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July  2021, 14(7): 2261-2272. doi: 10.3934/dcdss.2020176

## 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  July 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2261-2272. doi: 10.3934/dcdss.2020176
##### References:
 [1] J. Aguirre and D. Tully, Lake pollution model, (1999), Available from: https://mse.redwoods.edu/darnold/math55/DEProj/Sp99/DarJoel/lakepollution.pdf. [2] I. Aziz and S. Islam, New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math., 239 (2013), 333-345.  doi: 10.1016/j.cam.2012.08.031. [3] B. Benhammouda, H. Vazquez-Leal and L. Hernandez-Martinez, A collocation approach to solving the model of pollution for a system of lakes, Discrete Dyn. Nat. Soc., 2014 (2014), Art. ID 645726. [4] İ. Çelik, Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci., 18 (2012), 25-37.  doi: 10.1016/j.ajmsc.2011.08.003. [5] C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc. Control Theory Appl., 144 (1997), 87-94.  doi: 10.1049/ip-cta:19970702. [6] G. Hariharan and K. Kannan, Haar wavelet method for solving some nonlinear parabolic equations, J. Math. Chem., 48 (2010), 1044-1061.  doi: 10.1007/s10910-010-9724-0. [7] G. Hariharan, K. Kannan and K. R. Sharma, Haar wavelet in estimating depth profile of soil temperature, Appl. Math. Comput., 210 (2009), 119-125.  doi: 10.1016/j.amc.2008.12.036. [8] G. Hariharan, K. Kannan and K. R. Sharma, Haar wavelet method for solving Fisher's equation, Appl. Math. Comput., 211 (2009), 284-292.  doi: 10.1016/j.amc.2008.12.089. [9] S. Islam, B. Šarler, I. Aziz and F. Haq, Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems, Int. J. Therm. Sci., 50 (2011), 686-697. [10] M. A. Khanday, A. Rafiq and K. Nazir, Mathematical models for drug diffusion through the compartments of blood and tissue medium, Alexandria J. Med., 53 (2017), 245-249. [11] Ü. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simulation, 68 (2005), 127-143.  doi: 10.1016/j.matcom.2004.10.005. [12] Ü. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput., 176 (2006), 324-333.  doi: 10.1016/j.amc.2005.09.021. [13] Ü. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput., 185 (2007), 695-704.  doi: 10.1016/j.amc.2006.07.077. [14] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276-2285.  doi: 10.1016/j.amc.2010.03.063. [15] Ö. Oruç, F. Bulut and A. Esen, A numerical treatment based on Haar wavelets for coupled KdV equation, Int. J. Optim. Control. Theor. Appl. IJOCTA, 7 (2017), 195-204.  doi: 10.11121/ijocta.01.2017.00396. [16] M. Rehman and R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Model., 36 (2012), 894-907.  doi: 10.1016/j.apm.2011.07.045. [17] M. Rehman and R. A. Khan, Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Appl. Math. Model., 37 (2013), 5233-5244.  doi: 10.1016/j.apm.2012.10.045. [18] H. Saeedi, N. Mollahasani, M. Moghadam and G. Chuev, An operational Haar wavelet method for solving fractional Volterra integral equations, Int. J. Appl. Math. Comput. Sci., 21 (2011), 535-547.  doi: 10.2478/v10006-011-0042-x. [19] I. Singh and S. Kumar, Approximate solution of convection-diffusion equations using a Haar wavelet method, Ital. J. Pure Appl. Math., 35 (2015), 143-154. [20] J. Duintjer Tebbens, M. Azar, E. Friedmann, M. Lanzendörfer and P. Pávek, Mathematical models in the description of pregnane X receptor (PXR)-regulated cytochrome P450 enzyme induction, Int. J. Mol. Sci., 19 (2018), 1785. doi: 10.3390/ijms19061785. [21] S. G. Venkatesh, S. K. Ayyaswamy and G. Hariharan, Haar wavelet method for solving initial and boundary value problems of Bratu-type, Int. J. Comput. Math. Sci., 4 (2010), 286-289. [22] Ş. Yüzbaşı, N. Şahin and M. Sezer, A collocation approach to solving the model of pollution for a system of lakes, Math. Comput. Model., 55 (2012), 330-341.  doi: 10.1016/j.mcm.2011.08.007.

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##### References:
 [1] J. Aguirre and D. Tully, Lake pollution model, (1999), Available from: https://mse.redwoods.edu/darnold/math55/DEProj/Sp99/DarJoel/lakepollution.pdf. [2] I. Aziz and S. Islam, New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math., 239 (2013), 333-345.  doi: 10.1016/j.cam.2012.08.031. [3] B. Benhammouda, H. Vazquez-Leal and L. Hernandez-Martinez, A collocation approach to solving the model of pollution for a system of lakes, Discrete Dyn. Nat. Soc., 2014 (2014), Art. ID 645726. [4] İ. Çelik, Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci., 18 (2012), 25-37.  doi: 10.1016/j.ajmsc.2011.08.003. [5] C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc. Control Theory Appl., 144 (1997), 87-94.  doi: 10.1049/ip-cta:19970702. [6] G. Hariharan and K. Kannan, Haar wavelet method for solving some nonlinear parabolic equations, J. Math. Chem., 48 (2010), 1044-1061.  doi: 10.1007/s10910-010-9724-0. [7] G. Hariharan, K. Kannan and K. R. Sharma, Haar wavelet in estimating depth profile of soil temperature, Appl. Math. Comput., 210 (2009), 119-125.  doi: 10.1016/j.amc.2008.12.036. [8] G. Hariharan, K. Kannan and K. R. Sharma, Haar wavelet method for solving Fisher's equation, Appl. Math. Comput., 211 (2009), 284-292.  doi: 10.1016/j.amc.2008.12.089. [9] S. Islam, B. Šarler, I. Aziz and F. Haq, Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems, Int. J. Therm. Sci., 50 (2011), 686-697. [10] M. A. Khanday, A. Rafiq and K. Nazir, Mathematical models for drug diffusion through the compartments of blood and tissue medium, Alexandria J. Med., 53 (2017), 245-249. [11] Ü. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simulation, 68 (2005), 127-143.  doi: 10.1016/j.matcom.2004.10.005. [12] Ü. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput., 176 (2006), 324-333.  doi: 10.1016/j.amc.2005.09.021. [13] Ü. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput., 185 (2007), 695-704.  doi: 10.1016/j.amc.2006.07.077. [14] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276-2285.  doi: 10.1016/j.amc.2010.03.063. [15] Ö. Oruç, F. Bulut and A. Esen, A numerical treatment based on Haar wavelets for coupled KdV equation, Int. J. Optim. Control. Theor. Appl. IJOCTA, 7 (2017), 195-204.  doi: 10.11121/ijocta.01.2017.00396. [16] M. Rehman and R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Model., 36 (2012), 894-907.  doi: 10.1016/j.apm.2011.07.045. [17] M. Rehman and R. A. Khan, Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Appl. Math. Model., 37 (2013), 5233-5244.  doi: 10.1016/j.apm.2012.10.045. [18] H. Saeedi, N. Mollahasani, M. Moghadam and G. Chuev, An operational Haar wavelet method for solving fractional Volterra integral equations, Int. J. Appl. Math. Comput. Sci., 21 (2011), 535-547.  doi: 10.2478/v10006-011-0042-x. [19] I. Singh and S. Kumar, Approximate solution of convection-diffusion equations using a Haar wavelet method, Ital. J. Pure Appl. Math., 35 (2015), 143-154. [20] J. Duintjer Tebbens, M. Azar, E. Friedmann, M. Lanzendörfer and P. Pávek, Mathematical models in the description of pregnane X receptor (PXR)-regulated cytochrome P450 enzyme induction, Int. J. Mol. Sci., 19 (2018), 1785. doi: 10.3390/ijms19061785. [21] S. G. Venkatesh, S. K. Ayyaswamy and G. Hariharan, Haar wavelet method for solving initial and boundary value problems of Bratu-type, Int. J. Comput. Math. Sci., 4 (2010), 286-289. [22] Ş. Yüzbaşı, N. Şahin and M. Sezer, A collocation approach to solving the model of pollution for a system of lakes, Math. Comput. Model., 55 (2012), 330-341.  doi: 10.1016/j.mcm.2011.08.007.
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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