A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

Abstract

The physical systems are modeled by differential equations in engineering problems. If all the parameters involved in a physical system can be known, with the initial and boundary conditions, one can predict the time evolution of the physical system by solving the govern differential equations.But the more general case is that we do not know the parameters and conditions precisely because the real problems in engineering are all governed by some very complex nonlinear systems. So that the parameters and initial, boundary conditions may fluctuate randomly or at least it appears to us in that way.One way to deal with that situation is to use stochastic differential equations. The theory of stochastic ordinary differential equations has been well developed since K. Ito [2] introduced the stochastic integral and the stochastic integral equation in the mid-1940s. Recently, K. Yoshimi and T. Yamada [5] have used this method to study the uncertainty of discharge due to the random fluctuation in precipitation.The present study tried to general Ito's method to the shallow water equation which is a partial differential equation, to study the uncertainty of discharge due to the randomness effects in space by changing the partial differential equation to an ordinary differential equations using the Lagrange point of view. As a result, we proposed a Boltzmann-type equation to govern the time evolution of the probability density function of discharge and water level in river channel and show the result of simulation.