一类四阶降噪偏微分方程的对称性分析
Symmetry Analysis for a Fourth-order Noise-reduction Partial Differential Equation
摘要：
本文应用Lie对称性理论研究了一个四阶$1+2$演化偏微分方程，该方程已被提出用于图像处理降噪。特别是我们确定的李点对称性的特定1+2偏微分方程，我们应用不变函数来确定相似的解决方案。对于静态解，我们观察到，减少四阶常微分方程减少到二阶常微分方程是最大对称的。最后，非静态封闭形式的解决方案也确定。
英文摘要：
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.
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