On weighted Adams-Bashforth rules

Mohammad Masjed-Jamei, Gradimir Milovanović, Amir Hossein Salehi Shayegan


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.


weighted Adams-Bashforth rule; ordinary differential equation; linear multistep method; weight function

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ISSN: 1331-0623 (Print), 1848-8013 (Online)