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Dlamini, P. G.; Khumalo, M.

On the computation of blow-up solutions for semilinear ODEs and parabolic PDEs. (English) Zbl 1264.65208

Math. Probl. Eng. 2012, Article ID 162034, 15 p. (2012).
Summary: We introduce an adaptive numerical method for computing blow-up solutions for ODEs and well-known reaction-diffusion equations. The method is based on the implicit midpoint method and the implicit Euler method. We demonstrate that the method produces superior results to the adaptive PECE-implicit method and the MATLAB solver of comparable order.

Cited in 3 Documents

MSC:

65P20 Numerical chaos
34C11 Growth and boundedness of solutions to ordinary differential equations
35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B44 Blow-up in context of PDEs

Keywords:

MATLAB

Software:

Matlab
Cite Review PDF
Full Text: DOI

References:

[1] J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0692.35001
[2] S. Kaplan, “On the growth of solutions of quasi-linear parabolic equations,” Communications on Pure and Applied Mathematics, vol. 16, pp. 305-330, 1963. · Zbl 0156.33503 · doi:10.1002/cpa.3160160307
[3] H. Brunner, X. Wu, and J. Zhang, “Computational solution of blow-up problems for semilinear parabolic PDEs on unbounded domains,” SIAM Journal on Scientific Computing, vol. 31, no. 6, pp. 4478-4496, 2009/10. · Zbl 1208.65124 · doi:10.1137/090761367
[4] A. M. Stuart and M. S. Floater, “On the computation of blow-up,” European Journal of Applied Mathematics, vol. 1, no. 1, pp. 47-71, 1990. · Zbl 0701.76038 · doi:10.1017/S095679250000005X
[5] C. Bandle and H. Brunner, “Blowup in diffusion equations: a survey,” Journal of Computational and Applied Mathematics, vol. 97, no. 1-2, pp. 3-22, 1998. · Zbl 0932.65098 · doi:10.1016/S0377-0427(98)00100-9
[6] C. J. Budd, W. Huang, and R. D. Russell, “Moving mesh methods for problems with blow-up,” SIAM Journal on Scientific Computing, vol. 17, no. 2, pp. 305-327, 1996. · Zbl 0860.35050 · doi:10.1137/S1064827594272025
[7] W. Huang, J. Ma, and R. D. Russell, “A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations,” Journal of Computational Physics, vol. 227, no. 13, pp. 6532-6552, 2008. · Zbl 1145.65080 · doi:10.1016/j.jcp.2008.03.024
[8] J. Ma, Y. Jiang, and K. Xiang, “Numerical simulation of blowup in nonlocal reaction-diffusion equations using a moving mesh method,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 8-21, 2009. · Zbl 1166.65067 · doi:10.1016/j.cam.2008.10.063
[9] J. Ma, Y. Jiang, and K. Xiang, “On a moving mesh method for solving partial integro-differential equations,” Journal of Computational Mathematics, vol. 27, no. 6, pp. 713-728, 2009. · Zbl 1212.65519 · doi:10.4208/jcm.2009.09-m2852
[10] J. Chen, Numerical Study of Blowup Problems and Conservation Laws with Moving Mesh Methods, 1996, http://ir.lib.sfu.ca/handle/1892/8368.
[11] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1059.65122 · doi:10.1017/CBO9780511543234
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