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Monotonicity Results for a Compound Quadrature Method for Finite-Part Integrals  
 
  Authors: Kai Diethelm,  
  Keywords: Hadamard finite-part integral; Quadrature formula; Product trapezoidal formula; Monotonicity; Fractional differential equation; Discrete Gronwall inequality; Backward differentiation formula.  
  Date Received: 19/05/2003  
  Date Accepted: 30/03/2004  
  Subject Codes:

Primary 41A55; Secondary 65D30, 65L05.

  
  Editors: Gradimir V. Milovanovic,  
 
  Abstract:

Given a function $ f in C^3[0,1]$ and some $ q in (0,1)$, we look at the approximation for the Hadamard finite-part integral $ % mathop{= !!!!!! int}_0^1 x^{-q-1} f(x) dx$ based on a piecewise linear interpolant for $ f$ at $ n$ equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral in a monotonic way. An application of the results yields detailed information on the error term of a backward differentiation formula for a fractional differential equation.;


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