Bisheh-Niasar, M., Arab Ameri, M. (2018). Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod. Journal of Applied and Computational Mechanics, 4(3), 161-166. doi: 10.22055/jacm.2017.22854.1141

Morteza Bisheh-Niasar; Maryam Arab Ameri. "Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod". Journal of Applied and Computational Mechanics, 4, 3, 2018, 161-166. doi: 10.22055/jacm.2017.22854.1141

Bisheh-Niasar, M., Arab Ameri, M. (2018). 'Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod', Journal of Applied and Computational Mechanics, 4(3), pp. 161-166. doi: 10.22055/jacm.2017.22854.1141

Bisheh-Niasar, M., Arab Ameri, M. Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod. Journal of Applied and Computational Mechanics, 2018; 4(3): 161-166. doi: 10.22055/jacm.2017.22854.1141

Moving Mesh Non-standard Finite Difference Method for Non-linear Heat Transfer in a Thin Finite Rod

^{1}Department of Applied Mathematics, Faculty of Mathematical Science, University of
Kashan, Kashan, Iran.

^{2}Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.

Abstract

In this paper, a moving mesh technique and a non-standard finite difference method are combined, and a moving mesh non-standard finite difference (MMNSFD) method is developed to solve an initial boundary value problem involving a quartic nonlinearity that arises in heat transfer with thermal radiation. In this method, the moving spatial grid is obtained by a simple geometric adaptive algorithm to preserve stability. Moreover, it uses variable time steps to protect the positivity condition of the solution. The results of this computational technique are compared with the corresponding uniform mesh non-standard finite difference scheme. The simulations show that the presented method is efficient and applicable, and approximates the solutions well, while because of producing unreal solution, the corresponding uniform mesh non-standard finite difference fails.

[1] Jordan, P.M., A nonstandard finite difference scheme for a nonlinear heat transfer in a thin finite rod, Journal of Difference Equations and Applications, 9(11), 2003, 1015-102.

[2] Dai ,W., Su, S., A non-standard finite difference scheme for solving one dimensional nonlinear heat transfer, Journal of Difference Equations and Applications,10(11), 2004, 1025-1032.

[3] Mohammadi, A., Malek, A., Stable non-standard implicit finite difference schemes for non-linear heat transfer in a thin finite rod, Journal of Difference Equations and Applications, 15(7), 2009, 719-728.

[4] Qin, W., Wang, L., Ding, X., A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion, Journal of Difference Equations and Applications, 20(12), 2014, 1641-1651.

[5] Elsheikh, S., Ouifki, R., Patidar, K.C., A non-standard finite difference method to solve a model of HIV-Malaria co-infection, Journal of Difference Equations and Applications, 20(3), 2014, 354-378.

[6] Mickens, R.E., Nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 8(9), 2002, 823-847.

[7] Ehrhardt, M., Mickens, R.E., A nonstandard finite difference scheme for convection diffusion equations having constant coefficients, Applied Mathematics and Computation, 219(12), 2013, 6591-6604.

[8] Mickens, R.E., Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equation, 15(2), 1999, 201-2014.

[9] Mickens, R.E., Gumel, A.B., Construction and analysis of a nonstandard finite difference scheme for the Burgers-Fisher equation, Journal of Sound and Vibration, 257(4), 2002, 791-797.

[10] Sanz-Serna, J.M., Christie, I., A Simple Adaptive Technique for Nonlinear Wave Problems, Journal of Computational Physics, 67(2), 1986, 348-360.

[11] Mickens, R.E., Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.