半变分非线性p-Laplacian方程组特征对问题的数值解法

发布时间：2018-01-09 19:29

本文关键词：半变分非线性p-Laplacian方程组特征对问题的数值解法　出处：《中国科学技术大学》2015年硕士论文　论文类型：学位论文

【摘要】：偏微分方程中的特征对问题在物理学、力学、量子化学等等领域有着重要理论意义和广泛的应用价值。在有界弦的自由振动和热量的传导过程最关键的是求解特征对问题。而p-Laplacian方程在非线性扩散学、冰川学,非牛顿流体模型(non-Darcian)和气候学,湍流学、多孔介质和幂律材料流动学等等流体力学的数学建模中有着很广泛的应用。本文主要研究了半变分、非线性p-Laplacian方程组特征对问题的数值解法,讨论了将推广的局部极小正交算法从Hilbert空间应用于Banach空间中,并应用该算法解决了半变分、非线性p-Laplacian方程组特征对问题。首先应用Rayleigh商将特征对问题转化为能量泛函的“临界点”问题,并证明了Rayleigh商的“临界点”与“临界值”必是原特征对问题的解；因在全空间上寻求“临界点”一般是很困难的,故通过引入了一种L一⊥选择函数,并构造出一个子流形M,从而将原特征对问题转化为求子流形M上某些相对稳定的点；然后,在Banach空间中用伪梯度代替Hilbert空间中的梯度,并将伪梯度投影映射到相应的空间,再应用梯度下降法,结合步长引理更新搜索方向,寻找州上的那些相对稳定的点。最后,结合多个数值算例,成功地获得了方形区域和圆形区域上多重特征对。
[Abstract]:The characteristic pairs of problems in partial differential equations are in physics and mechanics. Quantum chemistry and other fields have important theoretical significance and wide application value. In the free vibration of bounded string and heat conduction process, the most important thing is to solve the eigenpair problem. Nonlinear diffusion. Glaciology, non-Newtonian fluid models and climatology, turbulence, porous media and power-law material flow are widely used in mathematical modeling of hydrodynamics. In this paper, the numerical solution of eigenpairs of semi-variational and nonlinear p-Laplacian equations is studied. This paper discusses the application of the generalized local minimal orthogonal algorithm from Hilbert space to Banach space, and solves the semi-variational problem by using this algorithm. The eigenpair problem of nonlinear p-Laplacian equations is first transformed into a "critical point" problem of energy functional by Rayleigh quotient. It is proved that the "critical point" and "critical value" of the Rayleigh quotient must be the solution of the original characteristic pair problem. Because it is very difficult to find "critical point" in the whole space, we introduce a L ~ (-)-Karabakh selection function and construct a submanifold M. Thus, the primordial characteristic pair problem is transformed into finding some relatively stable points on the submanifold M; Then, the pseudo gradient is used to replace the gradient in Hilbert space in Banach space, and the pseudo gradient projection is mapped to the corresponding space, and then gradient descent method is applied. The search direction is updated with the step Lemma to find the relatively stable points in the state. Finally, the multiple feature pairs on the square and circular regions are successfully obtained by combining several numerical examples.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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