d05by:: Integral Equations (NAG Toolbox)

nag_inteq_abel_weak_weights (d05by) 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 Further Comments.

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

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

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 (d05by). 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 (d05by):

(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).

Example

The following example generates the first

16

fractional convolution and

23

fractional starting weights generated by the fourth-order BDF method.

function d05by_example


fprintf('d05by example results\n\n');

iorder = int64(4);
iq = int64(3);
lenfw = int64(32);
[wt, sw, ifail] = d05by( ...
			 iorder, iq, lenfw);

fprintf('\nFractional convolution weights\n\n');
itiq = double(2^(iq+1));
n = [0:itiq-1]';
w(1:itiq) = wt(1:itiq,1);
fprintf('%3d    %10.4f\n',[n w']');

fprintf('\nFractional starting weights W\n\n');
ldsw = double(itiq+2*iorder-1);
n = [0:ldsw-1]';
fprintf('%5d%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f%9.4f\n',[n sw]');

d05by example results


Fractional convolution weights

  0        0.6928
  1        0.6651
  2        0.4589
  3        0.3175
  4        0.2622
  5        0.2451
  6        0.2323
  7        0.2164
  8        0.2006
  9        0.1878
 10        0.1780
 11        0.1700
 12        0.1629
 13        0.1566
 14        0.1508
 15        0.1457

Fractional starting weights W

    0   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
    1   0.0565   2.8928  -6.7497  11.6491 -11.1355   5.5374  -1.1223
    2   0.0371   1.7401  -2.8628   6.5207  -6.4058   3.2249  -0.6583
    3   0.0300   1.3207  -2.4642   6.3612  -5.4478   2.7025  -0.5481
    4   0.0258   1.1217  -2.2620   5.3683  -3.7553   2.2132  -0.4549
    5   0.0230   0.9862  -2.0034   4.5005  -3.2772   2.7262  -0.4320
    6   0.0208   0.9001  -1.8989   4.2847  -3.5881   2.8201   0.2253
    7   0.0190   0.8506  -1.9250   4.4164  -4.0181   2.7932   0.1564
    8   0.0173   0.8177  -1.9697   4.5348  -4.2425   2.7458  -0.0697
    9   0.0160   0.7886  -1.9781   4.5318  -4.2769   2.6997  -0.2127
   10   0.0149   0.7603  -1.9548   4.4545  -4.2332   2.6541  -0.2620
   11   0.0140   0.7338  -1.9198   4.3619  -4.1782   2.6059  -0.2716
   12   0.0132   0.7097  -1.8842   4.2754  -4.1246   2.5544  -0.2767
   13   0.0125   0.6880  -1.8497   4.1933  -4.0662   2.5011  -0.2845
   14   0.0119   0.6681  -1.8153   4.1109  -4.0004   2.4479  -0.2915
   15   0.0114   0.6497  -1.7805   4.0279  -3.9304   2.3962  -0.2951
   16   0.0110   0.6327  -1.7461   3.9463  -3.8598   2.3466  -0.2958
   17   0.0105   0.6168  -1.7126   3.8677  -3.7907   2.2990  -0.2950
   18   0.0102   0.6020  -1.6804   3.7926  -3.7238   2.2536  -0.2935
   19   0.0098   0.5882  -1.6495   3.7209  -3.6589   2.2101  -0.2917
   20   0.0095   0.5752  -1.6199   3.6523  -3.5961   2.1686  -0.2895
   21   0.0093   0.5631  -1.5916   3.5867  -3.5356   2.1291  -0.2871
   22   0.0090   0.5517  -1.5644   3.5240  -3.4774   2.0914  -0.2844

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 Toolbox: nag_inteq_abel_weak_weights (d05by)

▸▿ Contents

Purpose

Syntax

Description