Stochastic Partial Differential Equations in Hydrology

Abstract

Functional-analytic theory is introduced as a method to solve stochastic partial differential equations of the type appearing in groundwater flow problems. Equations are treated as abstract stochastic evolution equations for elliptic partial differential operators in an appropriate functional Sobolev space. Explicit forms of solutions are obtained by using the strongly continuous semigroup of the partial differential operator. Application of the solution to the randomly-forced equation is illustrated in the case of the one-dimensional groundwater problem. The solution is obtained by applying the concepts of semigroup and expressing the Wiener process as an infinite basis in a Hilbert space composed of independent unidimensional Wiener processes with incremental variance parameters. Ito’s lemma in Hilbert spaces is then outlined as a practical alternative to the problem of finding the equations satisfying the moments of a stochastic partial differential equation. The most important feature of the moments equations derived from lemma is that these deterministic equations can be solved by any analytical or numerical method available in the literature. This permits the analysis and solution of stochastic partial differential equations occurring in two-dimensional or three-dimensional domains of any geometrical shape. Potential application of the method is illustrated by regional groundwater flow analysis subject to general white noise disturbances.