具有非线性源的热传导方程的群分类
Group classification of heat conductivity equations with a nonlinear source
摘要：
我们提出了一个系统的程序分类偏微分方程不变的低维李代数。这个过程是无穷小李方法、等价变换技术和抽象低维李代数分类理论的综合。作为应用，我们考虑了具有非线性对流项和源项的一元热传导方程的分类问题。我们已经导出了这类非线性方程的一个完整分类，承认非平凡对称性。它表明，有三个，七个，二十八个和十二个不等价类的偏微分方程所考虑的类型是不变的一维，二维，三维和四维李代数，相应的。此外，我们证明了属于所研究的类，并承认对称群的维数大于4的任何偏微分方程是局部等价的线性方程。这种分类相比，现有的组分类的非线性热传导方程和结论之一是，所有这些都可以得到我们的方法的框架内。此外，一些新的不变方程的构造，具有丰富的对称性，因此，可用于数学建模，比如说，非线性传热过程。
英文摘要：
We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.
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