Main Article Content
This paper discusses a possible generalization of the transport model describing the chlorine concentration decay in pipes. The proposed generalized model is governed by a second-order fractional partial differential equation. The exact solution of the generalized model is obtained via the Laplace transform method and the method of residues. The exact solution reduces to the corresponding published one as the fractional order α tends to one. Analytical expression for the dimensionless cup-mixing average concentration is deduced. Influences of various parameters on the behavior of the dimensionless cup-mixing average concentration are discussed. It is shown that the physical interpretation of the dimensionless cup-mixing average concentration in view of the fractional calculus is completely different than its interpretation in the classical calculus.
- R.M. Clark, E.J. Read, J.C. Hoff, Analysis of Inactivation of Giardia Lamblia by Chlorine, J. Environ. Eng. 115 (1989), 80-90.
- M.W. LeChevallier, C.D. Cawthon, R.G. Lee, Inactivation of biofilm bacteria, Appl. Environ. Microbiol. 54 (1988), 2492-2499.
- B.F. Arnold, J.M. Colford, Treating Water With Chlorine at Point-of-Use to Improve Water Quality and Reduce Child Diarrhea in Developing Countries: A Systematic Review and Meta-Analysis, Amer. J. Trop. Med. Hyg. 76 (2007), 354-364.
- P. Biswas, C. Lu, R.M. Clark, A model for chlorine concentration decay in pipes, Water Res. 27 (1993), 1715-1724.
- J. Jakubowski, M. Wisniewolski, On matching diffusions, Laplace transforms and partial differential equations, Stoch. Proc. Appl. 125 (2015), 3663-3690.
- A. Ebaid, M. Al Sharif, Application of Laplace transform for the exact effect of a magnetic field on heat transfer of carbon-nanotubes suspended nanofluids, Z. Naturforsch., A, 70 (2015), 471-475.
- A. Ebaid, A.M. Wazwaz, E. Alali, B. Masaedeh, Hypergeometric Series Solution to a Class of Second-Order Boundary Value Problems via Laplace Transform with Applications to Nanouids, Commun. Theor. Phys. 67 (2017), 231.
- A. Ebaid, E. Alali, H. Saleh, The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, J. Assoc. Arab Univ. Basi Appl. Sci. 24 (2017), 156-159.
- S.M. Khaled, The exact effects of radiation and joule heating on magnetohydrodynamic Marangoni convection over a flat surface, Therm. Sci. 22 (2018), 63-72.
- H.O. Bakodah, A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018), 331.
- S. Handibag, B.D. Karande, Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives, Int. J. Comput. Eng. Res. 2 (2012), 1049-1052.
- N. Dogan, Solution of the system of ordinary differential equations by combined Laplace transform-Adomian decomposition method, Math. Comput. Appl. 17 (2012), 203-211.
- P. Rai, Application of Laplace Transforms to Solve ODE using MATLAB, J. Inform. Math. Sci. 7 (2015), 93-97.
- S.S. Handibag, B.D. Karande, Laplace Substitution Method for nth Order Linear and Non-Linear PDE's Involving Mixed Partial Derivatives, Int. Res. J. Eng. Technol. 2 (2015), 378-388.
- A.A. Alshikh, M.M.A. Mahgob, A Comparative Study Between Laplace Transform and Two New Integrals “ELzaki” Transform and “Aboodh” Transform, Pure Appl. Math. J. 5 (2016), 145-150.
- A. Atangana, B.S.T. Alkaltani, A novel double integral transform and its applications, J. Nonlinear Sci. Appl. 9 (2016), 424-434.
- X. Lianga, F. Gao, Y.-N. Gao, X.-J. Yang, Applications of a novel integral transform to partial differential equations, J. Nonlinear Sci. Appl. 10 (2017), 528-534.
- P.V. Pavani, U.L. Priya, B.A. Reddy, Solving Differential Equations by using Laplace Transforms, Int. J. Res. Anal. Rev. 5 (2018), 1796-1799.
- B.M. Faraj, F.W. Ahmed, On the MATLAB technique by using Laplace transform for solving second order ODE with initial conditions exactly, Matrix Sci. Math. 3 (2019), 8-10.
- A. Mousa, T.M. Elzaki, Solution of Volterra Integro-Differential Equations by Triple Laplace Transform, Irish Interdiscip. J. Sci. Res. 3 (2019), 67-72.
- R.R. Dhunde, G.L. Waghmare, Double Laplace iterative method for solving nonlinear partial differential equations, New Trends Math. Sci. 7 (2019), 138-149.
- D. Ziane, M.H. Cherif, C. Cattani, K. Belghaba, Yang-Laplace Decomposition Method for Nonlinear System of Local Fractional Partial Differential Equations, Appl. Math. Nonlinear Sci. 4 (2019), 489-502.
- S. Mastoi, W.A.M. Othman, N. Kumaresan, Randomly generated grids and Laplace Transform for partial differential equations, Int. J. Disaster Recovery Bus. Contin. 11 (2020), 1694-1702.
- H. Zhang, M. Nadeem, A. Rauf, Z. Guo Hui, A novel approach for the analytical solution of nonlinear time-fractional differential equations, Int. J. Numer. Meth. Heat Fluid Flow, 31 (2021) 1069-1084.
- M.R. Spiegel, Laplace transforms, McGraw-Hill. Inc., New York, 1965.