The purpose of this paper is to analyse mathematical models used in environmental modelling. Following a brief survey of the development in modelling scale- and time-dependent dispersion processes in the environment, this paper compares three similarity solutions, one of which is a solution of the generalized Feller equation （GF） with fractal parameters, and the other two for the newly-developed generalized Fokker-Planck equation （GFP）. The three solutions are derived with parameters having physical significance. Data from field experiments are used to verify the solutions. The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes, and simulate the realistic shapes of tracer breakthrough curves.
Su Ninghu School of Mathematics and Computational Science,Xiangtan University,Hunan 411105,China
Investigating the biological function of proteins is a key aspect of protein studies. Bioinformatic methods become important for studying the biological function of proteins. In this paper, we first give the chaos game representation （CGR） of randomly-linked functional protein sequences, then propose the use of the recurrent iterated function systems （RIFS） in fractal theory to simulate the measure based on their chaos game representations. This method helps to extract some features of functional protein sequences, and furthermore the biological functions of these proteins. Then multifractal analysis of the measures based on the CGRs of randomly-linked functional protein sequences are performed. We find that the CGRs have clear fractal patterns. The numerical results show that the RIFS can simulate the measure based on the CGR very well. The relative standard error and the estimated probability matrix in the RIFS do not depend on the order to link the functional protein sequences. The estimated probability matrices in the RIFS with different biological functions are evidently different. Hence the estimated probability matrices in the RIFS can be used to characterise the difference among linked functional protein sequences with different biological functions. From the values of the Dq curves, one sees that these functional protein sequences are not completely random. The Dq of all linked functional proteins studied are multifractal-like and sufficiently smooth for the Cq （analogous to specific heat） curves to be meaningful. Furthermore, the Dq curves of the measure μ based on their CCRs for different orders to link the functional protein sequences are almost identical if q 〉 0. Finally, the Ca curves of all linked functional proteins resemble a classical phase transition at a critical point.
