Okey Oseloka Onyejekwe^*

Show more
Computational Science Program, Addis Ababa University, Arat Kilo Campus, Addis Ababa, Ethiopia.
Show less

Abstract: This study is concerned with the numerical approximation of the extended Fisher-Kolmogorov equation with a modified boundary integral method. A key aspect of this formulation is that it relaxes the domain-driven approach of a typical boundary element (BEM) technique. While its discretization keeps faith with the second order accurate BEM formulation, its implementation is element-based. This leads to a local solution of all integral equation and their final assembly into a slender and banded coefficient matrix which is far easier to manipulate numerically. This outcome is much better than working with BEM’s fully populated coefficient matrices resulting from a numerical encounter with the problem domain especially for nonlinear, transient, and heterogeneous problems. Faithful results of high accuracy are achieved when the results obtained herein are compared with those available in literature.

Keywords: Boundary Element Method, Extended Fisher-Kolmogorov Equation Boundary Integral Formulation, Slender Coefficient Matrix, Hybridization, Domain-Driven

Cite this paper: Onyejekwe, O. (2015) A Direct Implementation of a Modified Boundary Integral Formulation for the Extended Fisher-Kolmogorov Equation. Journal of Applied Mathematics and Physics, 3, 1262-1269. doi: 10.4236/jamp.2015.310155.

References

[1]   Onyejekwe, O.O. (2004) A Green Element Method for Fourth Order Differential Equations. Advances in Engineering Software, 35, 517-525.
http://dx.doi.org/10.1016/j.advengsoft.2004.05.005

[2]   Dee, G.T. and Van Sarloos, W. (1988) Bistable Systems with Propagating Fronts Leading to Pattern Formation. Physical Review Letters, 60, 2641-2644.
http://dx.doi.org/10.1103/PhysRevLett.60.2641

[3]   Danumjaya, P. and Pani, A.K. (2006) Numerical Methods for Extended Fisher-Kolmogorov (EFK) Equation. International Journal of Numerical Analysis and Modeling, 3, 186-210.

[4]   Danumjaya, P. and Pani, A.K. (2005) Orthogonal Cubic Spline Collocation Method for the Extended Fisher Kolmogorov (EFK) Equation. Journal of Computational and Applied Mathematics, 174, 101-117.
http://dx.doi.org/10.1016/j.cam.2004.04.002

[5]   Zimmerman, W. (1991) Propagating Fronts near a Lifshitz Point. Physical Review Letters, 66, 1546.
http://dx.doi.org/10.1103/PhysRevLett.66.1546

[6]   Aronson, D.G. and Weinberger, H.F. (1978) Multidimensional Nonlinear Diffusion Arising in Population Genetics. Advances in Mathematics, 30, 33-67.
http://dx.doi.org/10.1016/0001-8708(78)90130-5

[7]   Peletier, L.A. and Troy, W.C. (1995) A Topological Shooting Method and Existence of Kinks of the Extended Fisher-Kolmogorov Equation. Topological Methods in Nonlinear Analysis, 6, 331-355.

[8]   Peletier, L.A., Troy, W.C. and Vander Vorst, R.C.A.M. (1995) Stationary Solutions of a Nonlinear Fourth Order Equation. Differential Equations, 31, 327-337.

[9]   Tersian, S. and Chaparova, J. (2001) Periodic and Homoclinic Solutions of Ex-tended Fisher Kolmogorov Equations. Journal of Mathematical Analysis and Applications, 260, 490-506.
http://dx.doi.org/10.1006/jmaa.2001.7470

[10]   Hong, L. (2011) Global Attractor of the Extended Fisher-Kolmogorov Equation in Hk Spaces. Boundary Value Problems, 2011, 39.
http://dx.doi.org/10.1186/1687-2770-2011-39

[11]   Kalies, W.D., Kwapisz, J. and VanderVorst, R.C.A.M. (1996) Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria. Preprint of Georgia Tech.

[12]   Noomen, K. and Omrani, K. (2011) Finite Difference Discretization of the Extended Fisher-Kolmogorov Equation in Two Dimensions. Computers & Mathematics with Applications, 62, 4151-4160.
http://dx.doi.org/10.1016/j.camwa.2011.09.065

[13]   Kadri, T. and Omrani, K. (2011) A Second Order Accurate Difference Scheme for an Extended Fisher-Kolmogorov Equation. Computers & Mathematics with Applications, 61, 451-459.
http://dx.doi.org/10.1016/j.camwa.2010.11.022

[14]   Taigbenu, A.E. and Onyejekwe, O.O. (1997) A Mixed Green Element Formulation for Transient Burgers’ Equation. International Journal for Numerical Methods in Fluids, 24, 563-578.

[15]   Onyejekwe, O.O. (2015) An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Applied and Computational Mathematics, 4, 83-99.
http://dx.doi.org/10.11648/j.acm.20150403.11

[16]   Onyejekwe, O.O. and Onyejekwe, O.N. (2013) A Nonlinear Integral Formulation for a Stream Aquifer Interaction Flow Problem. International Journal of Nonlinear Science, 15, 15-26.

[17]   Bagherinezhad, A. and Pishvaie, M.R. (2014) A New Approach to Counter-Current Spontaneous Imbibition Simulation Using Green Element Method. Journal of Petroleum Science and Engineering, 119, 163-168.
http://dx.doi.org/10.1016/j.petrol.2014.05.004

[18]   Portapila, M. and Power, H. (2008) Iterative Solution Schemes for Quadratic DRM-MD. Numerical Methods for Partial Differential Equations, 24, 1430-1459.

[19]   Hibersek, M. and Skerget, L. (1996) Domain Decomposition Methods for Fluid Flow Problems by Boundary Integral Method. Zeitschrift fur Angewandle Mathematik und Mechanik, 76, 115-139.

[20]   Sladeck, J., Sladeck, V. and Zhang, C. (2004) A Local BIEM for Analysis of Transient Heat Conduction with Nonlinear Source Terms in FMG’s. Engineering Analysis with Boundary Elements, 28, 1-11.

[21]   Archer, R., Horne, R.N. and Onyejekwe, O.O. (1999) Petroleum Reservoir Engineering Applications of the Dual Reciprocity Boundary Element Method and the Green Element Method. 21st World Conference on Boundary Element Method, 25-27 August 1999, Oxford University, Eng-land.

[22]   Grigoriev, M.M. and Dargush, G.F. (2003) Boundary Element Methods for Transient Convective Diffusion Part 1: General Formulation and 1D Implementation. Computer Methods in Applied Mechanics and Engineering, 192, 4281-4298.

[23]   Peratta, A. and Popov, V. (2004) Numerical Stability of the BEM for Advection-Diffusion Problems. Numerical Methods for Partial Differential Equations, 20, 675-702.
http://dx.doi.org/10.1002/num.20009

[24]   Onyejekwe, O.O. (2000) Experiences in Solutions of Nonlinear Transport Equations with Green Element Method. International Journal of Numerical Methods for Heat & Fluid Flow, 10, 675-686.
http://dx.doi.org/10.1108/09615530010350372

[25]   Onyejekwe, O.O. (1999) Numerical Properties of Green Element Unsteady Transport. Transactions on Modelling and Simulation, 24, 749-758.

[26]   Mittal, R.C. and Arora, G. (2009) Quintic B-Spline Collocation Method for Numerical Solution of the Extended Fisher-Kolmogorov Equation. International Journal of Applied Mathematics and Mechanics, 6, 74-85.