nag_inteq_abel_weak_weights (d05byc) : NAG Library, Mark 24

3 Description

nag_inteq_abel_weak_weights (d05byc) 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 9.

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 Arguments

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: omega[

2^{iq + 2}

] – doubleOutput

On exit: the first

2^{iq + 1}

elements of omega 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.

4: sw[

N \times (2 \times iorder - 1)

] – doubleOutput

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 fractional starting weights

W_{i, 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)

5: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_INT

On entry,

iorder = ⟨value⟩

.
Constraint:

4 \leq iorder \leq 6

On entry,

iq = ⟨value⟩

.
Constraint:

iq \geq 0

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.

9 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 nag_inteq_abel_weak_weights (d05byc). 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 nag_inteq_abel_weak_weights (d05byc):

(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 [j \times N + i] 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 [j \times N + i] 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} omega [0] K (i \times h, i \times h)) y_{i} = σ_{i} + \sqrt{h} \sum_{j = 2 \times iorder - 1}^{i - 1} omega [i - j] 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).

NAG Library Function Document

nag_inteq_abel_weak_weights (d05byc)

+− 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_abel_weak_weights (d05byc)

+− 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_abel_weak_weights (d05byc)