On Study of Generalized Novikov Equation by Reduced Differential Transform Method

Objectives: The object of the work is essentially to examine the generalization of NovikovPartial Differential Equations through differential transform algorithm. This work also shows that the method can allow us to construct explicit solutions highly nonlinear equations. We have also plotted the constructed solutions. Methods: We have constructed the approximate solutions of mentioned equation using a relatively new algorithm, known as reduced differential transform algorithm. Findings: It turns out that our solutions agree with the abstract findings known in key papers that we followed. Applications: Generalization of Novikov Partial Differential Equations models several physical phenomena such as shallow water flow, dynamics of enzymes in the human cells etc.


Introduction
This paper is intended to approximate an explicit solution to the following initial value problem, Where t ∈ ∞ [ , ) 0 , x ∈ and initial data will be suitably chosen from Sobelov space H 1 ( )  . There are several ways we can look at the above problem as generalized Nikov equation or as particular manifestation of following, where F is a homogeneous polynomial. This study gets its motivation from 1 where the authors studied the abstract well-posedness global weak solution of the above problem by arguing through viscosity vanishing method. Also the stability of weak solutions was proved in the case when the solutions have higher integrability. The equation was presented by 2 one of typical application of the Novikov equation is that it models shallow water flow.
Moreover, it possesses a bi-Hamiltonian structure and has ce x ct − − form of solution. Hamiltonian systems are the systems, admitting a complete sequence of first integrals. Bi Hamiltonian properties were first formulated in 3 the equation exhibits Bi-Hamiltonian structure, which means it is totallyintegrable like the Novikov equation. For more details on Novikov, CH equations and equations with Bi-Hamiltonian structure we refer to [4][5][6][7][8][9][10][11][12][13] . Now we give a breakdown of paper.

Description of Differential Transform Algorithm
This section has been devoted to give a precise description of the Reduced Differential Transform algorithm and how it works.Assume that we have a function u x t ( , ) with arguments x and t, that can expressed as the product two functions of x and t i.e., u x t f x g t ( , ) ( ) ( ) = . Then differential transform of the function u x t ( , ) can be explicitly written as, The more careful and precise definitions of transform of function u x t ( , ) is following, (cf. 14,15 ) Definition: Consider a function u x t ( , ) is C k -class with respect to time t ≥ 0 and space x ∈R . Then define the transform of u x t ( , ) as: Where theU x k ( ) can be treated as transformed u x t , ,

( )
and is essentially analogous the Taylor's coefficient in the 2D Taylor expansion. To recover the function u x t ( , ) from transformed functionsU x k ( ) , we define the following inverse of differential transform in the following manner.
Definition: Consider a function u x t ( , ) is C k -class with respect to time t ≥ 0 and space x ∈. Then define the transform of U x k ( ) , as: Or more explicitly, Next we discuss that how the above described transformation can be applied to solve the concrete nonlinear partial differential equations. Consider a nonlinear PDE in its generalized form, Subject to the initial condition: u x f x ,0 ( )= ( ) .
Here L denotes operator ∂ ∂t , Ru x t ( , ) denotes the linear part of PDE that contains the linear expressions of u and its derivatives, Nu x t ( , ) denotes the operator/ expression containing the nonlinear terms involving u and its derivatives operator, g x t ( , ) stands for an inhomogeneous term that can be treated a forcing factor in the model. Taking the differential transform of the eq. (4) leads to following recursive relation, and g x t ( , ) respectively. Hence the key computation that one need to is the computation of functions U U U 1 2 3 , ,  through recursive relation (5), by choosing , , ⊃ are found then we can write n -term approximate solution of PDE (4) as follows: Thus by increasing n more and more we get an exact solution of nonlinear PDE (4) Based on definition of the reduced differential transform algorithm following Table 1 of transformations can be proved. For the readers interested in the proofs we refer to [14][15] .

Solution of Generalized form of Novikov Equation by RDTM
For following Novikov equation, we are applying RDTM method to get an approximate solution: With initial condition chosen from On simplifying equation (8) we have By applying RDTM to equation (10), we have: (20) From the initial condition eq. (9), we solve the first, second and third partial derivatives of eq. (9), we have: Now taking k = 0 in Equation (11)

Conclusion
Novikovform of models is highly nonlinear in its structure. In this article we have constructed an explicit approximate solution to a highly nonlinear version of the generalized Novikovequation through a highly efficient algorithm knows as Reduced Differential Transform Algorithm. Our results are in agreement with some key abstract conclusions of 1 like blow up of evolution in finite times. Our work shows that Reduced differential transform algorithm is very efficient in constructing the explicit solutions of highly nonlinear problems.