nag_inteq_volterra_weights (d05bwc) computes the weights
and
for a family of quadrature rules related to the Adams' methods of orders three to six and the BDF methods of orders two to five, for approximating the integral:
with
, for
, for some given constant
.
In
(1),
is a uniform mesh,
is related to the order of the method being used and
,
are the starting and the convolution weights respectively. The mesh size
is determined as
, where
and
is the chosen number of convolution weights
, for
. A description of how these weights can be used in the solution of a Volterra integral equation of the second kind is given in
Section 9. For a general discussion of these methods, see
Wolkenfelt (1982) for more details.
Wolkenfelt P H M (1982) The construction of reducible quadrature rules for Volterra integral and integro-differential equations IMA J. Numer. Anal. 2 131–152
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
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- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_ENUM_INT
-
On entry, and .
Constraint: if , .
On entry, and .
Constraint: if , .
On entry, and .
Constraint:
if , .
On entry, and .
Constraint:
if , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
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 for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
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Not applicable.
Reducible quadrature rules are most appropriate for solving Volterra integral equations (and integro-differential equations). In this section, we propose the following algorithm which you may find useful in solving a linear Volterra integral equation of the form
using
nag_inteq_volterra_weights (d05bwc). In
(2),
and
are given and the solution
is sought on a uniform mesh of size
such that
. Discretization of
(2) yields
where
. We propose the following algorithm for computing
from
(3) after a call to
nag_inteq_volterra_weights (d05bwc):
(a) |
Equation (3) requires starting values, , for , with . These starting values can be computed by solving the linear system
|
(b) |
Compute the inhomogeneous terms
|
(c) |
Start the iteration for to compute from:
|
Note that for a nonlinear integral equation, the solution of a nonlinear algebraic system is required at step
(a) and a single nonlinear equation at step
(c).
The following example generates the first ten convolution and thirteen starting weights generated by the fourth-order BDF method.