Solving Non Linear Differential Equations By Using A G - Homotopy Analysis Method

This paper proposes an efficient hybrid analytical method named the Generalized Homotopy Analysis Method (GHAM) for solving nonlinear differential equations. The proposed hybrid method consists of topology based Homotopy Analysis Method (HAM) and Laplace typed integral transform (G-transform). Compared with HAM, GHAM does not require the effort to perform repeated integration and differentiation, when it is applied to solve higher order differential equations. Compared with G-transform, GHAM serves as an effective approach to tackle higher-order nonlinear differential equations. The effectiveness of GHAM is illustrated through the analysis of numerical examples, and the obtained results are graphically depicted. Also, GHAM is effectively utilized to analyze the density of the forest cover incorporating various parameters, which include the seed reproduction, seed deposition, seed establishment rates, old trees due to aging and the coefficients of mortality due to space variables.


Introduction
Last few decades, a number of analytical methods have been introduced to solve nonlinear differential equations in different fields. The list of analytical methods include the Homotopy Perturbation Method(HPM), Non-Perturbation Method as well as the Homotopy Analysis Method (HAM) which provides the solutions in the form of a convergent series. Though these methods are efficient in solving nonlinear differential equations, it possess inherent limitations. The use of HPM is subject to the existence of physical parameters involved in the differential equations. The convergence region yield by the solutions is also hard to be controlled. Liao (1992) introduced the HAM as a general analytical methodology to produce the solutions in a series form pertaining to a variety of linear as well as nonlinear equations. Liao (1997) proposed a strong solution methodology using HAM for analyzing nonlinear heat transfer problems that arise in the study of plate heating with a microwave. It is interesting to mention that, the HAM controls the convergence region and the rate of approximation of the series can also be handled effectively. In contrast to other analytical techniques, the HAM offers the ICMMCMSE 2020 Journal of Physics: Conference Series 1850 (2021) 012065 IOP Publishing doi: 10.1088/1742-6596/1850/1/012065 2 advantage of ensuring the convergence of the series solution pertaining to nonlinear problems. This is achieved by appropriately determining the convergence-control parameter c 0 . Liao (2004) studied the fundamental concepts related to HAM and applied to study a variety of nonlinear equations. They encompass ordinary differential equations, differential-integral equations, differential-difference equations, partial differential equations, as well as coupled equations. The HAM is effective in approximating the solution of nonlinear problems. This is achieved through determining suitable sets of base functions as well as suitable initial conditions.
In Abbasbandy (2007), a generalized Hirota-Satsuma coupled KdV equation was tackled by using HAM. This method is able to yield more accurate results when compared with HPM as well as ADM (i.e. Adomian Decomposition Method). The HAM is implemented to handle the discrete KdV differential-difference equations by Zou et al. (2007). On the other hand, the authors Nasabzadeh and Toutounian (2013), Lu and Liu (2014), Brociek et al (2016), Hariharan (2017), applied the HAM to undertake engineering related problems that involve variety of linear and nonlinear differential and integral equations.
Kim (2017) developed a Laplace typed integral transform that generalizes variety of transformation methods, e.g. Sumudu and Elzaki as well as Laplace methods. Recently, Abazari and Ganji (2011) extended two-dimensional differential transform method and their reduced form, by presenting and proving some theorems, to obtain the solution of partial differential equations (PDEs) with proportional delay. Ragab  Even though the HAM has many advantages, it is unavoidable to reduce the number of iterative differentiations and integrations when it is used to solve higher-order differential equations. In order to reduce the complexity, G-transform is incorporated with HAM and developed a computationally efficient hybrid method for solving linear as well as nonlinear differential equations. This paper contributes a new and computationally efficient analytical method, namely GHAM. This new method is able to produce the solutions with rapid convergence series. GHAM is useful for tackling the computable terms in various differential equations, even though the governing equations contain high degree of nonlinear terms. The results obtained by GHAM are more accurate than the results of HAM.
The following are the merits of the proposed Generalised Homotopy Analysis Method: • GHAM avoids multiple differentiation and integration that arise in HAM while solving higher order fractional differential equations .
• The proposed hybrid technique GHAM successfully overcomes the limitation of Laplace typed integral transform and can be effectively used to solve the complicated non linear fractional differential equations.
• The GHAM can be applied to solve various types of engineering problems and to obtain the

Preliminaries
This section provides a comprehensive description of HAM and detail explanation of the Laplace Typed Integral Transform, i.e., G-transform.

Homotopy Analysis Method
The concept and principle of 'homotopy' was initiated by Sen (1983). Later, the HAM was suggested by Liao (2004) to obtain an exact solution of a given differential equation.The HAM establishes a continuous mapping that involves an initial approximation based on the best guess, where the rules of solution expression and coefficient ergodicity are incorporated. As a result,the procedure for tackling physical problems can be simplified. Given the nonlinear differential equation where ′ N ′ is the nonlinear operator, the unknown function with an independent variable t is denoted as u(t).
The zeroth order deformation must be utilized to yield the higher order deformation. At the same time, the solution vector u n = u 0 (t), u 1 (t), ..., u n (t) can be obtained as follows.
The n-th order deformation equation is Given a nonlinear operator N , Equation(2.4) to represent the term R n ( u n−1 , t). This leads us to obtain the following solution, The next subsection presentsthe fundamental idea pertaining to the proposed HAM model for undertaking nonlinear partial differential equations.

Laplace Typed Integral Transform(G-transform)
The G-transform converts any time domain function u(t) into the associated frequency domain function G(ω) is defined as whereα is a suitable integerand ω is the frequency variable to be selected.The n-th derivative property of G-transformimply Readers are referred to Kim(2017) for the fundamentals of G-transform along with the standard functions.

Generalized Homotopy Analysis method (GHAM)
A systematic procedure for solving the nonlinear partial differential equation is illustrated in Saratha et al(2020). In the following section, ordinary differetial equation, linear and nonlinear partial differential equations are solved using GHAM. The results show an excellent agreement with those from some of the existing methods.

Numerical Examples
This sub-section illustrates the usefulness and computational efficiency of GHAM through the discussion of standard numerical examples.
Example 4.1: A fourth-order nonlinear differential equation is expressed [Abassy, 2010]as follows: The initial conditions are The G-transform is applied to both sides of Equation (4.1). This leads to, Using the properties of G-Transform, the following equation is obtained Applying GHAM, The above equation is solved with respect to n = 1, 2, 3, ......, i.e., (4.5) Figure 1 illustrate the h-curves. It can be observed that, for h ∈ [−1, 1], the method provides more accurate results. The solution curve u(t) of equation (4.1) is depicted in Figure 2. Figure  3 shows the comparison among GHAM, ADM and IADM. It shows an excellent agreement with the other existing methods.
The initial condition is u(x, 0) = f (x). Now, G-transform method is applied to both sides of equation ( By solving the above equation for n = 1, 2, 3, ... provides Similarly, we estimate u 3 , u 4 ,..., and this leads to the series solution Subject to the initial condition f (x) = x, the following solution is obtained.

Application
This sub-section illustrates an application of GHAM to a reaction-diffusion model pertaining to forest boundary. Consider the following seed dynamics model, where the density of a young class is u and the old class is v; the density of airborne seed is w;the coefficient of tree mortality and aging are g and f respectively; Other parameters include the rates of seed establishment(δ), deposition (β), and reproduction(α),as well as the mortality rate of young trees (γ(q)). In addition, d denotes the diffusion coefficient along with a space variable x.
Using GHAM, the above model of nonlinear equations (5.1), (5.2), (5.3) are solved. The solution of the forest model is obtained using GHAM is presented below.
w(x, t) =   Figure 13. The h-curve for u.   Figure 19. The h-curve for w.  2015) obtained the approximate solutions of this forest modelwith the use of iterative integrations. In this paper, more accurate solutions are obtained by applying the proposed GHAM approach. Furthermore, the h-curves corresponding to the solution curves of u, v, w are plotted, and the convergence region can be easily identified.

Conclusion
In this paper, a new hybrid model known as GHAM has been introduced. GHAMis useful for analyzing various higher-order nonlinear ordinary and partial differential equations. The GHAM offers a number of benefits. The need of multiple differentiations and integrations in HAM can be totally avoided while solving higher-order differential equations. The limitation of Laplace typed integral transform can be eliminated by using GHAM, and it is effective in solving complicated nonlinear differential equations. From the application perspective, GHAM is useful for analyzingvarious scientific problems in the engineering domain. Using the present method, the accurate solution of differential equations can be achieved by the appropriate selection ofthe auxiliary parameters.The usefulness of GHAM in tackling forest modelling problem is also clearly demonstrated. The exact solutions are obtained in a series form and the solution curves are graphically depicted, ascertaining the applicability of GHAM in the forest modelling case study.
In the future, one can apply GHAM to Bagley-Torvik equation, complicated Navier-Stokes equation and epidemic model etc.