A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Abstract

In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

Exact solutions of the Harry Dym Equation using Lie group method

“…To calculate the infinitesimals 2 1 ,   and  , applying (5) to (6), that is, 0 , 0 ) ( Pr ) 3 (      when (7) Eq. ( 7) is called condition of the invariant, and it leads to ,…”

“…Fonseca [4] solved the nonlinear Harry Dym equation using Lattice-Boltzmannn and a solitary wave methods. Ghiasi and Saleh [6] employed the homotopy analysis method (HAM) to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation.in [9] , Iyiola and Gaba used q-homotopy analysis method (q-HAM) to obtain analytical solutions of the to timefractional Harry Dym equation. Kumar et al [11]used homotopy perturbation Sumudu transform method (HPSTM) and Adomian decomposition method (ADM) to solve nonlinear fractional Harry Dym equation.…”

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

“…[15] and to obtain the solution of the HD equation with approximate analysis, Saleh and Ghiasi applied the homotopy analysis technique in Ref. [16]. Mokhtari used variational iteration, power series, and direct integration to obtain exact solutions for traveling waves of the HD equation [17], and this equation was solved by Fonseca through the Lattice-Boltzmannn technique in Ref.…”

“…Notice that we can obtain the value of N by the homogeneous balance principle. • Insert (16), along with (17), into (8). Collect all coefficients of the same power of…”