We apply the theory of Lie symmetries in order to study a fourth-order 1+2 evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.

中文翻译：

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四阶降噪偏微分方程的对称性分析

我们应用李对称理论来研究已被提出用于图像处理降噪的四阶 1+2 演化偏微分方程。特别地，我们确定特定 1+2 偏微分方程的李点对称性，并应用不变函数来确定相似性解。对于静态解，我们观察到简化的四阶常微分方程简化为最大对称的二阶常微分方程。最后，还确定了非静态封闭形式的解决方案。