A new technique of using homotopy analysis method for solving high-order nonlinear differential equations

Abstract

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high-order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth-order nonlinear differential equation to a system of n first-order equations. Second-and third-order problems are solved as illustration examples of the proposed method.

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth-order integro-differential equation

“…By means of the (8), we obtain directly the components of HAM in the series forms of (5). We can obtain the following: 6 C 0.000396825x 7 C 5.51146 10 7 x 10 .…”

“…From the initial conditions, it is straightforward to choose We can choose the auxiliary linear operator with property where c 1 , c 2 and c 3 are arbitrary constants that can be obtained by the initial conditions. According to Section , we can define the operator N as Thus, we can obtain the deformation equation , that is, …”

“…We can choose the auxiliary linear operator with property where c i , i = 1,2, … ,8, are arbitrary constants that can be obtained by the initial conditions. According to Section , we can define the operator N as Thus, we can obtain the deformation equation , that is, …”

An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method

“…In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering such as the viscous flows of non-Newtonian fluids [3--13], the KdV-type equations [14--18], nonlinear heat transfer [19--21], nonlinear water waves [22], groundwater flows [23], Burgers-Huxley equation [24], time-dependent Emden-Fowlertype equations [25], differential-difference equation [26], Laplace equation with Dirichlet and Neumann boundary conditions [27], MHD Falkner-Skan flow [28], the Sharma-Tasso-Olver equation [29], the Kawahara equation [30], for multiple solutions of nonlinear boundary value problems (BVPs) [31--35] and Abbasbandy et al [34] applied HAM to predict the multiplicity of the solutions of nonlinear BVPs and shows that convergence-control parameter h plays basic role in prediction of multiplicity of solutions of nonlinear problems. In [36] a new technique of HAM form introducing a change in the using of HAM in solving high-order nonlinear initial value problems. HAM enjoys great freedom in choosing initial approximations and auxiliary linear operators.…”

Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method

“…The HAM can guarantee the convergence of the series solutions by auxiliary parameters especially the so-called convergence-controller parameter ℏ [5][6]. In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering such as Troesch's problem [7], the Fitzhugh-Nagumo equation [8], heat radiation equations [9], MHD viscoelastic fluid flow [10], the Coupled nonlinear schrödinger equations [11], the Continuous population models for single and interacting species [12], the Boussinesq problem [13,14], the Sturm-Liouville problems [15], the wave propagation problems [16], Laplace equation with Dirichlet and Neumann boundary conditions [17], differential-difference equation [18], fractional equations [19]. Many authors trying to solve equation (1) by homotopy analysis method [7,20] but for λ=3 only.…”