Chebyschev iteration technique for solving second order singularly perturbed 1d reaction – diffusion equation

Abstract

In this thesis, Chebyschev iteration technique has been presented to solve second order singularly perturbed 1D reaction – diffusion equation for a very small perturbation parameter,  with both variable and constant coefficient of reaction term. The given problem of interest is discretized and the derivative of the given differential equation is replaced by finite central difference approximation to obtain system of algebraic equation. Chebyschev three – level scheme was developed from the two – level scheme to solve the obtained algebraic equation. To investigate the convergence of the proposed method, three examples were taken and compared with other methods listed in the literature and exact solution. The relationship between number of iteration number and the condition number is analyzed and found to be: the larger the condition number the slower is the rate of convergence. Finally, pointwise and maximum absolute error for each example was shown both by table and numerical approximation and exact solution is demonstrated on the same graph with different iteration number.