Premium
Some modified Adams‐Bashforth methods based upon the weighted Hermite quadrature rules
Author(s) -
MasjedJamei Mohammad,
Moalemi Zahra,
Srivastava Hari M.,
Area Iván
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5954
Subject(s) - linear multistep method , mathematics , quadrature (astronomy) , gauss–jacobi quadrature , numerical integration , hermite polynomials , mathematical analysis , gauss–kronrod quadrature formula , gauss–hermite quadrature , numerical analysis , cauchy distribution , gaussian quadrature , clenshaw–curtis quadrature , nyström method , ordinary differential equation , differential equation , integral equation , differential algebraic equation , engineering , electrical engineering
In this paper, we first introduce a modification of linear multistep methods, which contain, in particular, the modified Adams‐Bashforth methods for solving initial‐value problems. The improved method is achieved by applying the Hermite quadrature rule instead of the Newton‐Cotes quadrature formulas with equidistant nodes. The related coefficients of the method are then represented explicitly, the local error is given, and the order of the method is determined. If a numerical method is consistent and stable, then it is necessarily convergent. Moreover, a weighted type of the new method is introduced and proposed for solving a special case of the Cauchy problem for singular differential equations. Finally, several numerical examples and graphical representations are also given and compared.