International Journal of Mathematical, Engineering and Management Sciences Vol. 5, No. 2, 272-282, 2020 https://doi.org/10.33889/IJMEMS.2020.5.2.022 272 7 th -Order Caudrey-Dodd-Gibbon Equation and Fisher-Type Equation by Homotopy Analysis Method Ankita Sharma Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Saharanpur Campus, Saharanpur, India. Corresponding author: ank02dpt@iitr.ac.in Rajan Arora Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Saharanpur Campus, Saharanpur, India. E-mail: rajanfpt@iitr.ac.in (Received June 27, 2019; Accepted December 16, 2019) Abstract In this paper, we first describe the methodology of the Homotopy Analysis Method (HAM) which is an analytical technique and then employ it to some of the non-linear problems which are used in different fields of sciences like plasma physics, fluid dynamics, laser optics, biology, chemical kinetics, nucleation kinetics, physiology, etc. Approximate series solutions have been obtained and the results are compared with the closed form solutions of the equations, which show that this technique gives high accurate results. HAM is a reliable technique, easy to use and is widely applicable to a large class of non-linear differential equations. MATHEMATICA software package has been used for computations. Keywords- Homotopy analysis method, 7 th -order Caudrey-Dodd-Gibbon equation, Fisher-type equation. 1. Introduction Generally, the non-linear equations are not very easy to solve explicitly. Some techniques like perturbation techniques which involve some parameter (small or large), also known as the perturbation quantity, play an important role in solving the non-linear differential equations to some extent. But like other non-linear analytical methods, these perturbation techniques too, have their own drawbacks, viz., the dependency of these techniques on the perturbation quantities, which restricts the applications of them to a wide class of non-linear differential equations because it is not necessary that there always exists a perturbation quantity in every non-linear problem. The other drawback is that it seems to be a special art that requires a special technique in determining the small parameters. If an appropriate value of the small parameter is chosen, it will lead to the ideal results but if not then it may create a critical problem. Moreover, it has been observed that in most of the cases the approximated solutions obtained by these perturbation techniques are valid only for the smaller values of the perturbation quantities. Also, as the non- linearity becomes stronger, the analytic approximations start breaking down, thus, makes the perturbation approximations valid only for non-linear problems with weak non-linearity. Thus, in view of all above limitations one can conclude that it all arises due to the assumption of that small parameter which is known as the perturbation quantity. In order to overcome these limitations there were developed some non-perturbation techniques, e.g., Sinc-Collocation method (Zakeri and Navab, 2010), Sinc-Galerkin method (Rashidinia and Nabati, 2013), finite element method