A General Fractional Pollution Model for Lakes

doi:10.1007/s42967-021-00135-4

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 1105-1130.doi: 10.1007/s42967-021-00135-4

• ORIGINAL PAPER • Previous Articles Next Articles

A General Fractional Pollution Model for Lakes

Babak Shiri¹, Dumitru Baleanu^2,3

1. Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, Sichuan, China;
2. Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey;
3. Institute of Space Sciences, Magurele-Bucharest, Romania

Received:2020-11-30 Revised:2021-03-20 Online:2022-09-20 Published:2022-07-04
Contact: Babak Shiri,E-mail:shiri@tabrizu.ac.ir;Dumitru Baleanu,E-mail:dumitru@cankaya.edu.tr E-mail:shiri@tabrizu.ac.ir;dumitru@cankaya.edu.tr

Abstract

Abstract: 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.

Key words: Amount of pollution in lakes, System of ordinary differential equations, System of fractional differential equations, Explicit and implicit methods, Regularity of the solution

CLC Number:

34A08
45D05

Babak Shiri, Dumitru Baleanu. A General Fractional Pollution Model for Lakes[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1105-1130.

TrendMD

References

[1] Ahmed, M.E., Khan, M.A.: Modeling and analysis of the polluted lakes system with various fractional approaches. Chaos Solitons Fractals 134, 109720 (2020)
[2] Alijani, Z., Baleanu, D., Shiri, B., Wu, G.C.: Spline collocation methods for systems of fuzzy fractional differential equations. Chaos Solitons Fractals 131, 109510 (2020)
[3] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, New Jersey (2012)
[4] Baleanu, D., Muslih, S.I., Taş, K.: Fractional Hamiltonian analysis of higher order derivatives systems. J. Math. Phys. 47(10), 103503 (2006)
[5] Bildik, N., Deniz, S.: A new fractional analysis on the polluted lakes system. Chaos Solitons Fractals 122, 17–24 (2019)
[6] Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)
[7] Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, England (2016)
[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)
[9] Dassios, I.K., Baleanu, D.: Caputo and related fractional derivatives in singular systems. Appl. Math. Comput. 337, 591–606 (2018)
[10] Dassios, I., Tzounas, G., Milano, F.: Generalized fractional controller for singular systems of differential equations. J. Comput. Appl. Math. 378, 112919 (2020)
[11] Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)
[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)
[13] Hairer, E., Nørsett, S.P., Wanner G.: Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, Heidelberg (1993)
[14] Han, W., Atkinson, K.E.: Theoretical Numerical Analysis: a Functional Analysis Framework. Springer, New York (2009)
[15] Kachia, K., Solís-Pérez, J.E., Gómez-Aguilar, J.F.: 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)
[16] Khalid, M., Sultana, M., Zaidi, F., Khan, F.S.: Solving polluted lakes system by using perturbation-iteration method. Int. J. Comput. Appl. 114(4), 1–7 (2015)
[17] Khiabani, E.D., 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)
[18] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
[19] Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, Boca Raton (2015)
[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)
[21] Prakasha, D.G., Veeresha, P.: Analysis of lakes pollution model with Mittag-Leffler kernel. J. Ocean Eng. Sci. 5(4), 310–322 (2020)
[22] Sheng, H., Chen, Y.: FARIMA with stable innovations model of Great Salt Lake elevation time series. Signal Process. 91(3), 553–561 (2011)
[23] Shiri, B., Wu, G.C., Baleanu, D.: Collocation methods for terminal value problems of tempered fractional differential equations. Appl. Numer. Math. 156, 385–395 (2020)
[24] Sousa, J.V.C., dos Santos, M.N.N., Magna, L.A., Oliveira, E.C.: Validation of a fractional model for erythrocyte sedimentation rate. Comput. Appl. Math. 37, 6903–6919 (2018)
[25] Srivastava, H.M., Saad, K.M., Gómez-Aguilar, J.F., Almadiy, A.A.: Some new mathematical models of the fractional-order system of human immune against IAV infection. Math. Biosci. Eng. 17(5), 4942–4969 (2020)
[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)
[27] Zaky, M.A.: 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)

A General Fractional Pollution Model for Lakes

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 2

Recommended Articles

Metrics

Comments

[1]	Jin You, Mengrui Xu, Shurong Sun. Existence of Boundary Value Problems for Impulsive Fractional Diferential Equations with a Parameter [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 585-604.
[2]	Junying Cao, Ziqiang Wang, Chuanju Xu. A High-Order Scheme for Fractional Ordinary Diferential Equations with the Caputo-Fabrizio Derivative [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 179-199.