Three algorithms for Hadamard finite-part integrals and fractional derivatives
To evaluate the Hadamard finite part integral of the form \[ \int^{+1}_{-1} \bigl(f(t)/(1-t)^{1+ \alpha} \bigr)dt, \] \(\alpha\) a positive noninteger, the author develops three algorithms. The algorithms are based on the Chebyshev series expansion of \(f\), on interpolations to \(f\) at the zeros of Chebyshev polynomials of the first kind and again interpolation at these zeros under the additional assumption that \(f^{(k)}(-1) = f^{(k)} (1)=0\) for \(k=0,\dots,n\), \(n<\alpha <n+1\). The latter condition can always be achieved by subtracting from \(f \in C^{(n+1)} [-1,+1]\) an appropriate polynomial and enables the author to give a convergence proof without assuming that \(f\) is smoother than it needs to be for the Hadamard integral to exist. Convergence of the three algorithms is proved and numerical examples are given. Reviewer's annotation: The reader will encounter several formulas of classical special functions, which are utilized skilfully in the derivation of the rules and in the convergence proofs.
scientific article; zbMATH DE number 903478
scientific article