Mixed method for the product integral on the infinite interval

• Main Authors: Eshkuvatov, Zainidin K., Nik Long, Nik Mohd Asri, Muminov, Z. I., Khaldjigitov, Abduvali A.
• Format: Article
• Language: English
• Published: Institute for Mathematical Research, Universiti Putra Malaysia 2014
• Online Access: http://psasir.upm.edu.my/id/eprint/39069/1/39069.pdf; http://psasir.upm.edu.my/id/eprint/39069/; http://einspem.upm.edu.my/journal/fullpaper/vol8soct/7.%20Zainidin.pdf
• Institution: Universiti Putra Malaysia

In this note, quadrature formula is constructed for product integral on the infinite interval I(f) = ∫ 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 ≤ x ≤ N. For the approximate method we have reduced infinite interval x [a, ∞) into the interval t[0,1] and used the mixed method: Cubic Newton’s divided difference formula on [0, t3) and Romberg method on [t3,1] with equal step size, ti = t0+ih,i=0, …,n, h=1/n, where t0 = 0,tn=1. Error term is obtained for mixed method on different classes of functions. Finally, numerical examples are presented to validate the method presented.