We apply the method of preliminary group classification to a specific class of the nonlinear heat equation, ie, $u_t=f(x,u_x)u_{xx}+g(x,u_x)$. This results in an optimal system of one‐dimensional subalgebras, which will be used to find specific forms of the equation, which admit more symmetries and so allow us to find additional group invariant solutions. We extend this work by determining the potential symmetries of a closely related system, which gives rise to many new, additional solutions of the original equation.

中文翻译：

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关于应用于非线性热方程ut = f（x，ux）uxx + g（x，ux）的初步组分类方法

我们将初步组分类的方法应用于非线性热方程的特定类别，即 ${ü}_{Ť}=F（X，{ü}_X）{ü}_{XX}+G（X，{ü}_X）$。这样就产生了一个最佳的一维子代数系统，该系统将用于查找方程的特定形式，该形式允许更多的对称性，因此使我们能够找到其他的组不变解。我们通过确定密切相关的系统的潜在对称性来扩展这项工作，这引起了原始方程式的许多新的附加解。