Mathematical Research on Anomalous Diffusion Problem

假设两种金属的均匀混合物处于不稳定的非平衡状态，并且混合物从不稳定状态演变为由两相组成的更稳定的非均匀构型。这种现象被称为反常扩散问题，其时间演化由一个非线性偏微分方程Cahn-Hilliard方程描述，本研究的目的是建立一种有效的差分格式，稳定地数值求解这类非线性偏微分方程.本方法的思想是从所考虑的物理系统的能量的离散化开始，并应用适当的离散变分。本研究的关键点在于这种离散的变化。在能量离散化时，我们定义了与积分算子和积分算子一致的差分算子和求和算子，我们的方法不仅可以应用于Cahn-Hilliard方程，而且可以应用于更广泛的一类非线性偏微分方程，即Cahn-Hilliard方程。e.能量耗散系统和能量守恒系统的偏微分方程。耗散系统方程的例子和热方程和Cahn-Hilliard方程，以及波动方程和KdV方程是保守系统方程的例子。我们的方法已成功地应用于这些方程的数值解。在Cahn-Hilliard方程的情况下，我们从数学上证明了格式的稳定性和数值解收敛于原方程的精确解。

Suppose that a uniform mixture of two metals is in an unstable nonequilibrium state and that the mixture evolves from the unstable state to a more stable nonuniform configuration consisting of two phases. Such a phenomenon is call an anomalous diffusion problem and time evolution of such a phenomenon is described by a nonlinear partial differential equation called the Cahn-Hilliard equation.The purpose of the present research project is to establish an efficient difference scheme for stable numerical solution of such kind of nonlinear partial differential equations. The idea of the present method is to start from discretization of the energy of the physical system under consideration and to apply a suitable discrete variation. The key point of the present research exists in this discrete variation. When we discretize the energy we defined the difference operators and the summation operators consistently with the differetial operators and the integration operator.Our method can be applied not only to the Cahn-Hilliard equation but also to a wide class of nonlinear partial differential equations, i. e. partial differential equations for energy dissipative system and for energy conservative system. Examples of equations for the dissipative system and the heat equation and the Cahn-Hilliard equation, and the wave equation and the KdV equation are examples of the equations for the conservative system. Our method were applied successfully to numerical solution of these equations. In the case of the Cahn-Hilliard equation we proved mathematically the stability of the scheme and the convergence of the numerical solution to the exact solution of the original equation.