Malaysian Journal of Mathematical Sciences, October 2014, Vol. 8(S)
Special Issue: International Conference on Mathematical Sciences and Statistics 2013 (ICMSS2013)
Mixed Method for the Product Integral on the Infinite Interval
Z. K. Eshkuvatov, N. M. A. Nik Long, Z. I. Muminov and Abduvali A. Khaldjigitov
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
In this note, quadrature formula is constructed for product integral on the infinite interval $I(f)=\int_{a}^{\infty}w(x)f(x)dx$ where $w(x)$ is a weight function and $f(x)$ is a smooth decaying function for $x > N$ (large enough) and piecewise discontinuous function of the first kind on the interval $a \leq x \leq N$. For the approximate method we have reduced infinite interval $x \in [a,\infty)$ into the interval $t \in [0,1] $ and used the mixed method: Cubic Newton�s divided difference formula on $[0,t_{3}]$ and Romberg method on $[t_{3} ,1]$ with equal step size $t_{i}=t_{0}+ih$, $i=0,...,n$, $h=1/n$ where $t_{0}=0$, $t_{n}=1$. Error term is obtained for mixed method on different classes of functions. Finally, numerical examples are presented to validate the method presented.
Keywords: Product integral, Romberg method, mixed method, error estimate
