Water Pollution Models based on Stochastic Differential Equations

Water pollution has been a crucial problem in many countries and has attracted researcher’s attention from all over the world. In a non-tidal river, from the beginning of material input, mixing and reaction, to the end of material output, the system considered should be dynamic and stochastic. The stochastic differential equation (SDE) is a theory that studies stochastic process. It can describe the physical relationships of different factors, and can predict the effects of future loading and control policies on the environment. It has the advantage of water pollution control. This would then allow an attempt to formulate a general model on self-purification non-tidal rivers. Most water pollution models have a structure of input/output system. However these models do not assume explicit knowledge of pre-biochemical reaction, such as the explicit pre-reaction concentration. An internally descriptive model exploits the available information on the phenomena determining the system’s behaviours, e.g. the physical and biochemical mechanisms which control the internal descriptions. To exploit the information of pre-biochemical reactions is an essential task in modeling. This paper firstly, designs a structure of input/mixing/output model (we will call it the three-steps model); secondly, it represents each process of material input, mixing and output by an SDE respectively, where the additional equations representing mixing processes exploit the information of pre-biochemical reactions; and thirdly, it shows that the three-steps model has the advantage of prediction and control, using the numerical solutions of SDEs of the three-step model. The stochastic numerical methods applied here are discrete time approximation methods subject to the Ito-Taylor expansion schemes.