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Numerical study of Fisher's equation by a Petrov-Galerkin finite element method

Published online by Cambridge University Press:  17 February 2009

S. Tang  and
R. O. Weber
S. Tang
Affiliation:
Department of Mechanics, Peking University, Beijing 100871, China.
R. O. Weber
Affiliation:
Department of Mathematics, Australian Defence Force Academy, Canberra ACT 2600.
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Abstract

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Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by a Petrov-Galerkin finite element method. The results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are determined by the system itself and are independent of the initial values. Comparing with other studies, the numerical scheme used in this paper is satisfactory with regard to its accuracy and stability. It has the advantage of being much more concise.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 1 , July 1991 , pp. 27 - 38
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Canosa, J., “Diffusion in nonlinear multiplicative media”, J. Math. Phys., 10 (1969) 1862–8.CrossRefGoogle Scholar
[2]Canosa, J., “On a nonlinear diffusion equation describing population growth”, IBM J. Res. Develop, 17 (1973) 307–13.CrossRefGoogle Scholar
[3]Evans, D. J. and Sahimi, M. S., “The alternating group explicit (AGE) iterative method to solve parabolic and hyperbolic partial differential equations”, Ann. of Numerical Fluid Mechanics and Heat Transfer, 2 (1989) 283–389.Google Scholar
[4]Fisher, R. A., “The wave of advance of advantageous genes”, Ann. of Eugen, 7 (1936) 355–69.CrossRefGoogle Scholar
[5]Gazdag, J. and Canosa, J., “Numerical solutions of Fisher's equation”, J. Appl. Prob., 11 (1974) 445–57.CrossRefGoogle Scholar
[6]Grimshaw, R. and Tang, S., “The rotation-modified Kadomtsev-Petviashvili equation: an analytical and numerical study” (submitted) (1989).CrossRefGoogle Scholar
[7]Kolmogoroff, A., Petrovsky, I. and Piscounoff, N., “Study of the diffusion equation with growth of the quantity of matter and its application to biology problem”, Bulletin de l'Université d'élat ὰ Moscou, Série Internationale, Section A, 1 (1937).Google Scholar
[8]Smith, G. D., (1978), Numerical solution of partial differential equations, (Oxford University Press, 1978).Google Scholar
[9]Sun, H. and Tang, S., “Analytical and numerical studies on interaction of solitons of Modified KdV equations”, Chinese Journal of Computational Physics, (in press).Google Scholar
[10]Sachdev, P. L., Nonlinear diffusive waves, (Cambridge University Press, pp. 103–5, 1987).CrossRefGoogle Scholar
[11]Tang, S., “Numerical study on KdV solitons and their interactions by a Petrov-Galerkin finite element method”, Acta Mechanica, Sinica, 21 (1989) 354–9.Google Scholar
[12]Tang, S. and Wang, W., “Numerical study on regularized long wave equations”, Chinese Journal of Computational Physis, 3 (1989) 1989.Google Scholar
[13]Zeldovich, J. B. and Frank-Kamenetzki, D. A., (1938), “A theory of thermal propagation of flame”, Acta Physicochimica U.R.S.S., 9 (1938) No. 2.Google Scholar