| After the two reform of RMB exchange rate regime in 2005 and 2010, the process of exchange rate markctization has developed faster and faster. The trend of RMB exchange rate has become a key issue that affects the economic and trade relations between China and its trading partners, even the economic situation of the whole world. Therefore, how to discover the characteristics of the behavior of the exchange rate and predict the exchange rate in the future has become a focus of the international community. More and more scholars have joined the research of forecasting exchange rate.The first equation is the time-dependent variable-order fractional diffusion equation. The stability is analyzed by Fourier Methods.With further rescarch, scholars found that the behavior of exchange rate was not a simple linear time scries, some complex nonlinear characteristics had been showed up. The traditional theoretical models and linear models became useless gradually which prompted scholars to turn the research to nonlinear models. But what we can not ignore is that exchange rate time scries still has linear features, both pure linear models and pure nonlinear models can not predict the trend of exchange rate accurately. Therefore, a component model which contains both linear component and nonlinear component came into beingBased on the idea mentioned above, exchange rate time series were con-sidered to be composed of a linear autocorrelation structure and nonlinear structure. Finally, in this paper, the RMB exchange rate time scries arc considered to be composed of ARIMA and GPR. after analysis the features and advantages of the various types of linear and nonlinear models deeply The component model can take full advantage of the two parts both in linear space and nonlinear space. ARIMA is used to model the linear component of exchange rate time scries and the GPR model is applied to the non-linear residuals component prediction. The results of RMB exchange rate forecast-ing show that the proposed model, which integrates the unique strength of the two models in linear and nonlinear modeling,has the more forecasting accuracy than that of single model(including single ARIMA and single GPR). The composite forecasting model has a certain significance in forecasting RMB exchange rate.The last equation is the space variable-order fractional advection-diffusion e-quation with nonlinear source term. The stability and convergence are ana-lyzed by Maximum norm Methods. When q(x, t) is independent of t, we can analyze the convergence for the two equations above.In the end, a numerical example has demonstrated the effectiveness of the non-uniform meshes. |
