NAG Library Function Document
nag_inteq_abel_weak_weights (d05byc) computes the fractional quadrature weights associated with the Backward Differentiation Formulae (BDF) of orders , and . These weights can then be used in the solution of weakly singular equations of Abel type.
||nag_inteq_abel_weak_weights (Integer iorder,
nag_inteq_abel_weak_weights (d05byc) computes the weights
for a family of quadrature rules related to a BDF method for approximating the integral:
, for some given
. In (1)
is the order of the BDF method used and
are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of
is based on Newton's iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently
(see Baker and Derakhshan (1987)
and Henrici (1979)
for practical details and Lubich (1986)
for theoretical details). Some special functions can be represented as the fractional integrals of simpler functions and fractional quadratures can be employed for their computation (see Lubich (1986)
). A description of how these weights can be used in the solution of weakly singular equations of Abel type is given in Section 8
Baker C T H and Derakhshan M S (1987) Computational approximations to some power series Approximation Theory (eds L Collatz, G Meinardus and G Nürnberger) 81 11–20
Henrici P (1979) Fast Fourier methods in computational complex analysis SIAM Rev. 21 481–529
Lubich Ch (1986) Discretized fractional calculus SIAM J. Math. Anal. 17 704–719
iorder – IntegerInput
On entry: , the order of the BDF method to be used.
iq – IntegerInput
: determines the number of weights to be computed. By setting iq
to a value,
fractional convolution weights are computed.
omega – doubleOutput
: the first
elements of omega
contains the fractional convolution weights
. The remainder of the array is used as workspace.
sw – doubleOutput
On exit: contains the fractional starting weights
, for and , where .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
Fractional quadrature weights can be used for solving weakly singular integral equations of Abel type. In this section, we propose the following algorithm which you may find useful in solving a linear weakly singular integral equation of the form
using nag_inteq_abel_weak_weights (d05byc). In (2)
are given and the solution
is sought on a uniform mesh of size
. Discretization of (2)
. We propose the following algorithm for computing
after a call to nag_inteq_abel_weak_weights (d05byc):
||Set and .
||Equation (3) requires starting values, , for , with . These starting values can be computed by solving the system
||Compute the inhomogeneous terms
||Start the iteration for to compute from:
Note that for nonlinear weakly singular equations, the solution of a nonlinear algebraic system is required at step (b)
and a single nonlinear equation at step (d)
The following example generates the first fractional convolution and fractional starting weights generated by the fourth-order BDF method.
9.1 Program Text
Program Text (d05byce.c)
9.2 Program Data
9.3 Program Results
Program Results (d05byce.r)