The paper deals with a steady state version of a nonlocal nonlinear parabolic problem defined on a bounded polygonal domain. The nonlocal term involved in the strong formulation essentially increases the complexity of the problem and the necessary total computational work. The nonlinear weak formulation of the problem is reduced to a linear one suitable for applications of Newtonian type iterative methods. A discrete problem is obtained by the FEM. A fast and stable iterative method with ninth-order of convergence is applied for solving the discrete problem. The iterative algorithm is described by a pseudo-code. The method is computer implemented and the approximate solutions are presented graphically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Kumar and P. Kumar, “Computational method for finding various solutions for a quasilinear elliptic equation of Kirchhoff type,” Adv. Eng. Software, 40, 1104–1111 (2009) .
J. L. Lions, “On some questions in boundary value problems of mathematical physics,” in: G. de la Penha and L. A. Medeiros (editors), International Symposium on Continuum, Mechanics and Partial Differential Equations, Rio de Janeiro 1977, Mathematics Studies, North-Holland Amsterdam, 30, 284–346 (1978) .
A. Arosio and S. Panizzi, “On the well-posedness of the Kirchhoff string,” Trans. Am. Math. Soc., 348, 305–330 (1996) .
M. Chipot, V. Valente, and G.V. Caffarelli, “Remarks on a nonlocal problem involving the Dirichlet energy,” Rend. Sem. Mat. Univ. Padova, 110, 199–220 (2003).
C. O. Alves, F.J.S. Correa, and G.M. Figueiredo, “On a class of nonlocal elliptic problems with critical growth,” Differ. Equ. Appl., 2, 409–417 (2010).
T. F. Ma, “Remarks on an elliptic equation of Kirchhoff type,” Nonlinear Anal., 63, 1967–1977 (2005).
R. S. Cantrell and C. Cosner, “Spatial ecology via reaction–diffusion equation,” Wiley Series in Mathematical and Computational Biology, John Wiley SonsLtd (2003).
G. Liu, Y. Wang, and J. Shi, “Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong Allee effect,” Appl. Math. Mech., Engl. Ed., 30, 1461–1468 (2009).
J. Shi and R. Shivaji, “Persistence in reaction diffusion models with weak allee effect,” J. Math. Biol., 52, 807829 (2006).
Z. Hu, L. Guocai, and L. Tian, “An iterative method with ninth-order convergence for solving nonlinear equations,” Int. J. Contemp. Math. Sci., 6, 17–23 (2011).
F. J´ulio, S. A. Corr´ea, and G. M. Figueiredo, “On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method,” Bound. Value Probl., 2006, 1–10, Article ID 79679.
M. Kumar, A. Srivastava, and A. K. Singh, “Numerical simulation of singularly perturbed non-linear elliptic boundary value problems using finite element method,” Appl. Math. Comput., 219, 226–236 (2012).
M. Jung and T. D. Todorov, “Isoparametric multigrid method for reaction–diffusion equations,” Appl. Numer. Math., 56, 1570–1583 (2006).
T. D. Todorov, “The optimal mesh refinement strategies for 3-D simplicial meshes,” Comput. Math. Appl., 66, 1272–1283 (2013).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Srivastava, A., Kumar, M. & Todorov, T.D. A Ninth-Order Convergent Method for Solving the Steady State Reaction–Diffusion Model. Comput Math Model 26, 593–603 (2015). https://doi.org/10.1007/s10598-015-9296-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-015-9296-8