Boubaker Collocation Method for Approximate Solutions of the Model of Pollution for a System of Lakes

Abstract

This paper focuses on a numerical approach for the solution of the pollution problem for a system of lakes. The pollution problem consists of three lakes with interconnecting channels and this model corresponds to a system of linear differential equations. The main purpose of this study is to present a collocation method based on the Boubaker polynomials to obtain approximate solutions of this pollution model. Firstly, the approximation solutions are assumed in the forms of the truncated series of the Boubaker polynomials. The solution forms and their derivatives are written in the matrix forms. By means of these matrix forms, the matrix operations and the collocation points, the pollution model is reduced to a system of the algebraic linear equations. In addition, the error estimation method is presented by using the residual function. The parameters in the pollution model are selected according to the datas in the literature. For the selected parameters, the applications of the presented method are made by using a code written in MATLAB. The application results are compared with the results of other methods in the literature. The effectiveness and reliability of the presented method are observed from the obtained results.

Keywords

• Approximate solutions
• Boubaker polynomials
• collocation method
• error estimation
• the pollution model of lakes

References

1. Abdeljawad, T., Amin, R., Shah, K., Al-Mdallal, Q., Jarad, F., Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method, Alexandria Engineering Journal, 59(2020), 2391–2400.
2. Awoyemi, D.O., Idowu, O.M., A class of hybrid collocation methods for third-order ordinary differential equations, International Journal of Computer Mathematics, 82(2005), 1287–1293.
3. Benhammouda, B., Vazquez-Leal, H., Hernandez-Martinez, L., Modified differential transform method for solving the model of pollution for a system of lakes, Discrete Dynamics in Nature and Society, 2014(2014).
4. Biazar, J., Farrokhi, L., Islam, M.R., Modeling the pollution of a system of lakes, Applied Mathematics and Computation, 178(2006), 423–430.
5. Biazar, J., Shahbala, M., Ebrahimi, H., VIM for solving the pollution problem of a system of lakes, Journal of Control Science and Engineering, 2010(2010), 1–6.
6. Bildik, N., Deniz, S., Implementation of Taylor collocation and Adomian decomposition method for systems of ordinary differential equations, In AIP Conference Proceedings, 1648(2015), 370002.
7. D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B., Two-step almost collocation methods for ordinary differential equations, Numerical Algorithms, 53(2010), 195–217.
8. Dolapçi, İ. T., Chebyshev collocation method for solving linear differential equations, Mathematical and Computational Applications, 9(2004), 107–115.

9. Faghih, A., Mokhtary, P., A new fractional collocation method for a system of multi-order fractional differential equations with variable coefficients, Journal of Computational and Applied Mathematics, 383(2021), 113139.
10. Giordano, F.R., Weir, M.D., Differential Equations: A Modern Approach, Addison Wesley Publishing Company, 1991.
11. Guo, B.Y., Wang, Z.Q., A spectral collocation method for solving initial value problems of first order ordinary differential equations, Discrete & Continuous Dynamical Systems-B, 14(2010), 1029–1054.
12. Guo, B.Y., Wang, Z.Q., Legendre–Gauss collocation methods for ordinary differential equations, Advances in Computational Mathematics, 30(2009), 249–280.
13. Guo, B.Y., Yan, J.P., Legendre–Gauss collocation method for initial value problems of second order ordinary differential equations, Applied Numerical Mathematics, 59(2009), 1386–1408.
14. Labiadh, H., Boubaker K., A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials, Differential Equations and Control Processes, 2(2007), 117–133.
15. Merdan, M., A new application of modified differential transformation method for modelling the pollution of a system of lakes, Selc¸uk Journal of Applied Mathematics, 11(2010), 27–40.
16. Merdan, M., Homotopy perturbation method for solving modelling the pollution of a system of lakes, Suleyman Demirel University, Faculty of Science and Literature, Journal of Science, 4(2009), 99–111.
17. Sabermahani, S., Ordokhani, Y., An analytical method for solving the model of pollution for a system of lakes, (2016), Available at SSRN 3382429.
18. Sezer, M., Gülsu, M., Tanay, B., Rational Chebyshev collocation method for solving higher-order linear ordinary differential equations, Numerical Methods for Partial Differential Equations, 27(2011), 1130–1142.
19. Sezer, M., Karamete, A., Gülsu, M., Taylor polynomial solutions of systems of linear differential equations with variable coefficients, International Journal of Computer Mathematics, 82(2005), 755–764.
20. Sokhanvar, E., Yousefi, S., The Bernoulli Ritz-collocation method to the solution of modelling the pollution of a system of lakes, Caspian Journal of Mathematical Sciences (CJMS) peer, 3(2014), 253–265.
21. Verma, A.K., Kumar, N., Tiwari, D., Haar wavelets collocation method for a system of nonlinear singular differential equations, Engineering Computations, 38(2021), 659–698.
22. Wang, B., Meng, F., Fang, Y., Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations, Applied Numerical Mathematics, 119(2017), 164–178.
23. Wang, Z.Q., Guo, B.Y., Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, Journal of Scientific Computing, 52(2012), 226–255.
24. Wright, K., Chebyshev collocation methods for ordinary differential equations. The Computer Journal, 6(1964), 358–365.
25. Wu, X., Wang, B., Exponential Fourier collocation methods for first-order differential equations, In Recent Developments in Structure- Preserving Algorithms for Oscillatory Differential Equations (2018), 55–84.
26. Xiu, D., Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing, 27(2005), 1118–1139.
27. Yahaya, Y.A., Badmus, A.M., A class of collocation methods for general second order ordinary differential equations, African Journal of Mathematics and Computer Science Research, 2(2009), 069–072.
28. Yalçınbaş, S., Akkaya, T., A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases, Ain Shams Engineering Journal, 3(2012), 153–161.
29. Yap, L. K., Ismail, F., Senu, N., An accurate block hybrid collocation method for third order ordinary differential equations, Journal of Applied Mathematics, 2014(2014).
30. Yüzbaşı, Ş., Sezer, M., An exponential matrix method for solving systems of linear differential equations, Mathematical Methods in the Applied Sciences, 36(2013), 336–348.
31. Yüzbaşı, Ş., Şahin, N., Sezer, M., A collocation approach to solving the model of pollution for a system of lakes, Mathematical and Computer Modelling, 55(2012), 330–341.
32. Yüzbaşı, Ş., Yıldırım, G., A Laguerre approach for solving of the systems of linear differential equations and residual improvement, Computational Methods for Differential Equations, 9(2021), 553–576.
33. Yüzbaşı, Ş., Yıldırım, G., Laguerre Collocation Method for Solutions of Systems of First Order Linear Differential Equations, Turkish Journal of Mathematics and Computer Science, 10(2018), 222–241.
34. Zaky, M.A., An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Applied Numerical Mathematics, 154(2020), 205–222.