Error analysis of solutions of second-order boundary value problems obtained with residual correction method

Abstract

Based on the maximum principle of differential equations and with the aid of asymptotic iteration technique, this paper tries to establish monotonic relation of second-order obstacle boundary value problems with their approximate solutions to eventually obtain the upper and lower approximate solutions of the exact solution. To obtain numerical solutions, the cubic spline approximation method is applied to discretize equations, and then according to the ‘residual correction method’ proposed in this paper, residual correction values are added into discretized grid points to translate once complex inequalities’ constraint mathematical programming problems into simple equational iteration problems. The numerical results also show that such method has the characteristic of correcting residual values to symmetrical values for such problems, as a result, the mean approximate solutions obtained even with a considerably small quantity of grid points still quite approximate the exact solution. Furthermore, the error range of approximate solutions can be identified very easily by using the obtained upper and lower approximate solutions, even if the exact solution is unknown.

Keywords:

• maximum principle
• cubic spline
• residual
• upper and lower solutions
• obstacle problems

2000 AMS Subject Classifications :

• 11Y35
• 65N11
• 35J25
• 35T60
• 35J67

Acknowledgements

The author thanks for the subsidy of the outlay NSC95-2221-E-432-006 given by the National Science Council, the Republic of China, which helped in the successful completion of this research.