On weighted Adams-Bashforth rules

Sa？etak One class of the linear multistep methods for solving the Cauchy problems of the form $ y'=F(x,y) $, $ y(x_{0})=y_{0} $, contains Adams-Bashforth rules of the form $y_{n+1}=y_{n}+h\sum_{i=0}^{k-1} B_i^{(k)} F(x_{n-i},y_{n-i})$, 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,\[A(x)y'+B(x)y=G(x,y), \quad y(x_0)=y_0,\]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