基于对称约化的偏微分方程相似解研究

Study on similarity solution of partial differential equations based on symmetry reduction

由于非线性系统的复杂性,对于其求解问题的研究目前还没有通用的方法,为了丰富非线性系统的求解方法,在此通过偏微分方程的决定方程确定点对称无穷小生成元,结合对称约化中的非经典Lie群法得到热方程新的相似解,并基于符号计算系统Maple给出相应的符号计算方法和实现步骤。结果表明,该算法能够有效求解PDEs的相似解,并且不需要显示地求解对应于不变曲面条件的特征方程,同时也适用于其他的发展方程。

Because of the complexity of nonlinear systems,the general method to solve the systems has not been found. In order to enrich the method for solving nonlinear systems,the point symmetry infinitesimal generator was determined by the deci-sion equations of partial differential equations,and the new similarity solution of a heat conduction equation was obtained in combination with the non-classical Lie group approach in the symmetry reduction. The corresponding symbolic computation meth-od and implementation steps are given according to the symbolic computation system Maple. The results demonstrate the method can solve the similarity solutions of PDEs effectively without the need to solve the characteristic equation corresponding to the in-variant curved surface. It can also be applied to other evolution equations.