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

Yıl 2020, Cilt: 10 Sayı: 2, 415 - 427, 01.03.2020

R. K. Mohanty S. Sharma

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• Bataineh, A. S., Noorani, M. S. M. and Hashim, I., (2009), Analytical treatment of generalized
• Burgers-Huxley equation by homotopy analysis method.Bull. Malaya Math. Sci. Soc. 32, pp. 233–243.
• Bratsos, A. G., (2011), A fourth order improved numerical scheme for the generalized Burgers-Huxley equation. Am. J. Comput. Math. 1, pp. 152–158.
• Burgers, J. M., (1948), A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. , pp. 171–199.
• Gao, H. and Zhao, R., (2010), New exact solutions to the generalized Burgers-Huxley equation, Appl. Math. Comput. 217, pp. 1598–1603.
• 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.
• Ismail, H. N. A., Raslan, K. and Rabboh, A. A. A., (2004), Adomian decomposition method for
• Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput. 159, pp. 291–301.
• 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.
• 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.
• Javidi, M., (2006), A numerical solution of the generalized Burgers-Huxley equation by spectral col- location method. Appl. Math. Comput. 178, pp. 338–344.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Wang, X., (1985), Nerve propagation and wall in liquid crystals, Phys. Lett. 112A, pp. 402–406.