Stochastic partial differential equations in groundwater hydrology
 T. E. Unny
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Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.
Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.
There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the nonhomogeneous part becomes random and iv) the random parameter problem.
Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.
This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.
 Basawa, I.V.; Prakasa Rao, B.L.S. 1980: Statistical inference for stochastic processes, Academic Press, N.Y.
 Bear, J. 1972: Dynamics of fluids in porous media, American Elsevier, New York, N.Y.
 Bensoussan, A. 1978: Control of stochastic partial differential equations, in distributed parameter systems, edited by W.H. Ray and D.G. Lainiotis. Marcel Dekker Inc., 209–246
 Butzer, P.L.; Berens, H. 1967: Semigroups of operators and approximations. SpringerVerlag
 Cannon, J.R. 1984: The onedimensional heat equation. In: Encyclopedia of Mathematics and its Applications, Rota, GC (ed.). AddisonWesley Publishing Co.
 Carslaw, H.S.; Jaeger, J.C. 1971: Conduction of heat in solids. Oxford University Press
 Crank, J. 1970: The mathematics of diffusion. Oxford University Press
 Curtain, R.F.; Pritchard, A.J. 1978: Infinite dimensional linear systems theory. Lecture Notes in Control and Information Sciences 8, edited by A.V. Balakrishnan and M. Thoma. SpringerVerlag
 Curtain, R.F.; Falb, P.L. 1971: Stochastic differential equations in Hilbert space. J. Diff. Eqs. 10, 412–430
 Cushman, J.H. 1987: Development of stochastic partial differential equations in subsurface hydrology. J. Stoch. Hydrol. Hydraul. 1, 241–262
 Heunis, A.J. 1989: On the stochastic differential equations of filtering theory, Special issue of applied mathematics and computation devoted to stochastic differential equations, to appear in 1989
 Jazwinski, A.H. 1970: Stochastic processes and filtering theory, Academic Press, N.Y.
 Kuo, H. 1972: Stochastic integrals in abstract Wiener space, Pac. J., Math., Vol 41, No. 2, 1972
 Kutoyants, Yu.A. 1984: Parameter estimation for stochastic processes, Herderman Verlag, Berlin
 Ladas, G.; Lakshmikantham, V. 1972: Differential equations in abstract spaces. Academic Press
 Lipster, R.S.; Shirayev, A.N. 1978: Statistics of random processes II: Applications, Springer, N.Y.
 Lipster, R.S.; Shirayev, A.N. 1977: Statistics of random processes I: General Theory, Springer N.Y.
 Pardoux, E. 1979: Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3
 Serrano, S.E.; Unny, T.E. 1988a.: General solution of random advectivedispersive transport equation in prous media. Part I: Stochasticity in the sources and the Boundaries., J. Stoch. Hydrol. and Hydr., 2(3), 1–20
 Serrano, S.E.; Unny, T.E. 1988b: General solution of random advectivedispersive transport equation in porous media. Part II: Stochasticity in the ources and the Boundaries., J. Stoch. Hydrol. and Hydr., 2(5), 20–34
 Serrano, S.E.; Unny, T.E. 1989: Random evolution equations in hydrology, Special issue of Applied mathematics and computation, to appear in 1989
 Serrano, S.E.; Unny, T.E. 1987a: Semigroup solutions of the unsteady groundwater flow equation with stochastic parameters. J. Stoch. Hydrol. Hydraul. 1, 281–296
 Serrano, S.E. Unny, T.E. 1987b: Predicting groundwater flow in a phreatic aquifer. J. of Hydrol. 95, 241–268
 Serrano, S.E.; Unny, T.E. 1986: Boundary element solution of the twodimensional groundwater flow equation with stochastic freesurface boundary condition. Num. Meth. for Part. Diff. Eqs. 2, 237–258
 Unny, T.E. 1984: Numerical integration of stochastic differential equations in Catchment modeling, Water resources research, Vol. 20(3).
 Zauderer, E. 1983: Partial differential equations of applied mathematics. John Wiley & Sons
 Title
 Stochastic partial differential equations in groundwater hydrology
 Journal

Stochastic Hydrology and Hydraulics
Volume 3, Issue 2 , pp 135153
 Cover Date
 19890601
 DOI
 10.1007/BF01544077
 Print ISSN
 09311955
 Online ISSN
 1435151X
 Publisher
 SpringerVerlag
 Additional Links
 Topics

 Math. Applications in Geosciences
 Probability Theory and Stochastic Processes
 Statistics for Engineering, Physics, Computer Science, Chemistry & Geosciences
 Numerical and Computational Methods in Engineering
 Math. Appl. in Environmental Science
 Waste Water Technology / Water Pollution Control / Water Management / Aquatic Pollution
 Keywords

 Stochastic partial differential equations
 maximum likelihood estimation
 parameter estimation
 moment equations
 stodhastic contaminant transport
 Industry Sectors
 Authors

 T. E. Unny ^{(1)}
 Author Affiliations

 1. Dept. of System Design Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada