Homotopy perturbation method with Laplace Transform (LT-HPM) for solving Lane–Emden type differential equations (LETDEs)

Abstract

In this communication, we describe the Homotopy Perturbation Method with Laplace Transform (LT-HPM), which is used to solve the Lane–Emden type differential equations. It’s very difficult to solve numerically the Lane–Emden types of the differential equation. Here we implemented this method for two linear homogeneous, two linear nonhomogeneous, and four nonlinear homogeneous Lane–Emden type differential equations and use their appropriate comparisons with exact solutions. In the current study, some examples are better than other existing methods with their nearer results in the form of power series. The Laplace transform used to accelerate the convergence of power series and the results are shown in the tables and graphs which have good agreement with the other existing method in the literature. The results show that LT-HPM is very effective and easy to implement.

An algorithm for solving the Burgers–Huxley equation using the Elzaki transform

“…By taking the Elzaki transform of (32) By taking the inverse Elzaki transform of (35) We now apply the homotopy perturbation method as the next algorithm to Eq. (38).…”

“…In quest of seeking exact solutions for model/equations, mathematicians have also come up with hybrid methods (coupling two distinct methods) to obtain exact solutions of some models, viz. Sumudu decomposition method [32][33][34], homotopy perturbation transformation method [35,36], homotopy variational iteration method [37], Elzaki differential transform [38], Elzaki projected differential transform [39,40], Elzaki homotopy transformation perturbation method [41][42][43], Laplace Adomian decomposition method [44] and Laplace variational iteration method [45][46][47].…”

Variational homotopy perturbation method for solving Lane-Emden initial value problems

“…In [1] Lane-Emden differential equation was solved by combining homotopy perturbation and reproducing kernel Hilbert space methods. Optimal homotopy asymptotic method was applied for singular Lane-Emden type equation in [21].…”

“…In this paper, we will present a method based on combining the VIM and HPM, the Variational Homotopy Perturbation Method (VHPM) is proposed to solve many kinds of differential equations, for example, Bratu-Type Problems [1], Riccati Differential Problems [12] and Stiff Systems of ODEs [11]. Also, VHPM was applied on partial differential equations such as Fisher's Equation [21], Burgers' equations [9], Klein-Gordon and sine-Gordon equations [25]. The results reveal that the proposed method is very effective and simple.…”

Homotopy perturbations method: Theoretical as­pects and applications

“…Finally, let us point out that HPM has been improving, developing and modifying, until to the present time. Consequently, today exist various solver techniques that are based, to a greater or lesser extent, on the basic HPM assumptions (Noor & Khan, 2012;Zhang et al, 2015;Tripathi & Mishra, 2016;Bota & Caruntu, 2017). Here is given a brief theoretical background about the HPM and some sufficient conditions to its convergence.…”