Application of Homotopy Analysis Method and Variational Iteration Method for Shock Wave Equation

INTRODUCTION

Nonlinear phenomena play a crucial role in applied mathematics and physics. These phenomenons can be described by the means of their nonlinear governing equations if those are valid, so happens in shock waves. The shock waves appear in explosion, traffic flow, glacier waves and airplanes breaking the sound barrier and so on with partial nonlinear differential equation form. The analytical solution of the nonlinear equations of such nonlinear phenomena can help the authors know the described process deeply but except a limited number, most of them do not have precise analytical solutions. Therefore, these nonlinear equations are to be solved using other methods. In recent decades, numerical analysis (Burden and Faires, 1993) has remarkably developed for nonlinear partial equations related to fully developed shock wave equation. Also there are many standard semi-exact analytical methods for solving nonlinear partial differential equations; for instance, Backland transformation method (Rogers and Shadwich, 1982), Lie group method (Olver, 1986), Adomian’s decomposition method (Adomian, 1988) and variational iteration method. But as numerical calculation methods were improving, semi-exact analytical methods were, too. Most scientists believe that the combination of numerical and semi-exact analytical methods can also end with remarkable results. Homotopy Analysis Method (HAM) is one of the well-known and resent methods to solve the nonlinear equations, which was introduced by Liao (1992, 1995, 1997, 1999, 2003, 2004, 2005). This method eliminated the classical perturbation method problem, because of the existence a small parameter in the equation.

Recently, Allan and Al-Khaled, (in press) applied ADM solution for shock wave equation and achieved better results compared to numerical approaches (Allan and Al-Khaled, 2007). In this survey, we extend this work by applying HAM and VIM which are one of the latest analytical methods for solving linear and nonlinear equations to analysis the shock wave equation.

BASIC OF THE HE’S VARIATIONAL ITERATION METHOD (VIM)

To clarify the basic ideas of VIM, we consider the following differential equation:

(1)

where, L is a linear operator, N a nonlinear operator and g(t) an inhomogeneous term.

According to VIM, we can write down a correction functional as follows:

(2)

where, λ is a general lagrangian multiplier which can be identified optimally via the variational theory Liao (1999, 2005). The subscript n indicates the nth approximation and is considered as a restricted variation (He, 1997, 1998a, b, 1999), i.e.,.

BASIC OF THE LIAO’S HOMOTOPY ANALYSIS METHOD (HAM)

Consider the following differential equation (Liao, 1992, 1995, 1997, 1999, 2003, 2004, 2005):

(3)

where, N is a nonlinear operator, τ denotes an independent variable, u(τ) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the traditional homotopy method, Liao (2003) constructed the so-called zero-order deformation equation as:

(4)

where, pε[0, 1] is the embedding parameter, h≠0 a nonzero auxiliary parameter, h(τ)≠0 an auxiliary function, L an auxiliary linear operator, μ₀(τ) an initial guess of μ(τ) and φ(τ; p) is an unknown function. It is important to have enough freedom to choose auxiliary unknowns in HAM. Obviously, when p = 0 and p = 1 it holds:

Thus, as p increases from 0 to 1, the solution φ(τ; p) varies from the initial guess μ₀(τ) to the solution μ(τ). Expanding φ(τ; p) in Taylor series with respect to p, we have:

(5)

Where:

(6)

If the auxiliary linear operator, the initial guess, the auxiliary parameter h and the auxiliary function are quite properly chosen, the series of Eq. 5 converges at p = 1 then we have:

(7)

This must be one of the solutions of the original nonlinear equation, as proved by He (1997). While h = -1 and H(τ) = 1, Eq. 4 becomes:

(8)

This is mostly used in Homotopy Perturbation Method (HPM); whereas the solution can be obtained directly without using Taylor series (He, 2006).

According to the Eq. 5, the governing equation can be deduced from the zero-order deformation Eq. 4. The vector is defined as:

Differentiating Eq. 4 for m times with respect to the embedding parameter p, then setting p = 0 and finally dividing them by m, we will have the so-called mth order deformation equation as:

(9)

Where:

(10)

And

It should be emphasized that u_m(τ) for m≥1 is governed by the linear Eq. 8 with the linear boundary conditions coming from the original problem, which can be easily solved using symbolic computation software.

APPLICATIONS

In the following, we apply HAM and VIM to solve shock wave equation.

The shock wave equation which describes the flow of most gases is given by Chow (1979), Al-Khled (1996) and Smaller (1983):

(11)

where, c₀, γ are constant and γ is the specific heat. Study the case under c₀ = 2 and γ = 3/2 which corresponds to the flow of air, results to the following shock wave equation:

(12)

And the initial condition is assumed to be:

(13)

Implementation of VIM: In order to solve Eq. 12 by means of VIM, we construct a correction functional, as follows:

(14)

The Lagrange multiplier can therefore be simply identified as λ = 1 and the following iteration formula can be obtained:

(15)

Now we start with an arbitrary initial approximation that satisfies the initial conditions, as follows:

By the above iteration formula (15) we can obtain the other components as follows:

(16)

In the same manner the rest of components of the iteration formula Eq. 15 can be obtained. The excellent agreement between the solution of Eq. 12 which is obtained from VIM after three iterations and the Runge-Kutta’s algorithm will be shown in results and discussion.

Implementation of HAM: Now we apply homotopy analysis method Eq. 4 to 12. We construct a homotopy in the following form:

(17)

Consider Eq. 17 and let us solve them through HAM with proper assignment of H(τ) = 1 subject to the initial condition and assuming m = 2.

By two times derivation of Eq. 17 with respect to the p, two equations will be obtained. Consider the following equation:

(18)

Substitution Eq. 18 into both of them, we have following equations:

(19)

(20)

Subject to the boundary condition (13) gives us:

(21)

Thus;

(22)

(23)

(24)

Further from Eq. 24 it will be shown that HAM requires only two steps to achieve accurate results.