Institute of Computational Mathematics and Mathematical Geophysics

Abstracts

Mathematical modeling of environmental protection problems

The increasing demand groundwater as a drinking water resource requires the development of remediation methods. The damage caused by contaminant or sludge particles to the waste water treatment plant varies spatially due to the sludge settling and sludge velocity, depending on time. In this report we examine some mathematical problems associated with a modeling of sludge particles in the water treatment plant (settler), when the residence time of in sludge particles in the settler is considered. In an effort to understand optimal control of parabolic partial differential equations governing age-dependent transport-diffusion problem, we consider the case of one spatial dimension. The model is motivated by modeling of a settler as an environmental remedation process.

In the case of "average constant velocity" we reduce the nonlocal nonlinear identification problem for parabolic equation to the two point boundary value problem for the second order nonlinear ordinary differential equation with respect to $c=c(x)$. Although this problem may be solved by nonlinear solver methods, we transform this problem to nonlinear equation, i.e. integral representation for the function $c=c(x)$. We present iterative procedure for the nonlinear equation and derive sufficient condition for its convergence. We also present a simple engineering approach for estimation of sludge concentration in a settler, in the case of "average constant velocity".

Effectiveness of this approach, as well as convergence of the iteration procedure for the nonlinear equation with respect to $c=c(x)$, is demonstrated on concrete examples. As we demonstrate in the final session, by using the presented approach and numerical methods a comprehensive numerical investigation of the function sludge concentration $c(x)$, as well as the density $u(x,t)$ of sludge particles in a settler, can be successfully performed for different input data.

References
[1] Chancelier JP, Cohen de Lara M, Pacard F. Analysis of a conservation PDE with discontinuous flux: a model of settler. SIAM J Appl Math 1994;54:954-995.
[2] Chancelier JP, Cohen de Lara M, Pacard F. Existence of a solution in an age-dependent transport-diffusion PDE: a model of settler. Math Models and Methods in Appl Sci 1995;5(3):267-278.
[3] Cohn S, Pfabe K, Redepenning J. A similarity solution to a problem in nonlinear ion transport with a nonlocal condition. Math Mod Meth Appl Sci 1999; 9(3): 445-461.
[4] Hasanov A. Variational approach to a nonlocal identification problem related to nonlinear ion transport. Math Methods in Appl Sci 1998; 21: 1195-1206.
[5] Hasanov A, Mueler J, Cohn S, Redepenning J. Numerical solution of a nonlocal identification problem related to nonlinear ion transport. Computers $&$ math with appl 2000; 39: 225-235.

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