# American Institute of Mathematical Sciences

## A novel model for the contamination of a system of three artificial lakes

 Muğla Sıtkı Koçman University, Muğla, 48300, Turkey

Received  April 2019 Revised  May 2019 Published  December 2019

In this study, a new model has been developed to monitor the contamination in connected three lakes. The model has been motivated by two biological models, i.e. cell compartment model and lake pollution model. Haar wavelet collocation method has been proposed for the numerical solutions of the model containing a system of three linear differential equations. In addition to the solutions of the system, convergence analysis has been briefly given for the proposed method. The contamination in each lake has been investigated by considering three different pollutant input cases, namely impulse imposed pollutant source, exponentially decaying imposed pollutant source, and periodic imposed pollutant source. Each case has been illustrated with a numerical example and results are compared with the exact ones. Regarding the results in each case it has been seen that, Haar wavelet collocation method is an efficient algorithm to monitor the contamination of a system of lakes problem.

Citation: Veysel Fuat Hatipoğlu. A novel model for the contamination of a system of three artificial lakes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020176
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Illustration of the interconnected Lakes 1, 2, 3, and flow $F_{12}$, $F_{13}$, $F_{21}$, $F_{23}$, $F_{31}$, $F_{32}$
Graphical representation of approximate and exact solutions of Example 4.1 for $m = 8$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Graphical representation of approximate and exact solutions of Example 4.1 for $m = 256$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Graphical representation of approximate and exact solutions of Example 4.2 for $m = 8$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Graphical representation of approximate and exact solutions of Example 4.2 for $m = 256$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Graphical representation of approximate and exact solutions of Example 4.3 for $m = 8$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Graphical representation of approximate and exact solutions of Example 4.3 for $m = 256$ of (a) the function $u_1(t)$ (pollution in Lake 1), (b) the function $u_2(t)$ (pollution in Lake 2), and (c) the function $u_3(t)$ (pollution in Lake 3)
Numerical results for the case of the impulse input imposed pollutant source for $m = 8$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 22.9684 3.00002 0.0164893 0.0000103978 0.0151401 $5.40755\times 10^{-6}$ 0.4 41.8738 2.00003 0.0657719 0.0000194067 0.0604641 0.0000101545 0.6 57.7167 1.99995 0.147536 0.0000303268 0.135809 0.0000159672 0.8 76.4975 2.99994 0.26148 0.0000389783 0.241017 0.0000206502 1 99.2168 0.0000745642 0.407297 0.0000486384 0.375927 0.0000259258
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 22.9684 3.00002 0.0164893 0.0000103978 0.0151401 $5.40755\times 10^{-6}$ 0.4 41.8738 2.00003 0.0657719 0.0000194067 0.0604641 0.0000101545 0.6 57.7167 1.99995 0.147536 0.0000303268 0.135809 0.0000159672 0.8 76.4975 2.99994 0.26148 0.0000389783 0.241017 0.0000206502 1 99.2168 0.0000745642 0.407297 0.0000486384 0.375927 0.0000259258
Numerical results for the case of the impulse input imposed pollutant source for $m = 256$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 20.0309 0.0625 0.0164996 $9.84289\times 10^{-9}$ 0.0151455 $5.11793\times 10^{-9}$ 0.4 39.78 0.09375 0.0657913 $1.95302\times 10^{-8}$ 0.0604743 $1.02186\times 10^{-8}$ 0.6 59.8104 0.09375 0.147566 $2.90455\times 10^{-8}$ 0.135825 $1.52923\times 10^{-8}$ 0.8 79.4349 0.0624999 0.261519 $3.83733\times 10^{-8}$ 0.241038 $2.03287\times 10^{-8}$ 1 99.2167 $7.28163\times 10^{-8}$ 0.407345 $4.74981\times 10^{-8}$ 0.375953 $2.53183\times 10^{-8}$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 20.0309 0.0625 0.0164996 $9.84289\times 10^{-9}$ 0.0151455 $5.11793\times 10^{-9}$ 0.4 39.78 0.09375 0.0657913 $1.95302\times 10^{-8}$ 0.0604743 $1.02186\times 10^{-8}$ 0.6 59.8104 0.09375 0.147566 $2.90455\times 10^{-8}$ 0.135825 $1.52923\times 10^{-8}$ 0.8 79.4349 0.0624999 0.261519 $3.83733\times 10^{-8}$ 0.241038 $2.03287\times 10^{-8}$ 1 99.2167 $7.28163\times 10^{-8}$ 0.407345 $4.74981\times 10^{-8}$ 0.375953 $2.53183\times 10^{-8}$
Numerical results for the case of the pollutant source is exponential decaying for $m = 8$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 21.1544 3.89703 0.0158795 0.00284471 0.0145808 0.00261002 0.4 21.6476 2.10907 0.0445744 0.00499602 0.0409975 0.00459647 0.6 15.2365 4.55673 0.0747691 0.00702142 0.0688998 0.00647562 0.8 13.554 6.22018 0.104924 0.00900802 0.0968779 0.00832662 1 18.4945 1.2239 0.134828 0.0109746 0.124735 0.0101665
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 21.1544 3.89703 0.0158795 0.00284471 0.0145808 0.00261002 0.4 21.6476 2.10907 0.0445744 0.00499602 0.0409975 0.00459647 0.6 15.2365 4.55673 0.0747691 0.00702142 0.0688998 0.00647562 0.8 13.554 6.22018 0.104924 0.00900802 0.0968779 0.00832662 1 18.4945 1.2239 0.134828 0.0109746 0.124735 0.0101665
Numerical results for the case of the pollutant source is exponential decaying for $m = 256$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 17.3813 0.123881 0.0187212 $3.00622\times 10^{-6}$ 0.0171881 $2.75854\times 10^{-6}$ 0.4 19.3498 0.188703 0.0495652 $5.197\times 10^{-6}$ 0.0455892 $4.78187\times 10^{-6}$ 0.6 19.9795 0.18621 0.0817833 $7.25942\times 10^{-6}$ 0.0753687 $6.69578\times 10^{-6}$ 0.8 19.6479 0.126229 0.113923 $9.28787\times 10^{-6}$ 0.105196 $8.58611\times 10^{-6}$ 1 19.7171 0.00124969 0.145791 0.0000112953 0.134891 0.0000104645
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 17.3813 0.123881 0.0187212 $3.00622\times 10^{-6}$ 0.0171881 $2.75854\times 10^{-6}$ 0.4 19.3498 0.188703 0.0495652 $5.197\times 10^{-6}$ 0.0455892 $4.78187\times 10^{-6}$ 0.6 19.9795 0.18621 0.0817833 $7.25942\times 10^{-6}$ 0.0753687 $6.69578\times 10^{-6}$ 0.8 19.6479 0.126229 0.113923 $9.28787\times 10^{-6}$ 0.105196 $8.58611\times 10^{-6}$ 1 19.7171 0.00124969 0.145791 0.0000112953 0.134891 0.0000104645
Numerical results for the case of the periodic imposed pollutant source for $m = 8$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 0.249602 0.0300057 0.000178068 $2.08508\times 10^{-6}$ 0.000163497 $1.96112\times 10^{-6}$ 0.4 0.497558 0.0200493 0.000748935 $3.8634\times 10^{-6}$ 0.000688452 $3.6487\times 10^{-6}$ 0.6 0.751366 0.0199072 0.0017721 $5.90008\times 10^{-6}$ 0.001631 $5.59805\times 10^{-6}$ 0.8 1.06715 0.0298126 0.00330008 $7.43142\times 10^{-6}$ 0.00304105 $7.08325\times 10^{-6}$ 1 1.44966 0.000281844 0.00537877 $8.97633\times 10^{-6}$ 0.00496263 $8.59806\times 10^{-6}$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 0.249602 0.0300057 0.000178068 $2.08508\times 10^{-6}$ 0.000163497 $1.96112\times 10^{-6}$ 0.4 0.497558 0.0200493 0.000748935 $3.8634\times 10^{-6}$ 0.000688452 $3.6487\times 10^{-6}$ 0.6 0.751366 0.0199072 0.0017721 $5.90008\times 10^{-6}$ 0.001631 $5.59805\times 10^{-6}$ 0.8 1.06715 0.0298126 0.00330008 $7.43142\times 10^{-6}$ 0.00304105 $7.08325\times 10^{-6}$ 1 1.44966 0.000281844 0.00537877 $8.97633\times 10^{-6}$ 0.00496263 $8.59806\times 10^{-6}$
Numerical results for the case of the periodic imposed pollutant source for $m = 256$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 0.220221 0.000625009 0.000175985 $1.97563\times 10^{-9}$ 0.000161538 $1.8579\times 10^{-9}$ 0.4 0.476572 0.000937457 0.000745076 $3.87845\times 10^{-9}$ 0.000684807 $3.66306\times 10^{-9}$ 0.6 0.77221 0.0009376 0.00176621 $5.66633\times 10^{-9}$ 0.00162541 $5.37572\times 10^{-9}$ 0.8 1.09634 0.000624822 0.00329265 $7.30236\times 10^{-9}$ 0.00303397 $6.96049\times 10^{-9}$ 1 1.44937 $2.75122\times 10^{-7}$ 0.0053698 $8.75757\times 10^{-9}$ 0.00495404 $8.38886\times 10^{-9}$
 $t$ App. sol. of $u_1$ Abs. error in $u_1$ App. sol. of $u_2$ Abs. error in $u_2$ App. sol. of $u_3$ Abs. error in $u_3$ 0 0 0 0 0 0 0 0.2 0.220221 0.000625009 0.000175985 $1.97563\times 10^{-9}$ 0.000161538 $1.8579\times 10^{-9}$ 0.4 0.476572 0.000937457 0.000745076 $3.87845\times 10^{-9}$ 0.000684807 $3.66306\times 10^{-9}$ 0.6 0.77221 0.0009376 0.00176621 $5.66633\times 10^{-9}$ 0.00162541 $5.37572\times 10^{-9}$ 0.8 1.09634 0.000624822 0.00329265 $7.30236\times 10^{-9}$ 0.00303397 $6.96049\times 10^{-9}$ 1 1.44937 $2.75122\times 10^{-7}$ 0.0053698 $8.75757\times 10^{-9}$ 0.00495404 $8.38886\times 10^{-9}$
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