Mathematical Research on Anomalous Diffusion Problem

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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.