The energy method to derive various properties of the solution from its energy-structure of the equations is one of the classical methods for partial differential equations. When the numerical scheme also possesses the energy-structure, the energy method can be applied to the scheme in the same way. In this article we introduce an application of energy method for the structure-preserving finite difference schemes through proofs of the existence of the solution and error estimate.
The usages of already popular words (ex. “cat”, “handle”) look like very stable. Then, how stable are they? In this paper, first, by analysing nation-wide text data such as 3 billion Japanese blog articles, we find that the uses of already popular words are changing logarithmically. In addition, this logarithmic behaviour is also observed in various languages and media such as newspaper articles, page views of Wikipedia (English, Chinese etc.). Second, we show that empirical properties are basically explained the random walk with power law forgetting, which is related to the fractional calculus. In the model, the parameter characterizes the speed of forgetting corresponds to the border between the stationary and the non-stationary.