Analysis of stochastic groundwater flow problems. Part III: Approximate solution of stochastic partial differential equations

Abstract Following the theory presented in Part I and Part II of these series of articles, functional analysis theory and a formulation of the Ito's lemma in Hilbert spaces are outlined as a practical alternative to the problem of finding the equations satisfying the moments of a stochastic partial differential equation of the type appearing in groundwater flow. By combining the moments equations derived from Ito's lemma and the strongly continuous semigroup associated with a particular partial differential operator in a Sobolev space, very simple solutions of the moments equations can be obtained. The most important feature of the moments equations derived from Ito's 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. The method provides a rigorous bridge between the abstract-theoretical analysis of stochastic partial differential equations and computer-oriented numerical techniques. An illustrative example showed the potential applications of the method in regional groundwater flow analysis subject to general white noise disturbances. The example used the present method in combination with the boundary integral equation method to solve the problem of regional groundwater flow in a two-dimensional domain subject to a random phreatic surface. Given the stochastic properties of the boundary condition, the first two moments of the potential as well as sample functions were found.

[1]  G. Adomian,et al.  Random Operator Equations in Mathematical Physics. I , 1970 .

[2]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[3]  Carlos Alberto Brebbia,et al.  The Boundary Element Method for Engineers , 1978 .

[4]  M. Popescu.  Boundary element methods in solid mechanics , 1985 .

[5]  D. Brillinger.  Introduction to random differential equations and their applications , 1971 .

[6]  R. Curtain.  Infinite-Dimensional Linear Systems Theory , 1978 .

[7]  J. Tóth.  A Theoretical Analysis of Groundwater Flow in Small Drainage Basins , 1963 .

[8]  J. Tóth.  Reply [to “Discussion of a paper by J. Tóth, ‘A theory of groundwater motion in small drainage basins in central Alberta, Canada’”] , 1963 .

[9]  B. Sagar,et al.  Galerkin Finite Element Procedure for analyzing flow through random media , 1978 .

[10]  Yoshikazu Sawaragi,et al.  Modeling, estimation, and their applications for distributed parameter systems , 1978 .

[11]  P. Chow.  Perturbation Methods in Stochastic Wave Propagation , 1975 .

[12]  M. Lax.  Method of moments approximate solutions of random linear integral equations , 1977 .

[13]  R. Bertram,et al.  Stochastic Systems , 2008, Control Theory for Physicists.

[14]  William E. Boyce,et al.  A Numerical Analysis of Some First Order Stochastic Initial Value Problems , 1974 .

[15]  S. Davis.  Discussion of a paper by J. Tóth, ‘A theory of groundwater motion in small drainage basins in central Alberta, Canada’ , 1963 .

[16]  T. E. Unny,et al.  Numerical Integration of Stochastic Differential Equations in Catchment Modeling , 1984 .

[17]  D. Griffel.  Applied functional analysis , 1982 .

[18]  G. Bécus.  Random generalized solutions to the heat equation , 1977 .

[19]  M. Lax,et al.  The method of moments for linear random initial value problems , 1976 .

[20]  P. Witherspoon,et al.  THEORETICAL ANALYSIS OF REGIONAL GROUNDWATER FLOW.. , 1966 .

[21]  Mary P. Anderson.  The Boundary Integral Equation Method for Porous Media Flow , 1984 .

[22]  A. Vandenberg.  Regional groundwater motion in response to an oscillating water table , 1980 .

[23]  B. Sagar.  Solution of linearized boussinesq equation with stochastic boundaries and recharge , 1979 .

[24]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[25]  R. Curtain,et al.  Functional Analysis in Modern Applied Mathematics , 1977 .

[26]  T. Sun,et al.  A Finite Element Method for Random Differential Equations with Random Coefficients , 1979 .

[27]  G. Adomian.  Nonlinear stochastic differential equations , 1976 .

[28]  N. J. Rao,et al.  Numerical Solution of Ito Integral Equations , 1974 .

[29]  P. Chow.  Stochastic partial differential equations in turbulence related problems , 1978 .

[30]  Karmeshu,et al.  Stochastic nature of outputs from conceptual reservoir model cascades , 1984 .

[31]  M. Lax.  The Method of Moments for Linear Random Boundary Value Problems , 1976 .

[32]  Chris P. Tsokos,et al.  Random integral equations with applications to life sciences and engineering , 1974 .

[33]  T. T. Soong,et al.  Solutions of a Class of Random Differential Equations , 1973 .

[34]  J. Tóth,et al.  A theory of groundwater motion in small drainage basins in central Alberta, Canada , 1962 .

[35]  VARIATIONAL FORMULATION OF SOME PROBLEMS FOR THE RANDOM HEAT EQUATION , 1980 .

[36]  Albert Turner Bharucha-Reid.  A survey on the theory of random equations , 1964 .

[37]  G. Adomian,et al.  Linear Random Operator Equations in Mathematical Physics. III , 1971 .

[38]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[39]  Approximate solution of random differential equation , 1987 .

[40]  Mark J. Beran,et al.  Statistical Continuum Theories , 1968 .

[41]  M. R. Spiegel.  Advanced mathematics: for engineers and scientists. , 1971 .

[42]  B. Sagar.  Analysis of dynamic aquifers with stochastic forcing function , 1978 .

[43]  G. N. Mil’shtejn.  Approximate Integration of Stochastic Differential Equations , 1975 .

[44]  M. Lax,et al.  APPROXIMATE SOLUTION OF RANDOM DIFFERENTIAL AND INTEGRAL EQUATIONS , 1980 .

[45]  J. Bear.  Hydraulics of Groundwater , 1979 .

[46]  P. Chow,et al.  Applications of Function Space Integrals to Problems in Wave Propagation in Random Media , 1972 .