Some problems are also discussed. AMS subject classifications Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions. Introduction In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations.
As we all know, the symmetry method is the most universal method, and many traditional methods become its special cases. During the end of the nineteenth century, in order to unify and expand the methods used in solving the ordinary differential equations ODE , Norwegian mathematician Sophus Lie — firstly proposed the symmetry theory of differential equations Lie At present, using the symmetry method and others, such as the analytic solutions method, the approximate analytic solutions method and the numerical method with the aid of thorough considering mutual complementarity and availability to solve NLPDE are the new research subjects.
The premise of applying the symmetry method is to determine the all kinds of symmetries of the partial differential equations PDEs. Therefore, Wu-differential characteristic set algorithm extended and constructed by Temuer Chaolu can partially solve the above-mentioned problems Temuer ; Temuer and Bai Some other investigators use the symmetry method, the variational iterative method and the homotopy perturbation method to solve the boundary value problem based on Wu-differential characteristic set algorithm Lu and Temuer a , b ; EerDun and Temuer In addition, it will also effectively popularize the range of application and advance the efficiency of using method.
Symmetries of Partial Differential Equations
Next, we will give the process of calculating the symmetry and reduction of Eq. The symmetry group of Eq. According to the Lie algorithm, we obtain the determining equations of symmetry 2 , but it is too difficult to get its solutions. However, we can obtain the followig system of equations corresponding to the characteristic set which is equivalent to the determining equations by using Wu-differential characteristic set algorithm Temuer Obviously, X has five one-parameter point symmetries, then the corresponding infinitesimal vectors are as follow:.
To facilitate solve the Eq. The resulting reduced PDE is fewer independent variable than Eq. From the above Eqs. This method is further improved by the generalized Riccati equation and introducing its twenty seven new solutions, these solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, respectively.
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When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Taking Eq. Doing the travelling wave transformations.
In order to look for the travelling wave solutions of Eq. Then we reduce Eq. Choosing the expression of solution. The ODE 15 has four cases of solutions as follows. Determining the coefficients. By substituting 14 into Eq.
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With the aid of mathematica, we get the solutions as follows:. Acquiring the exact travelling wave solutions. The solitary wave solutions can be obtained see Fig. The solitary wave solutions of the exact solutions U 1.
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The homotopy perturbation method is proposed by He , and it has successfully been applied to solve many types of linear and nonlinear functional equations. This method, which is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions for a wide variety of problems arising indifferent fields. In recent years, the application of the homotopy perturbation method in nonlinear problems has been developed by scientists and engineers He , ; Olga ; Ebaid ; Najafi and Edalatpanah Next, we construct the approximate analytic solutions of Eq.
According to the homotopy perturbation method He , we construct the following homotopy. In order to be convenient for computing, we choose the following initial value approximation. By substituting 20 and 21 into Eq. We choose the initial conditions as follows:.
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Then the second-order approximate solutions of Eq. According to the figure and table, the exact property of the homotopy perturbation method has been showed successfully. In this paper, we studied that construct the exact solutions and the approximate analytic solutions of NLPDE by using the Lie symmetry, the extended tanh method and the homotopy perturbation method. Lie symmetry, the extended tanh method and the homotopy perturbation method are effective methods which applied to solve PDEs.
Hence, comprehensive use of them will advance their availability. The Wu-differential characteristic set algorithm is a key factor which influence the calculating the symmetry of PDEs. At present, combining the Wu-differential characteristic set algorithm, symmetry method and others to solve NLPDE has been regarded as a hot research topic and widened the application of symmetry and the Wu-differential characteristic set algorithm. This investigation is valuable in advanced research and development. All authors contributed extensively in the development and completion of this article.
All authors read and approved the final manuscript. The authors would like to thank the Editor and anonymous referees for their helpful suggestions and valuable comments. Litao Gai, Email: nc. Sudao Bilige, Email: moc. Yingmo Jie, Email: moc.
National Center for Biotechnology Information , U. Published online Aug 5. Author information Article notes Copyright and License information Disclaimer. Corresponding author. Received May 7; Accepted Jul Keywords: Symmetry, Wu-differential characteristic set algorithm, The extended tanh method, The homotopy perturbation method. Background Recently, the nonlinear phenomenon has been extensively appeared in the fields of mathematical physics and engineering technology. The reduction of Eq. Step 1 Doing the travelling wave transformations.
Step 2 Choosing the expression of solution. Step 3 Determining the coefficients.
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Step 4 Acquiring the exact travelling wave solutions. Open in a separate window. The approximate analytic solutions of Eq.