标题： (1+2)维精细柯尔莫哥洛夫后向方程的扩展对称性分析 显示英文标题

标题： Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation

摘要： 在(1+2)维超抛物线线性方程的类中，我们区分出一个具有二次扩散率的精细柯尔莫哥洛夫后向方程。 在点等价下，它是该类中唯一一个其本质李不变代数是五维且不可解的方程。 使用直接方法，我们计算了该方程的点对称伪群并分析了其结构。 特别是，我们指出了其本质子群并分类了其离散元素。 我们详尽地分类了相应本质李不变代数的所有子代数，考虑内自同构以及本质点对称群的作用。 这使我们能够对该考虑中的方程进行李约化和李不变解的分类。 我们还讨论了利用点对称性和线性广义对称性生成其解，并进行了其特有的广义约化。 结果，我们构造了由(1+1)维线性热方程的任意有限数量的任意解，或由(1+1)维线性热方程带有倒数平方势的任意一个或两个解参数化的广泛解族。 显示英文摘要

摘要： Within the class of (1+2)-dimensional ultraparabolic linear equations, we distinguish a fine Kolmogorov backward equation with a quadratic diffusivity. Modulo the point equivalence, it is a unique equation within the class whose essential Lie invariance algebra is five-dimensional and nonsolvable. Using the direct method, we compute the point symmetry pseudogroup of this equation and analyze its structure. In particular, we single out its essential subgroup and classify its discrete elements. We exhaustively classify all subalgebras of the corresponding essential Lie invariance algebra up to inner automorphisms and up to the action of the essential point-symmetry group. This allowed us to classify Lie reductions and Lie invariant solutions of the equation under consideration. We also discuss the generation of its solutions using point and linear generalized symmetries and carry out its peculiar generalized reductions. As a result, we construct wide families of its solutions parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation or one or two arbitrary solutions of (1+1)-dimensional linear heat equations with inverse square potentials.