Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30184
Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method

Authors: Gülnihal Meral

Abstract:

In this study, the density dependent nonlinear reactiondiffusion equation, which arises in the insect dispersal models, is solved using the combined application of differential quadrature method(DQM) and implicit Euler method. The polynomial based DQM is used to discretize the spatial derivatives of the problem. The resulting time-dependent nonlinear system of ordinary differential equations(ODE-s) is solved by using implicit Euler method. The computations are carried out for a Cauchy problem defined by a onedimensional density dependent nonlinear reaction-diffusion equation which has an exact solution. The DQM solution is found to be in a very good agreement with the exact solution in terms of maximum absolute error. The DQM solution exhibits superior accuracy at large time levels tending to steady-state. Furthermore, using an implicit method in the solution procedure leads to stable solutions and larger time steps could be used.

Keywords: Density Dependent Nonlinear Reaction-Diffusion Equation, Differential Quadrature Method, Implicit Euler Method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1071770

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1842

References:


[1] J. D. Murray, Mathematical Biology, I: An Introduction. New York: Springer-Verlag, 2002.
[2] W. Yuan-Ming, Petrov-Galerkin Methods for Nonlinear Reaction- Diffusion Equations, International Journal of Computer Mathematics, vol. 69, pp. 123-145, 1998.
[3] G. Meral, M. Tezer-Sezgin, Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation type time integration, International Journal of Computer Mathematics, vol. 86, no. 3, pp.451-463, 2009.
[4] G. Meral, M. Tezer-Sezgin, The differential quadrature solution of nonlinear reaction-diffusion and wave equations using several timeintegration schemes, Communications in Numerical Methods in Engineering, to be published(DOI: 10.1002/cnm.1305), 2009.
[5] J. Satsuma, Explicit solutions of nonlinear equations with density dependent diffusion, Journal of Physical Society of Japan, vol. 56, no. 6, pp. 1947-1950, 1987.
[6] Z. Biro, Attractors in a density-dependent Diffusion-Reaction Model, Nonlinear Analysis, Theory, Methods and Applications, vol. 29, no. 5, pp. 485-499, 1997.
[7] R. Bellman, J. Casti, Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications, vol. 34, pp. 235- 238, 1971.
[8] R. Bellman, B. G. Kashef, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics, vol. 10, pp. 40-52, 1972.
[9] C. Shu, Differential quadrature and its applications in engineering. London: Springer Verlag, 2000.
[10] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, New-York: Springer Verlag, 2002.