Variational iteration method for solving non-linear partial differential equations
Introduction
Differential equations are widely used to describe physical problems. In most cases, these problems may be too complicated to solve exactly. Alternatively, the numerical methods can provide approximate solutions rather than the exact solutions. In fact there are many of methods to solve these problems numerically such as: finite difference methods, multi-grid methods, perturbation methods, but most of these methods are of low accuracy and are of high expensive. In this paper, the variational iteration method has been shown to solve effectively, easily and accurately a large class of linear and non-linear problems with approximations converging rapidly to accurate solutions. In 1978, Inokuti et al. [1] proposed a general Lagrange multiplier method to solve analytically non-linear problems, at first the proposed method used to solve problems in quantum mechanics. The main feature of the proposed method is: the solution of a mathematical problem with linearization assumption is used as initial approximation (trial-function), then a more highly precise approximation at some special point can be obtained. More details of this method is illustrated in the following problems. Problem 1 Duffing equation This equation is used widely by many perturbation techniques to verify their effectiveness, here we shall use Duffing equation with non-linearity of seventh order to illustrate the general evaluation process of the proposed method. Considering:
The stationary conditions for this problem takes the form:The multiplier, therefore, can be identified as λ = sin(τ −t), and the following variational iteration formula can be obtained:Starting with the trial function: u0 = cos αt, where α ≠ 0 is a non-zero unknown constant. By the variational iteration formulae (4) we have:The constant α can be identified by various methods such as least square method. To simplify the integral, we set:Therefore, we have:with α defined by Eq. (6), which is more accurate than the results can be obtained by perturbations. Problem 2 In this problem the proposed technique can be also extended to the following partial differential equation (PDE) [4], [5], [6], [7], [8]:with the boundary and initial conditions: Problem 3 Again the proposed technique can be also extended to PDEs with mixed derivative terms. Consider the following problem:with the boundary and initial conditions:
As in Eq. (13) the Lagrange multipliers can be easily identified:The iteration formulae in x- and y-directions can be, therefore expressed, respectively, as follows:Now we start with the arbitrary initial approximation: u0 = A + Bx, where A and B are constants to be determined, by using the initial conditions (20a), thus we can obtain:By the variational iteration formula in x-direction (24), we have:which is the first approximation, in the same way we can obtain the following second approximation:which is an exact solution. The approximation can also be obtained by y- direction or alternate use of x- and y-directions iteration formulae. Problem 4 Combined KdV–MKdV equation [9], [10] The KdV and MKdV equations are most popular solution equations. The non-linear terms of KdV and MKdV equations often simultaneously exist in practical problems such as fluid physics, physics and quantum field theory, and form the following so-called combined KdV–MKdV equation:
This equation may describe the wave propagation of the bound particle, sound wave and thermal pulse. In this example we shall solve this equation:Its correction variational functional in x- and t-directions can be expressed, respectively, as follows:where is a restricted variation. The stationary conditions for this problem takes the form:The Lagrange multipliers can be easily identified:The iteration formulae in x- and t-directions can be written as:Now we start with the arbitrary initial approximation: u0 = At + B, where A and B are functions to be determined, by the iteration formula in x-direction (35), we have:Imposing the boundary conditions at t = 0 and x = 0, we have:Solving Eqs. (38), (39) for A and B, we have:Substituting A and B into Eq. (37) we have the following solution:which is an approximate solution satisfies the given boundary conditions. Problem 5 Camassa–Holm equation [9], [11] The Camassa–Holm equationAppeared first in a physical context as describing the shallow water approximation in invisible hydrodynamics. The variable represents the fluid velocity in the horizontal direction and p is a constant. Much of the interest on this equation come from the facts that it is completely integrable and possess peaked solution [9], [11]. The proposed method can be used to solve this equation: Remark Camassa–Holm equation has other exact solutions which can be obtained if we change the boundary conditions.
Section snippets
Conclusion
In this paper we have studied some problems, which does not require small parameter in an equation as the perturbation techniques do. The Obtained results show that:
- 1.
A correction functional can be easily constructed by a general Lagrange multiplier, and the multiplier can be optimally identified by variational theory.
- 2.
The initial approximation can be freely chosen with unknown constants or functions, which can be determined via various methods.
- 3.
The approximations obtained by this method are valid
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