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 non-homogeneous 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. Springer-Verlag
- Cannon, J.R. 1984: The one-dimensional heat equation. In: Encyclopedia of Mathematics and its Applications, Rota, G-C (ed.). Addison-Wesley 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. Springer-Verlag
- 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 advective-dispersive 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 advective-dispersive 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 two-dimensional groundwater flow equation with stochastic free-surface 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
- Stochastic partial differential equations in groundwater hydrology
Stochastic Hydrology and Hydraulics
Volume 3, Issue 2 , pp 135-153
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- 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
- Stochastic partial differential equations
- maximum likelihood estimation
- parameter estimation
- moment equations
- stodhastic contaminant transport
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- T. E. Unny (1)
- Author Affiliations
- 1. Dept. of System Design Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada