The Finite Element Approximation in Hyperbolic Equation and Its Application—The Pollution of the Water in the West of Algeria as an Example

Affiliation(s)

University of Mohamed Boudiaf USTO,El’ Monour Oran, Algeria.

Hydrometeorological Institute of Formation and Research, Seddikia, Oran, Algeria.

University of Oran, Oran, Algeria.

University of Mohamed Boudiaf USTO,El’ Monour Oran, Algeria.

Hydrometeorological Institute of Formation and Research, Seddikia, Oran, Algeria.

University of Oran, Oran, Algeria.

Abstract

Hyperbolic variational equations are discussed and their existence and uniqueness of weak solution is established over in the last six decades. In this paper the hyperbolic equations (strong formula) can be transformed into a Hyperbolic variational equations. In this research, we propose a time-space discretization to show the existence and uniqueness of the discrete solution and how we apply it in the transport problem. The proposed approach stands on a discrete*L*^{∞}-stability property with respect to the right-hand side and the boundary conditions of our problem which has been proposed. Furthermore the numerical example is given for the pollution in the smooth fluid as water and we have taken the pollution of the water in the west of Algeria as an example.

Hyperbolic variational equations are discussed and their existence and uniqueness of weak solution is established over in the last six decades. In this paper the hyperbolic equations (strong formula) can be transformed into a Hyperbolic variational equations. In this research, we propose a time-space discretization to show the existence and uniqueness of the discrete solution and how we apply it in the transport problem. The proposed approach stands on a discrete

Cite this paper

S. Boulaaras, K. Mahdi, H. Abderrahim, A. Hamou and S. Kabli, "The Finite Element Approximation in Hyperbolic Equation and Its Application—The Pollution of the Water in the West of Algeria as an Example,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 456-463. doi: 10.4236/am.2013.43068.

S. Boulaaras, K. Mahdi, H. Abderrahim, A. Hamou and S. Kabli, "The Finite Element Approximation in Hyperbolic Equation and Its Application—The Pollution of the Water in the West of Algeria as an Example,"

References

[1] A. Quarteroni and A.Valli, “Numerical Approximation of Partial Differential Equations,” Springer Series in Computational Mathematics, Springer, Berlin and Heidelberg, 2008. doi:10.1007/978-3-540-85268-1

[2] H. M. Duwairi and A. Rablhi, “Fluid over a Vertical Surface,” International Journal of Fluid Mechanics, Vol. 32, No. 1, 2005, pp. 81-87.
doi:10.1615/InterJFluidMechRes.v32.i3.10

[3] J. Z. Jordan, “Network Simulation Method Applied to Radiation and Viscous Dissipation Effects on MHD Unsteady Free Convection over Vertical Porous Plate,” Applied Mathematical Modeling, Vol. 31, No. 9, 2007, pp. 2019-2033. doi:10.1016/j.apm.2006.08.004

[4] S. Boulaaras and M. Haiour, “*L*^{∞}-Asymptotic Behavior for a Finite Element Approximation in Parabolic Quasi-Variational Inequalities Related to Impulse Control Problem,” Applied Mathematics and Computation, Vol. 217, No. 3, 2011, pp. 6443-6450.
doi:10.1016/j.amc.2011.01.025

[5] S. Boulaaras and M. Haiour, “A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities,” ISRN Mathematical Analysis, Vol. 2011, 2011, Article ID: 703670.
doi:10.5402/2011/703670

[6] S. Boulaaras and M. Haiour, “The Finite Element Approximation of Evolutionary Hamilton-Jacobi-Bellman Equations with Nonlinear Source Terms,” Indagationes Mathematicae, Vol. 24, No. 1, 2013, pp. 161-173.
doi:10.1016/j.indag.2012.07.005

[7] M. Haiour and S. Boulaaras, “The Finite Element approximation for a Class of Parabolic Quasi-Variational Inequalities with Non Linear Source Terms,” International Journal of Numerical Analysis and Application, Vol. 5, No. 1, 2011, pp. 123-139.