Motivated by our recent publication in ([9]), which was supported during the academic year 2016/1437 by the Deanship of Scientific Research in Project 1175, we will develop the previous mathematical model into a two-dimensional model of the transition equations of some pollutants and their rapid spread in a particular liquid, taking into account the given stability analysis of the numerical method which based on finite difference time scheme combined with Galerkin special approximation. This method has been given by our published work in ([12]).
S.Abdel-Khalek, Y.S.El-Saman and M.Abdel-Aty, Geometric phase of a moving three-level atom, Optics Communications283(9) (2010), 1826–1831.
2.
M.Abdel-Aty, Quantum information entropy and multi-qubit entanglement, Progress in Quantum Electronics31(1) (2007), 1–49.
3.
M.Abdel-Aty, An investigation of entanglement and quasiprobability distribution in a generalized Jaynes-Cummings model, Journal of Mathematical Physics44(4) (2003), 1457–1471.
4.
J.Batle, C.R.Ooi, A.Farouk, M.Abutalib and S.Abdalla, Do multipartite correlations speed up adiabatic quantum computation or quantum annealing?Quantum Information Processing15(8) (2016), 3081–3099.
5.
J.Batle, A.Farouk, O.Tarawneh and S.Abdalla, Multipartite quantum correlations among atoms in QED cavities, Frontiers of Physics13(1) (2018), 130305.
6.
J.Batle, C.R.Ooi, A.Farouk, M.S.Alkhambashi and S.Abdalla, Global versus local quantum correlations in the Grover search algorithm, Quantum Information Processing15(2) (2016), 833–849.
7.
J.Batle, M.Naseri, M.Ghoranneviss, A.Farouk, M.Alkhambashi and M.Elhoseny, Shareability of correlations in multiqubit states: Optimization of nonlocal monogamy inequalities, Physical Review A95(3) (2017), 032123.
8.
J.Batle, O.Ciftja, M.Naseri, M.Ghoranneviss, A.Farouk and M.Elhoseny, Equilibrium and uniform charge distribution of a classical two-dimensional system of point charges with hard-wall confinement, Physica Scripta92(5) (2017), 055801.
9.
S.Boulaaras and A.Allahem, A new mathematical model and its application in the pollution of air and water, Appl Math Inf Sci11(6) (2017), 1615–1624. DOI: 10.18576/amis/11060
10.
M.Ainsworth, Dispersive and dissipative behavior of high order Discontinuous Galerkin finite element methods, J Comp Phy198 (2004), 106–130.
11.
S.Boulaaras and R.Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Math Meth Appl Sci41 (2018), 5203–5210.
12.
S.Boulaaras and M.Haiour, L1-asymptotic behavior for a finite element approximation in parabolic quasi-variational inequalities related to impulse control problem, Appl Math Comp217 (2011), 6443–6450.
13.
S.Boulaaras and M.Haiour, A General case for the maximum norm analysis of an overlapping Schwarz methods of evolutionary HJB equation with nonlinear source terms with the mixed boundary conditions622, App Math Inf Sci9 (2015), 1247–1257.
14.
S.Boulaaras and A.Allahem, Existence of positive solutions of nonlocal p(x)-Kirchhoff evolutionary systems via Sub-Super Solutions Concept, Symmetry11 (2019), 1–11.
15.
M.L.Batteen and Y.-J.Han, On the computational noise of finitedifference schemes used in ocean models Tellus, 33 (1981), 387–396.
16.
J.-M.Beckers and E.Deleersnijder, Stability of a fbtcs scheme applied to the propagation of shallow-water inertia-gravity waves on various space grids, J Comp Phy108 (1993), 95–104.
17.
S.Boulaaras and M.Haiour, The theta time scheme combined with a finite element spatial approximation of Hamilton-Jacobi-Bellman equation, Comp Math Mod25 (2014), 423–438.
18.
S.Boulaaras and M.Haiour, The finite element approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms, Ind Math24 (2013), 161–173.
19.
R.Glowinski, Numerecal methods for Nolinear variational problems, Springer-Verlag, Germny, 1982.
20.
W.Gao, J.L.G.Guirao, M.Abdel-Aty and W.Xi, An independent set degree condition for fractional critical deleted graphs, Discrete and Continuous Dynamical Systems - Series S12(4-5) (2019), 877–886.
21.
M.Haiour and S.Boulaaras, Overlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem622, Pro Math Sci121 (2011), 481–493.
22.
A.Majda, Introduction to PDE's and waves for the atmosphere and ocean, Amer Math Soci, USA (2003).
23.
K.Nagata, T.Nakamura, H.Geurdes, J.Batle, S.Abdalla and A.Farouk, Creating very true quantum algorithms for quantum energy based computing, International Journal of Theoretical Physics57(4) (2018), 973–980.
24.
R.Guefaifia and S.Boulaaras, Existence of positive solution for a class of (p(x), q(x))-Laplacian systems, Rend Circ Mat Palermo, II Ser67 (2018), 93–103.
25.
V.Rostand, D.Y.Le RouxRaviart-Thomas and Brezzi-Douglas-Marini, Finite element approximations of the shallow water equations, Int J Num Meth Fluids57 (2008), 951–976.
26.
L.White and E.Deleersnijder, Diagnoses of vertical transport in a three-dimensional finite element model of the tidal circulation around an island, Est Coastal and Shelf Sci74 (2007), 655–669.
