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A discontinuous Galerkin least-squares finite element method for solving Fisher's equation

Pages: 489 - 497, Issue special, November 2013

 Abstract        References        Full Text (266.6K)          

Runchang Lin - Department of Engineering, Mathematics, and Physics, Texas A&M International University, Laredo, TX 78041, United States (email)
Huiqing Zhu - Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406, United States (email)

1 M.J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), no. 6, pp. 835-840.       
2 K. Al-Khaled, Numerical study of Fishers reaction-diffusion equation by the sinc collocation method, J. Comput. Appl. Math., 137 (2001), pp. 245-255.       
3 J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Develop., 17 (1973), pp. 307-313.       
4 G.F. Carey and Y. Shen, Least-squares finite element approximation of Fishers reactiondiffusion equation, Numer. Methods Partial Differential Equations, 11 (1995), pp. 175-186.       
5 I. Daǧ, A. Şahin, and A. Korkmaz, Numerical investigation of the solution of Fisher's equation via the B-spline Galerkin method, Numer. Methods Partial Differential Equations 26 (2010), no. 6, pp. 1483-1503.       
6 R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 355-369.
7 J. Gazdag and J. Canosa, Numerical solution of Fisher's equation, J. Appl. Probab., 11 (1974), pp. 445-457.       
8 B.Y. Guo and Z.X. Chen, Analytic solutions of the Fisher equation, J. Phys. A, 24 (1991), no. 3, pp. 645-650.       
9 P.S. Hagan, Traveling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal. 13 (1982), no. 5, pp. 717-738.       
10 T. Hagstrom and H.B. Keller, The numerical calculation of traveling wave solutions of nonlinear parabolic equations, SIAM J. Sci. Statist. Comput., 7 (1986), no. 3, pp. 978-988.       
11 A. Kolmogorov, I. Petrovshy, and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou Ser. Int. Sect. A Math. et Mecan., 1 (1937), pp. 1-25.       
12 D.A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34 (1978), no. 1, pp. 93-103.       
13 S. Li, L. Petzold, and Y. Ren, Stability of moving mesh systems of partial differential equations, SIAM J. Sci. Comput., 20 (1998), no. 2, pp. 719-738.       
14 R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal. 47 (2008/09), no. 1, pp. 89-108.       
15 R. Lin, Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities, Numer. Math. 112 (2009), no. 2, pp. 295-318.       
16 J.D. Logan, "An introduction to nonlinear partial differential equations,'' second edition, Wiley-Interscience, John Wiley & Sons, Hoboken, NJ, 2008.       
17 R.E. Mickens, A best finite-difference scheme for the Fisher equation, Numer. Methods Partial Differential Equations 10 (1994), no. 5, pp. 581-585.       
18 J.D. Murray, "Mathematical biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989.       
19 D. Olmos and B.D. Shizgal, A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math., 193 (2006), pp. 219-242.       
20 N. Parekh and S. Puri, A new numerical scheme for the Fisher equation, J. Phys. A: Math. Gen., 23 (1990), pp. L1085-L1091.       
21 Y. Qiu and D.M. Sloan, Numerical solution of Fisher's equation using a moving mesh method, J. Comput. Phys., 146 (1998), pp. 726-746.       
22 Rizwan-uddin, Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation, SIAM. J. Sci. Comput., 22 (2000), pp. 1926-1942.       
23 J. Roessler and H. Hüssner, Numerical solution of the $1+2$ dimensional Fisher's equation by finite elements and the Galerkin method, Math. Comput. Modelling, 25 (1997), pp. 57-67.       
24 S. Tang and R.O. Weber, Numerical study of Fisher's equation by a Petrov-Galerkin finite element method, J. Austral. Math. Soc. Sci. B, 33 (1991) pp. 27-38.       
25 V. Thomée, "Galerkin finite element methods for parabolic problems,'' second edition, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.       
26 S. Zhao and G.W. Wei, Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation, SIAM J. Sci. Comput., 25 (2003) pp. 127-147.       

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