A general fractional pollution model for lakes. (English) Zbl 1513.34331
Summary: A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 45D05 | Volterra integral equations |
| 92D40 | Ecology |
| 34K37 | Functional-differential equations with fractional derivatives |
Keywords:
amount of pollution in lakes; system of ordinary differential equations; system of fractional differential equations; explicit and implicit methods; regularity of the solutionReferences:
| [1] | Ahmed, ME; Khan, MA, Modeling and analysis of the polluted lakes system with various fractional approaches, Chaos Solitons Fractals, 134, 109720 (2020) · Zbl 1483.92153 · doi:10.1016/j.chaos.2020.109720 |
| [2] | Alijani, Z.; Baleanu, D.; Shiri, B.; Wu, GC, Spline collocation methods for systems of fuzzy fractional differential equations, Chaos Solitons Fractals, 131, 109510 (2020) · Zbl 1495.65017 · doi:10.1016/j.chaos.2019.109510 |
| [3] | Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, JJ, Fractional Calculus: Models and Numerical Methods (2012), New Jersey: World Scientific, New Jersey · Zbl 1248.26011 · doi:10.1142/8180 |
| [4] | Baleanu, D.; Muslih, SI; Taş, K., Fractional Hamiltonian analysis of higher order derivatives systems, J. Math. Phys., 47, 10, 103503 (2006) · Zbl 1112.81074 · doi:10.1063/1.2356797 |
| [5] | Bildik, N.; Deniz, S., A new fractional analysis on the polluted lakes system, Chaos Solitons Fractals, 122, 17-24 (2019) · Zbl 1448.92369 · doi:10.1016/j.chaos.2019.02.001 |
| [6] | Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 |
| [7] | Butcher, JC, Numerical Methods for Ordinary Differential Equations (2016), England: Wiley, England · Zbl 1354.65004 · doi:10.1002/9781119121534 |
| [8] | Dadkhah, E.; Shiri, B.; Ghaffarzadeh, H.; Baleanu, D., Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods, J. Appl. Math. Comput., 63, 29-57 (2020) · Zbl 1490.74071 · doi:10.1007/s12190-019-01307-5 |
| [9] | Dassios, IK; Baleanu, D., Caputo and related fractional derivatives in singular systems, Appl. Math. Comput., 337, 591-606 (2018) · Zbl 1427.34007 |
| [10] | Dassios, I.; Tzounas, G.; Milano, F., Generalized fractional controller for singular systems of differential equations, J. Comput. Appl. Math., 378, 112919 (2020) · Zbl 1447.93139 · doi:10.1016/j.cam.2020.112919 |
| [11] | Diethelm, K., The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type (2010), Berlin: Springer, Berlin · Zbl 1215.34001 |
| [12] | Einstein, A., On the movement of small particles suspended in stationary liquids required by the molecular kinetic theory of heat, Ann. Phys., 17, 549-560 (1905) · JFM 36.0975.01 · doi:10.1002/andp.19053220806 |
| [13] | Hairer, E., Nørsett, S.P., Wanner G.: Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, Heidelberg (1993) · Zbl 0789.65048 |
| [14] | Han, W.; Atkinson, KE, Theoretical Numerical Analysis: a Functional Analysis Framework (2009), New York: Springer, New York · Zbl 1181.47078 · doi:10.1007/978-1-4419-0458-4 |
| [15] | Kachia, K.; Solís-Pérez, JE; Gómez-Aguilar, JF, Chaos in a three-cell population cancer model with variable-order fractional derivative with power, exponential and Mittag-Leffler memories, Chaos Solitons Fractals, 140, 110177 (2020) · Zbl 1495.92030 · doi:10.1016/j.chaos.2020.110177 |
| [16] | Khalid, M.; Sultana, M.; Zaidi, F.; Khan, FS, Solving polluted lakes system by using perturbation-iteration method, Int. J. Comput. Appl., 114, 4, 1-7 (2015) |
| [17] | Khiabani, ED; Ghaffarzadeh, H.; Shiri, B.; Katebi, J., Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models, J. Vib. Control, 26, 17-18, 1445-1462 (2020) · doi:10.1177/1077546319898570 |
| [18] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and Applications of Fractional Differential Equations (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [19] | Li, C.; Zeng, F., Numerical Methods for Fractional Calculus (2015), Boca Raton: Chapman and Hall/CRC, Boca Raton · Zbl 1326.65033 · doi:10.1201/b18503 |
| [20] | Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. Math. Gen., 37, 31, 161-208 (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01 |
| [21] | Prakasha, DG; Veeresha, P., Analysis of lakes pollution model with Mittag-Leffler kernel, J. Ocean Eng. Sci., 5, 4, 310-322 (2020) · doi:10.1016/j.joes.2020.01.004 |
| [22] | Sheng, H.; Chen, Y., FARIMA with stable innovations model of Great Salt Lake elevation time series, Signal Process., 91, 3, 553-561 (2011) · Zbl 1203.94053 · doi:10.1016/j.sigpro.2010.01.023 |
| [23] | Shiri, B.; Wu, GC; Baleanu, D., Collocation methods for terminal value problems of tempered fractional differential equations, Appl. Numer. Math., 156, 385-395 (2020) · Zbl 1455.65238 · doi:10.1016/j.apnum.2020.05.007 |
| [24] | Sousa, JVC; dos Santos, MNN; Magna, LA; Oliveira, EC, Validation of a fractional model for erythrocyte sedimentation rate, Comput. Appl. Math., 37, 6903-6919 (2018) · Zbl 1438.92033 · doi:10.1007/s40314-018-0717-0 |
| [25] | Srivastava, HM; Saad, KM; Gómez-Aguilar, JF; Almadiy, AA, Some new mathematical models of the fractional-order system of human immune against IAV infection, Math. Biosci. Eng., 17, 5, 4942-4969 (2020) · Zbl 1470.92088 · doi:10.3934/mbe.2020268 |
| [26] | Yüzbaşı, Ş.; Şahin, N.; Sezer, M., A collocation approach to solving the model of pollution for a system of lakes, Math. Comput. Model., 55, 3-4, 330-341 (2012) · Zbl 1255.65152 · doi:10.1016/j.mcm.2011.08.007 |
| [27] | Zaky, MA, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357, 103-122 (2019) · Zbl 1415.41001 · doi:10.1016/j.cam.2019.01.046 |
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