A modified mechanical quadrature formula and its extensions

Abstract

The typical mechanical quadrature formula is modified as

$$ \int_{a}^{b}f(x)dx\approx\sum_{i=0}^{n}\sum_{j=0}^{m}A_{(i,j)}f^{(j)}(x_{i}), $$

where $f(x)\in C^{(m)}[a,b]$ and $A_{(i,j)}$ are the quadrature weights. Based on the Taylor-series expansion technique, the methods for determining the quadrature weights $A_{(i,j)}$ with the known quadrature points $x_{i}$ are given. The corresponding convergence and error estimate are made, then a sequence of Romberg-like quadrature formulae are analyzed. The modified mechanical quadrature formulae are further extended to solve the Riemann-Liouville fractional integral. Numerical results are carried out to show the effectiveness of the proposed methods by comparing some known methods. The proposed methods can be used to solve various linear and nonlinear integral equations with continuous and weakly singular kernels arising in practical physics, mechanics and engineering.

https://doi.org/10.28919/jmcs/3419