Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

作者：Hasegawa Takemitsu; Sugiura Hiroshi^*

来源：Journal of Computational and Applied Mathematics, 2011, 236(2): 243-252.

DOI：10.1016/j.cam.2011.06.027

• Quadrature rule
• Hilbert transform
• Principal value integral
• Finite-part integral
• End-point singularities of Jacobi type
• Chebyshey interpolation

Algorithms are proposed for the numerical evaluation of Cauchy principal value integrals f(-1)(1)w(t)f(t)/(t - x)dt, -1 < x < 1, with weight functions of Jacobi type singularities w(t) = (1 - t)(alpha) (1 t)(beta), where alpha = /- 1/2 and beta = /- 1/2, for a given function f (t) and Hadamard finite-part integrals f(-1)(1) w(t)f(t)/(t - x)(2)dt. The function f is interpolated by using a finite sum of Chebyshey polynomials. The present algorithms require O(N log N) arithmetic operations, where N is the order of the interpolation polynomial. It is shown that the present scheme gives uniform approximations, namely the errors are bounded independently of x, and is very efficient for smooth f. Further, we discuss approximations of hyper-singular integrals f-(-1)(1)w(t)f(t)/(t - x)(n)dt, it >= 3, and show their uniform convergences. Numerical examples are given to demonstrate the performance of the present schemes.

• 出版日期2011-8-15