Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57981", author = "Gülnihal Meral", title = "Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method", 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.", volume = "4", number = "5", pages = "569-6", }