On weighted Adams-Bashforth rules

Abstract

One class of the linear multistep methods for solving the Cauchy problems of the form $y\u27=F(x,y)$, $y(x_{0})=y_{0}$, contains Adams-Bashforth rules of the form $y_{n+1}=y_{n}+hsum_{i=0}^{k-1} B_i^{(k)} F(x_{n-i},y_{n-i})$, where ${ B_i^{(k)}} _{i = 0}^{k - 1}$ are fixed numbers. In this paper, we propose an idea for weighted type of Adams-Bashforth rules for solving the Cauchy problem for singular differential equations,[A(x)y\u27+B(x)y=G(x,y), quad y(x_0)=y_0,]where $A$ and $B$ are two polynomials determining the well-known classical weight functions in the theory of orthogonal polynomials. Some numerical examples are also included