On the convergence of finite difference methods for weakly regular singular boundary value problems

The second order finite difference methods M 1 based on a non-uniform mesh and M 2 based on an uniform mesh developed by Chawla and Katti [Finite difference methods and their convergence for a class of singular two point boundary value problems, Numer. Math. 39 (1982) 341–350] for weakly regular singular boundary value problems ( p ( x ) y ′ ) ′ = f ( x , y ) , 0 < x ⩽ 1 , with p ( x ) = x b 0 , 0 ⩽ b 0 < 1 , and boundary conditions y ( 0 ) = A , y ( 1 ) = B (A, B are finite constants) have been extended for general class of nonnegative functions p ( x ) = x b 0 g ( x ) , 0 ⩽ b 0 < 1 , and the boundary conditions y ( 0 ) = A , α y ( 1 ) + β y ′ ( 1 ) = γ , or, y ′ ( 0 ) = 0 , α y ( 1 ) + β y ′ ( 1 ) = γ . The second order convergence of the methods have been established for general non-negative function p ( x ) and under quite general conditions on f ( x , y ) . Both methods reduce to classical methods in the case b 0 = 0 and g ( x ) = 1 except for the method based on a uniform mesh with boundary condition y ′ ( 0 ) = 0 . Numerical examples for general nonnegative function p ( x ) illustrate the order of convergence of both methods.