nag_inteq_volterra_weights (d05bwc) : NAG Library, Mark 24

nag_inteq_volterra_weights (d05bwc) computes the quadrature weights associated with the Adams' methods of orders three to six and the Backward Differentiation Formulae (BDF) methods of orders two to five. These rules, which are referred to as reducible quadrature rules, can then be used in the solution of Volterra integral and integro-differential equations.

nag_inteq_volterra_weights (d05bwc) computes the weights

W_{i, j}

and

ω_{i}

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:

\int_{0}^{t} ϕ (s) d s ≃ h \sum_{j = 0}^{p - 1} W_{i, j} ϕ (j \times h) + h \sum_{j = p}^{i} ω_{i - j} ϕ (j \times h), 0 \leq t \leq T,

(1)

with

t = i \times h

, for

i = 0, 1, \dots, n

, for some given constant

h

In (1),

h

is a uniform mesh,

p

is related to the order of the method being used and

W_{i, j}

ω_{i}

are the starting and the convolution weights respectively. The mesh size

h

is determined as

h = \frac{T}{n}

, where

n = n_{w} + p - 1

and

n_{w}

is the chosen number of convolution weights

w_{j}

, for

j = 1, 2, \dots, n_{w} - 1

. 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.

Lambert J D (1973) Computational Methods in Ordinary Differential Equations John Wiley

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

On entry: the type of method to be used.

$method = Nag_Adams$: For Adams' type formulae.
$method = Nag_BDF$: For Backward Differentiation Formulae.

Constraint:

method = Nag_Adams

Nag_BDF

On entry: the order of the method to be used. The number of starting weights,

p

is determined by method and iorder.

method = Nag_Adams

p = iorder - 1

method = Nag_BDF

p = iorder

Constraints:

if $method = Nag_Adams$ , $3 \leq iorder \leq 6$ ;
if $method = Nag_BDF$ , $2 \leq iorder \leq 5$ .

On entry: the number of convolution weights,

n_{w}

Constraint:

nomg \geq 1

On exit: contains the first nomg convolution weights.

Note: the

(i, j)

th element of the matrix is stored in

sw [(j - 1) \times n + i - 1]

On exit:

sw [j \times n + i - 1]

contains the weights

W_{i, j}

, for

i = 1, 2, \dots, n

and

j = 0, 1, \dots, p - 1

, where

n

is as defined in Section 3.

The NAG error argument (see Section 3.6 in the Essential Introduction).

On entry, argument

⟨value⟩

had an illegal value.

On entry,

method = Nag_Adams

and

iorder = 2

.
Constraint: if

method = Nag_Adams

3 \leq iorder \leq 6

On entry,

method = Nag_BDF

and

iorder = 6

.
Constraint: if

method = Nag_BDF

2 \leq iorder \leq 5

On entry,

method = ⟨value⟩

and

iorder = ⟨value⟩

.
Constraint: if

method = Nag_Adams

3 \leq iorder \leq 6

On entry,

method = ⟨value⟩

and

iorder = ⟨value⟩

.
Constraint: if

method = Nag_BDF

2 \leq iorder \leq 5

On entry,

iorder = ⟨value⟩

.
Constraint:

2 \leq iorder \leq 6

On entry,

nomg = ⟨value⟩

.
Constraint:

nomg \geq 1

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.

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

y (t) = f (t) + \int_{0}^{t} K (t, s) y (s) d s, 0 \leq t \leq T,

(2)

using nag_inteq_volterra_weights (d05bwc). In (2),

K (t, s)

and

f (t)

are given and the solution

y (t)

is sought on a uniform mesh of size

h

such that

T = n h

. Discretization of (2) yields

y_{i} = f (i \times h) + h \sum_{j = 0}^{p - 1} W_{i, j} K (i, h, j, h) y_{j} + h \sum_{j = p}^{i} ω_{i - j} K (i, h, j, h) y_{j},

(3)

where

y_{i} ≃ y (i \times h)

. We propose the following algorithm for computing

y_{i}

from (3) after a call to nag_inteq_volterra_weights (d05bwc):

(a)

Equation (3) requires starting values,

y_{j}

, for

j = 1, 2, \dots, p - 1

, with

y_{0} = f (0)

. These starting values can be computed by solving the linear system

y_{i} = f (i \times h) + h \sum_{j = 0}^{p - 1} sw [j \times n + i - 1] K (i, h, j, h) y_{j}, i = 1, 2, \dots, p - 1 .

(b)

Compute the inhomogeneous terms

σ_{i} = f (i \times h) + h \sum_{j = 0}^{p - 1} sw [j \times n + i - 1] K (i, h, j, h) y_{j}, i = p, p + 1, \dots, n .

(c)

Start the iteration for

i = p, p + 1, \dots, n

to compute

y_{i}

from:

(1 - h \times omega [0] K (i, h, i, h)) y_{i} = σ_{i} + h \sum_{j = p}^{i - 1} omega [i - j] K (i, h, j, h) y_{j} .

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.

NAG Library Function Document

nag_inteq_volterra_weights (d05bwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_inteq_volterra_weights (d05bwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_inteq_volterra_weights (d05bwc)