AM  Vol.6 No.5 , May 2015
On the Computation of Extinction Time for Some Nonlinear Parabolic Equations
ABSTRACT
The phenomenon of extinction is an important property of solutions for many evolutionary equa-tions. In this paper, a numerical simulation for computing the extinction time of nonnegative solu-tions for some nonlinear parabolic equations on general domains is presented. The solution algo-rithm utilizes the Donor-cell scheme in space and Euler’s method in time. Finally, we will give some numerical experiments to illustrate our algorithm.

Cite this paper
Ngarmadji, K. , Ndeuzoumbet, S. , Nkounkou, H. and Mampassi, B. (2015) On the Computation of Extinction Time for Some Nonlinear Parabolic Equations. Applied Mathematics, 6, 754-763. doi: 10.4236/am.2015.65071.
References
[1]   Kalashnikov, A.S. (1974) The Propagation of Disturbances in Problems of Non-Linear Heat Conduction with Absorption. USSR Computational Mathematics and Mathematical Physics, 14, 70-85.
http://dx.doi.org/10.1016/0041-5553(74)90073-1

[2]   Galaktionov, V.A. and Vasquez, J.L. (2002) The Problem of Blow-Up in Nonlinear Parabolic Equations. Discrete and Continuous Dynamics Systems, 8, 399-433.
http://dx.doi.org/10.3934/dcds.2002.8.399

[3]   Levine, H.A. (1985) The Phenomenon of Quenching: A Servey. North-Holland Mathematics Studied, 110, 275-286.
http://dx.doi.org/10.1016/S0304-0208(08)72720-8

[4]   Diaz, J.I. (2001) Qualitative Study of Nonlinear Parabolic Equations: An Introduction. Extracta Mathematicae, 16, 303-341.

[5]   Friedman, A. and Herrero, M.A. (1987) Extinction Properties of Semilinear HEAT Equations with Strong Absorption. Journal of Mathematical Analysis and Applications, 124, 530-546.
http://dx.doi.org/10.1016/0022-247X(87)90013-8

[6]   Gu, Y.G. (1994) Necessary and Sufficient Conditions for Extinction of Solutions to Parabolic Equations. Acta Mathematica Sinica, 37, 73-79.

[7]   Lair, A.V. (1993) Finite Extinction Time for Solutions of Nonlinear Parabolic Equations. Nonlinear Analysis, Theory, Methods and Applications, 21, 1-8.

[8]   Boni, T.K. (2001) Extinction for Dicretizations of Some Semilinear Parabolic Equations. Comptes Rendus de l’Aca- dmie des Sciences de Paris, Serie I, Mathmatique, 333, 795-800.

[9]   Mikula, K.B. (1995) Numerical Solution of Nonlinear Diffusion with Finite Extinction Phenomenom. Acta Mathematica Universitatis Comenianae, LXIV, 173-184.

[10]   Nabongo, D. and Boni, T.K. (2008) Numerical Quenching for a Semilinear Parabolic Equation. Mathematical Modelling and Analysis, 13, 521-538.
http://dx.doi.org/10.3846/1392-6292.2008.13.521-538

[11]   Nabongo, D. and Boni, T.K. (2008) Quenching for Semidiscretization of a Semilinear Heat Equation with Dirichlet and Neumann Boundary Condition. Commentationes Mathematicae Universitatis Carolinae, 49, 463-475.

[12]   Lair, A.V. and Oxley, M.K. (1996) Anisotropic Nonlinear Diffusion with Absorption: Existence and Extinction. International Journal of Mathematics and Mathematical Sciences, 19, 427-434.
http://dx.doi.org/10.1155/S0161171296000610

[13]   Dumitrache, A. (2007) A Numerical Method to Approximate the Solutions of Nonlinear Absorption Diffusion Equation. Proceeding in Applied Mathematics and Mechanics, 7, 4070041-4070042.
http://dx.doi.org/10.1002/pamm.200701050

[14]   Kim, D. and Proskurowski, W. (2004) An Efficient Approach for Solving a Class of Nonlinear 2D Parabolic PDEs. International Journal of Mathematics and Mathematical Sciences, 2004, 881-899.

[15]   Griebel, M., Dornseifer, T. and Neunhoeffer, T. (1998) Numerical Simulation in Fluid Dynamics. A Practical Guide, SIAM, Philadephia.

 
 
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