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This paper describes the optimal fish-feeding in a three-dimensional calm freshwater pond based on the concentrations of seven water quality variables. A certain number of baby fishes are inserted into the pond simultaneously. They are then taken out of the pond simultaneously for harvest after having gone through a feeding program. This feeding program creates additional loads of water quality variables in the pond, which becomes pollutants. Thus, an optimal fish-feeding problem is formulated to maximize the final weight of the fishes, subject to the restrictions that the fishes are not under-fed and over-fed and the concentrations of the pollutants created by the fish-feeding program are not too large. A computational scheme using the finite element Galerkin scheme for the three-dimensional cubic domain and the control parameterization method is developed for solving the problem. Finally, a numerical example is solved.
Citation: |
Table 1. Weight of the fishes at the final time
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Table 2. Maximum concentration at various places of the pond in the fish-feeding water pollution (FFWP) model obtained by using optimal fish-feeding rate
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Table A1. Processes described in the Three-Dimensional Water Pollution Model and their equations
D. Decay of PHY |
where |
(ⅰ) |
(ⅱ) |
G. Photosynthesis (Growth of PHY, |
where |
(ⅰ) |
is the growth rate of PHY; |
is the effect of the temperature on the growth of PHY at temperature |
with |
with |
(ⅱ) |
A. Adsorption of NAM |
where |
(ⅰ) |
(ⅱ) |
Set. Settling of PHY |
where |
(ⅰ) |
(ii) |
Am. Ammonification - Mineralization of NOR |
(ⅰ) |
(ⅱ) |
is the saturation constant for Nitrogen mineralization at |
(ⅲ) |
N. Nitrification |
where |
(ⅰ) |
(ⅱ) |
(ⅲ) |
R. Endogenous respiration of PHY |
where |
|
(ⅱ) |
where the formula for |
(ⅲ) |
P. Mineralization of POR |
where |
(ⅰ) |
(ⅱ) |
is the mineralization rate of |
(ⅲ) |
Table A2. Parameters of the three-dimensional water pollution model and their values
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Boundary conditions of the water pollution model
The Front Part of the Fish Pond
The global node point of the fish pond
Instantaneous average concentration in the No-Fish-Feeding Water Pollution (NFFWP) sub-model
Comparison of the instantaneous average concentration between the NFFWP sub-model, the FFWP sub-model with
The instantaneous optimal control (i.e. the instantaneous optimal fishes' feeding rate)
Comparison of the instantaneous average concentration between the NFFWP sub-model and the FFWP sub-model obtained by using the optimal control