D05BYF : NAG Library, Mark 24

3 Description

D05BYF computes the weights

W_{i, j}

and

ω_{i}

for a family of quadrature rules related to a BDF method for approximating the integral:

\frac{1}{\sqrt{π}} \int_{0}^{t} \frac{ϕ (s)}{\sqrt{t - s}} d s ≃ \sqrt{h} \sum_{j = 0}^{2 p - 2} W_{i, j} ϕ (j \times h) + \sqrt{h} \sum_{j = 2 p - 1}^{i} ω_{i - j} ϕ (j \times h), 0 \leq t \leq T,

(1)

with

t = i \times h (i \geq 0)

, for some given

h

. In (1),

p

is the order of the BDF method used and

W_{i, j}

ω_{i}

are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of

ω_{i}

is based on Newton's iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently

W_{i, j}

(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.

4 References

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

5 Parameters

1: IORDER – INTEGERInput

On entry:

p

, the order of the BDF method to be used.

Constraint:

4 \leq IORDER \leq 6

2: IQ – INTEGERInput

On entry: determines the number of weights to be computed. By setting IQ to a value,

2^{IQ + 1}

fractional convolution weights are computed.

Constraint:

IQ \geq 0

3: LENFW – INTEGERInput

On entry: the dimension of the array WT as declared in the (sub)program from which D05BYF is called.

Constraint:

LENFW \geq 2^{IQ + 2}

4: WT(LENFW) – REAL (KIND=nag_wp) arrayOutput

On exit: the first

2^{IQ + 1}

elements of WT contains the fractional convolution weights

ω_{i}

, for

i = 0, 1, \dots, 2^{IQ + 1} - 1

. The remainder of the array is used as workspace.

5: SW(LDSW,

2 \times IORDER - 1

) – REAL (KIND=nag_wp) arrayOutput

On exit:

SW (i, j + 1)

contains the fractional starting weights

W_{i - 1, j}

, for

i = 1, 2, \dots, N

and

j = 0, 1, \dots, 2 \times IORDER - 2

, where

N = (2^{IQ + 1} + 2 \times IORDER - 1)

6: LDSW – INTEGERInput

On entry: the first dimension of the array SW as declared in the (sub)program from which D05BYF is called.

Constraint:

LDSW \geq 2^{IQ + 1} + 2 \times IORDER - 1

7: WORK(LWK) – REAL (KIND=nag_wp) arrayWorkspace

8: LWK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which D05BYF is called.

Constraint:

LWK \geq 2^{IQ + 3}

9: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

8 Further Comments

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

y (t) = f (t) + \frac{1}{\sqrt{π}} \int_{0}^{t} \frac{K (t, s) y (s)}{\sqrt{t - s}} d s, 0 \leq t \leq T,

(2)

using D05BYF. 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 \times h

. Discretization of (2) yields

y_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 p - 2} W_{i, j} K (i \times h, j \times h) y_{j} + \sqrt{h} \sum_{j = 2 p - 1}^{i} ω_{i - j} K (i \times h, j \times h) y_{j},

(3)

where

y_{i} ≃ y (i \times h)

, for

i = 1, 2, \dots, N

. We propose the following algorithm for computing

y_{i}

from (3) after a call to D05BYF:

(a)

Set

N = 2^{IQ + 1} + 2 \times IORDER - 2

and

h = T / N

(b)

Equation (3) requires

2 \times IORDER - 2

starting values,

y_{j}

, for

j = 1, 2, \dots, 2 \times IORDER - 2

, with

y_{0} = f (0)

. These starting values can be computed by solving the system

y_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times IORDER - 2} SW (i + 1, j + 1) K (i \times h, j \times h) y_{j}, i = 1, 2, \dots, 2 \times IORDER - 2 .

(c)

Compute the inhomogeneous terms

σ_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times IORDER - 2} SW (i + 1, j + 1) K (i \times h, j \times h) y_{j}, i = 2 \times IORDER - 1, 2 \times IORDER, \dots, N .

(d)

Start the iteration for

i = 2 \times IORDER - 1, 2 \times IORDER, \dots, N

to compute

y_{i}

from:

(1 - \sqrt{h} WT (1) K (i \times h, i \times h)) y_{i} = σ_{i} + \sqrt{h} \sum_{j = 2 \times IORDER - 1}^{i - 1} WT (i - j + 1) K (i \times h, j \times h) y_{j} .

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).

On entry,	$IORDER < 4$ or $IORDER > 6$ ,
or	$IQ < 0$ ,
or	$LENFW < 2^{IQ + 2}$ ,
or	$LDSW < 2^{IQ + 1} + 2 \times IORDER - 1$ ,
or	$LWK < 2^{IQ + 3}$ .

NAG Library Routine Document

D05BYF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentD05BYF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D05BYF