THERMAL SCIENCE
International Scientific Journal
SINGULARLY PERTURBED BURGERS-HUXLEY EQUATION BY A MESHLESS METHOD
ABSTRACT
A meshless method based upon radial basis function is utilized to approximate the singularly perturbed Burgers-Huxley equation with the viscosity coefficient ε. The proposed method shows that the obtained solutions are reliable and accurate. Convergence analysis of method was analyzed in a numerical way for different small values of singularity parameter.
KEYWORDS
PAPER SUBMITTED: 2015-12-24
PAPER REVISED: 2016-05-07
PAPER ACCEPTED: 2016-05-09
PUBLISHED ONLINE: 2016-05-30
THERMAL SCIENCE YEAR
2017, VOLUME
21, ISSUE
Issue 6, PAGES [2689 - 2698]
- Kyrychko. Y.N., et al., Persistence of traveling wave solution of a fourth order diffusion system, J. Comput. Appl. Math., 176 (2005), pp. 433-443.
- Loskutov. A.Y., Mikhailov. A.S., Introduction to Synergetics, Nauka, Moscow, 1990.
- Farrel. P.A., et al., Robust computational techniques for boundary layers, Chapman & Hall, London, 2000.
- Miller. J.J.H., et al., Fitted numerical methods for singular perturbation problems, World-Scientific, Singapore, 1996.
- Roos. H.G., et al., Numerical methods for singularly perturbed differential equations: convectio ndiffusion and flow problems, Springer-Verlag, Berlin, 1996.
- Molabahrami. A., Khani. F., The homotopy analysis method to solve the Burgers-Huxley equation equation, Nonlinear Anal. Real., 10 (2009), pp. 589-600.
- Deng. X., Travelling wave solutions for the generalized Burgers-Huxley equation, Appl. Math. Comput.204 (2008), pp. 733-737.
- Batiha. B., et al., Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Soliton Fract., 36 (2008), pp. 660-663.
- Liu. J., et al., New Multi-Soliton solitions for generalized Burgers-Huxley equation, Thermal Science., 17 (2013), pp. 1486-1489.
- Wazwaz. A.M., Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations, Appl. Math. Comput., 195 (2008), pp. 754-761.
- Wang. X.Y., et al., Solitary wave solutions of the generalized Burger's-Huxley equation, J. Phys. A: Math. Gen., 23 (1990), pp. 271-274.
- Liu. J., et al., New multi-Soliton solutions for generalized Burgers-Huxley equation, Thermal Science, 17 (2013), pp. 1486-1489.
- Javidi. M., A numerical solution of the generalized Burger's-Huxley equation by pseudospectral method and Darvishi's preconditioning, Appl. Math. Comput., 175 (2006), pp. 1619-1628.
- Khattak. A.J., A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33 (2009), pp. 3718-3729.
- Rathish Kumar. B.V., et al., A numerical study of singularly perturbed generalized Burgers-Huxley equation using three-step Taylor-Galerkin method, Comput. Math. Appl., 62 (2011), pp. 776-786.
- Xie. H., Li. D., A meshless method for Burgers equation using MQ-RBF and high-order temporal approximation, Appl. Math. Model., 37 (2013), pp. 9215-9222.
- Roohani Ghehsareh. H., et al., A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation, Eng. Anal. Bound. Elem., 61 (2015), pp. 52-60.
- Shivanian. E., Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation, Math. Meth. Appl. Sci., 39 (2016), pp. 1820-1835.
- Shivanian. E., et al., Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions, Inter. J. Comput. Math. doi: 10.1080/00207160.2015.1085032.
- Abbasbandy. S., et al., A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation, Eng. Anal. Bound. Elem., 37 (2013), pp. 885-898.
- Hon. Y.-C., et al., Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface, Eng. Anal. Bound. Elem., 57 (2015), pp. 2-8.
- Sileimani. S., et al., Meshless local RBF-DG for 2-D heat conduction: A comparative study, Thermal Science, 15 (2011), pp. S117-S121.
- Benton. E., Platzman. G.W., A table of solutions of the one-dimensional Burgers equations, Quart. Appl. Math., 30 (1972), pp. 195-212.
- Hassanein. I.A., et al., Forth-order finite difference method for solving Burgers equation, Appl. Math. Comput., 170 (2005), pp. 781-800.
- Kansa. E.J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II. Solution to parabolic, hyperbolic and elliptic partial differential equations, Computers Math. Applic., 19 (1990), pp. 147-161.
- Hon. Y.C., Mao. X.Z., An efficient numerical scheme for Burgers-Huxley equation, Appl. Math. Comput., 95 (1998), pp. 37-50.
- Wang. X.Y., et al., Solitary wave solutions of the generalised Burger-Huxley equation, J. Phys. A Math. Gen., 23 (1990), pp. 271-274.
- Gupta. V., Kadalbajoo. M.K., A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun Nonlinear Sci Numer Simulat., 16 (2011), pp. 1825-1844.