FOURTH-ORDER ACCURATE METHOD BASED ON HALF-STEP CUBIC SPLINE APPROXIMATIONS FOR THE 1D TIME-DEPENDENT QUASILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

In this article, we discuss a fourth-order accurate scheme based on cubic spline approximations for the solution of quasilinear parabolic partial differential equations PDE . The stability of the scheme is discussed using a model linear PDE. The proposed method is tested on Burgers’ equations in polar coordinates and Burgers-Huxley equation.

Keywords

• Quasi-linear parabolic equations,Uniform mesh,Cubic Spline approximations,Burgers-Huxley equations

References

1. Bataineh, A. S., Noorani, M. S. M. and Hashim, I., (2009), Analytical treatment of generalized
2. Burgers-Huxley equation by homotopy analysis method.Bull. Malaya Math. Sci. Soc. 32, pp. 233–243.
3. Bratsos, A. G., (2011), A fourth order improved numerical scheme for the generalized Burgers-Huxley equation. Am. J. Comput. Math. 1, pp. 152–158.
4. Burgers, J. M., (1948), A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. , pp. 171–199.
5. Gao, H. and Zhao, R., (2010), New exact solutions to the generalized Burgers-Huxley equation, Appl. Math. Comput. 217, pp. 1598–1603.
6. Hashim, I., Noorani, M. S. M. and Batiha, B., (2006), A note on the adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput. 181, pp. 1439–1445.
7. Ismail, H. N. A., Raslan, K. and Rabboh, A. A. A., (2004), Adomian decomposition method for
8. Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput. 159, pp. 291–301.

9. Jain, M. K., Jain R. K. and Mohanty, R. K., (1990), Fourth order difference method for the one- dimensional general quasi-linear parabolic partial differential equation, Numer. Meth. Partial Diff. Eqn. 6, pp. 311–319.
10. Jain, M. K., Jain, R. K. and Mohanty, R. K., (1990), High order difference methods for system of 1-D non-linear parabolic partial differential equations. Int. J. Comput. Math. 37, pp. 105–112.
11. Javidi, M., (2006), A numerical solution of the generalized Burgers-Huxley equation by spectral col- location method. Appl. Math. Comput. 178, pp. 338–344.
12. Mohanty, R. K., Dai, W. and Liu, D., (2015), Operator compact method of accuracy two in time and four in space for the solution of time independent Burgers-Huxley equation, Numer. Algor. 70, pp. –605.
13. Mohanty, R. K. and Jain, M. K., (2009), High-accuracy cubic spline alternating group explicit methods for 1D quasilinear parabolic equations, Int. J. Comput. Math. 86, pp. 1556–1571.
14. Mohanty, R. K. and Sharma, S., (2017), High accuracy quasi-variable mesh method for the system of 1D quasi-linear parabolic partial differential equations based on off-step spline in compression approximations, Adv. Diff. Eqn., 2017:212.
15. Mohanty, R. K. and Sharma, S., (2019) A new two-level implicit scheme for the system of 1D quasi- linear parabolic partial differential equations using spline in compression approximations, Differ. Equ. Dyn. Syst. 27, pp. 327–356.
16. Molabahrami, A. and Khami, F., (2009), The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Anal. Real World Appl. 10, pp. 589–600.
17. Rashidinia, J. and Mohammadi, R., (2008), Non-polynomial cubic spline methods for the solution of parabolic equations, Int. J. Comput. Math. 85, pp. 843–850.
18. Wang, X., (1985), Nerve propagation and wall in liquid crystals, Phys. Lett. 112A, pp. 402–406.