d05 Chapter Contents
d05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_inteq_abel_weak_weights (d05byc)

## 1  Purpose

nag_inteq_abel_weak_weights (d05byc) computes the fractional quadrature weights associated with the Backward Differentiation Formulae (BDF) of orders $4$, $5$ and $6$. These weights can then be used in the solution of weakly singular equations of Abel type.

## 2  Specification

 #include #include
 void nag_inteq_abel_weak_weights (Integer iorder, Integer iq, double omega[], double sw[], NagError *fail)

## 3  Description

nag_inteq_abel_weak_weights (d05byc) computes the weights ${W}_{i,j}$ and ${\omega }_{i}$ for a family of quadrature rules related to a BDF method for approximating the integral:
 $1π∫0tϕs t-s ds≃h∑j=0 2p-2Wi,jϕj×h+h∑j=2p-1iωi-jϕj×h, 0≤t≤T,$ (1)
with $t=i×h\left(i\ge 0\right)$, for some given $h$. In (1), $p$ is the order of the BDF method used and ${W}_{i,j}$, ${\omega }_{i}$ are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of ${\omega }_{i}$ 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  Arguments

1:     iorderIntegerInput
On entry: $p$, the order of the BDF method to be used.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
2:     iqIntegerInput
On entry: determines the number of weights to be computed. By setting iq to a value, ${2}^{{\mathbf{iq}}+1}$ fractional convolution weights are computed.
Constraint: ${\mathbf{iq}}\ge 0$.
3:     omega[${2}^{{\mathbf{iq}}+2}$]doubleOutput
On exit: the first ${2}^{{\mathbf{iq}}+1}$ elements of omega contains the fractional convolution weights ${\omega }_{i}$, for $\mathit{i}=0,1,\dots ,{2}^{{\mathbf{iq}}+1}-1$. The remainder of the array is used as workspace.
4:     sw[$\mathit{n}×\left(2×{\mathbf{iorder}}-1\right)$]doubleOutput
On exit: ${\mathbf{sw}}\left[\mathit{j}×\mathit{n}+\mathit{i}-1\right]$ contains the fractional starting weights ${W}_{\mathit{i},\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=0,1,\dots ,2×{\mathbf{iorder}}-2$, where $\mathit{n}=\left({2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-1\right)$.
5:     failNagError *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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{iq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{iq}}\ge 0$.
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.

## 7  Accuracy

Not applicable.

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
 $yt=ft+1π∫0tKt,sys t-s ds, 0≤t≤T,$ (2)
using nag_inteq_abel_weak_weights (d05byc). In (2), $K\left(t,s\right)$ and $f\left(t\right)$ are given and the solution $y\left(t\right)$ is sought on a uniform mesh of size $h$ such that $T=\mathit{n}×h$. Discretization of (2) yields
 $yi = fi×h + h ∑ j=0 2p-2 W i,j K i×h,j×h yj + h ∑ j=2p-1 i ωi-j K i×h,j×h yj ,$ (3)
where ${y}_{\mathit{i}}\simeq y\left(\mathit{i}×h\right)$, for $\mathit{i}=1,2,\dots ,\mathit{n}$. We propose the following algorithm for computing ${y}_{i}$ from (3) after a call to nag_inteq_abel_weak_weights (d05byc):
(a) Set $\mathit{n}={2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-2$ and $h=T/\mathit{n}$.
(b) Equation (3) requires $2×{\mathbf{iorder}}-2$ starting values, ${y}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,2×{\mathbf{iorder}}-2$, with ${y}_{0}=f\left(0\right)$. These starting values can be computed by solving the system
 $yi = fi×h + h ∑ j=0 2×iorder-2 sw[j×n+i] K i×h,j×h yj , i=1,2,…,2×iorder-2 .$
(c) Compute the inhomogeneous terms
 $σi = fi×h + h ∑ j=0 2×iorder- 2 sw[j×n+i] K i×h,j×h yj , i = 2 × iorder-1 , 2×iorder , … , n .$
(d) Start the iteration for $i=2×{\mathbf{iorder}}-1,2×{\mathbf{iorder}},\dots ,\mathit{n}$ to compute ${y}_{i}$ from:
 $1 - h omega[0] K i×h,i×h yi = σi + h ∑ j=2×iorder-1 i-1 omega[i-j] K i×h,j×h yj .$
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).

## 9  Example

The following example generates the first $16$ fractional convolution and $23$ fractional starting weights generated by the fourth-order BDF method.

### 9.1  Program Text

Program Text (d05byce.c)

None.

### 9.3  Program Results

Program Results (d05byce.r)