Mixed method for the product integral on the infinite interval - Eshkuvatov, Zainidin K.; Nik Long, Nik Mohd Asri; Muminov, Z. I.; Khaldjigitov, Abduvali A. (Institute for Mathematical Research, Universiti Putra Malaysia, 2014)
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.