Abstract
Elementary approximate formulae for numerical integration of functions containing oscillating factors of a special form with a parameter have been proposed in the paper. In this case general quadrature formulae can be used only at sufficiently small values of the parameter. Therefore, it is necessary to consider in advance presence of strongly oscillating factors in order to obtain formulae for numerical integration which are suitable in the case when the parameter is changing within wide limits. This can be done by taking into account such factors as weighting functions. Moreover, since the parameter can take values which cannot always be predicted in advance, approximate formulae for calculation of such integrals should be constructed in such a way that they contain this parameter in a letter format and they are suitable for calculation at any and particularly large values of the parameter. Computational rules with such properties are generally obtained by dividing an interval of integration into elementary while making successive approximation of the integral density at each elementary interval with polynomials of the first, second and third degrees and taking the oscillating factors as weighting functions. The paper considers the variant when density of the integrals at each elementary interval is approximated by a polynomial of zero degree that is a constant which is equal to the value of density in the middle of the interval. At the same time one approximate formula for calculation of an improper integral with infinite interval of the function with oscillating factor of a special type has been constructed in the paper. In this case it has been assumed that density of the improper integral rather quickly goes to zero when an argument module is increasing indefinitely. In other words it is considered as small to negligible outside some finite interval. Uniforms in parameter used for evaluation of errors in approximate formulae have been obtained in the paper and they make it possible to calculate integrals with the required accuracy.
Highlights
Естественные науки an interval of integration into elementary while making successive approximation of the integral density at each elementary interval with polynomials of the first, second and third degrees and taking the oscillating factors as weighting functions
The paper considers the variant when density of the integrals at each elementary interval is approximated by a polynomial of zero degree that is a constant which is equal to the value of density in the middle of the interval
At the same time one approximate formula for calculation of an improper integral with infinite interval of the function with oscillating factor of a special type has been constructed in the paper
Summary
Естественные науки an interval of integration into elementary while making successive approximation of the integral density at each elementary interval with polynomials of the first, second and third degrees and taking the oscillating factors as weighting functions. At the same time one approximate formula for calculation of an improper integral with infinite interval of the function with oscillating factor of a special type has been constructed in the paper. И θk (t) = 0 , когда t ∉ tk h, 2 tk Нетрудно убедиться, что если f(t) непрерывна на отрезке [− T , T ], то f (t) − f (t) ≤ ω( f ; h), t ∈[−T , T ], (6) Если же f(t) – непрерывно дифференцируемая функция на этом отрезке, то с помощью формулы Тейлора легко установить, что f (t) − ~f (t) ≤ M1 h, t ∈[−T , T ], (7)
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