Numerical Integration with Singularity by Taylor Series

H. Hirayama ()
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H. Hirayama: Kanagawa Institute of Technology, Department of Vehicle System Engineering, Faculty of Creative Engineering

Chapter Chapter 14 in Integral Methods in Science and Engineering, 2013, pp 195-204 from Springer

Abstract: Abstract We consider the integration of the product of a smooth function f(x) and a function K(x; c) with a singularity in the finite integration interval [a, b]; that is, 14.1 $$\displaystyle{ I(a,b,c) =\int _{ a}^{b}K(x;c)f(x)\,dx. }$$ This type of integral is difficult to evaluate by the usual numerical methods when K(x; c) is a singular function such as $$\vert x - c{\vert }^{\alpha }{(\log \vert x - c\vert )}^{n}$$ , with α > − 1 a real number and n > 0 an integer, or $${(x - c)}^{-1}$$ (the Cauchy principal-value case) or $${(x - c)}^{-n}$$ , with n > 1 an integer (the Hadamard finite-part case).

Keywords: Usual Numerical Methods; Hadamard Finite-part Integral; Cauchy Principal Value Integrals; Double Exponential Method; Logarithmic Singular Points (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-7828-7_14

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