable derivative. These types of solutions have not been proclaimed so far. 2D and 3D graphical patterns of some solutions are also given for clarification of physical features. The conformable derivative is one of the excellent choices to solve the nonlinear conformable problems arising in theory of solitons and many other areas. The results are new and very interesting for the large community of researchers working in the field of mathematics and mathematical physics.

Highlights

1 Introduction Fractional calculus [1, 2] containing differential equations of fractional order in numerous physical phenomena has put a revolutionary impact since these particular equations reflect generalization of evolution equations of integer order
Approximate and numerical techniques have been used for finding exact solutions of nonlinear differential equations including homotopy analysis transform method [3], homotopy analysis samadu transform method [4], fractional homotopy analysis transform method [5], the differential transform method [6], F-expansion method [7], first integral method [8], fractional sub-equation method [9], sine-cosine method [10], Bibi et al Advances in Difference Equations
We will find the relationship between the exact solutions obtained by the generalized Riccati equation mapping method with Atangana–Baleanu fractional derivative applied on (CDG)

Summary

Introduction

Fractional calculus [1, 2] containing differential equations of fractional order in numerous physical phenomena has put a revolutionary impact since these particular equations reflect generalization of evolution equations of integer order. By studying the generalized Riccati equation mapping method [46], the conformable derivative of Atangana’s version finds the hyperbolic, rational, and trigonometric solutions of (CDG) equation, not brought to light before in the literature so far. Section gives some new solutions of traveling wave form of the space–time conformable Caudrey–Dodd–Gibbon (CDG) equation. 2. We have implemented the same mapping for attaining new exact soliton solutions to the time and space conformable Caudrey–Dodd–Gibbon (CDG) equation in Sect. 4 Discussion and physical interpretation This section gives visualization of our obtained solutions of the space–time conformable Caudrey–Dodd–Gibbon (CDG) equation. Using the generalized Riccati equation mapping (GREM) method, we came up with the solitary wave solutions of the (CDG) equation These are generalized and closed form solutions of traveling wave type.

Comparison of the obtained solutions

Conclusion

Methods