# NAG Library Routine Document

## 1Purpose

d05bdf computes the solution of a weakly singular nonlinear convolution Volterra–Abel integral equation of the second kind using a fractional Backward Differentiation Formulae (BDF) method.

## 2Specification

Fortran Interface
 Subroutine d05bdf ( ck, cf, cg, tlim, yn, work, lwk, nct,
 Integer, Intent (In) :: iorder, nmesh, lwk Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: nct(nmesh/32+1) Real (Kind=nag_wp), External :: ck, cf, cg Real (Kind=nag_wp), Intent (In) :: tlim, tolnl Real (Kind=nag_wp), Intent (Inout) :: work(lwk) Real (Kind=nag_wp), Intent (Out) :: yn(nmesh) Character (1), Intent (In) :: initwt
#include nagmk26.h
 void d05bdf_ (double (NAG_CALL *ck)(const double *t),double (NAG_CALL *cf)(const double *t),double (NAG_CALL *cg)(const double *s, const double *y),const char *initwt, const Integer *iorder, const double *tlim, const double *tolnl, const Integer *nmesh, double yn[], double work[], const Integer *lwk, Integer nct[], Integer *ifail, const Charlen length_initwt)

## 3Description

d05bdf computes the numerical solution of the weakly singular convolution Volterra–Abel integral equation of the second kind
 $yt = ft + 1π ∫0t kt-s t-s g s,ys ds , 0≤t≤T .$ (1)
Note the constant $\frac{1}{\sqrt{\pi }}$ in (1). It is assumed that the functions involved in (1) are sufficiently smooth.
The routine uses a fractional BDF linear multi-step method to generate a family of quadrature rules (see d05byf). The BDF methods available in d05bdf are of orders $4$, $5$ and $6$ ($\text{}=p$ say). For a description of the theoretical and practical background to these methods we refer to Lubich (1985) and to Baker and Derakhshan (1987) and Hairer et al. (1988) respectively.
The algorithm is based on computing the solution $y\left(t\right)$ in a step-by-step fashion on a mesh of equispaced points. The size of the mesh is given by $T/\left(N-1\right)$, $N$ being the number of points at which the solution is sought. These methods require $2p-1$ (including $y\left(0\right)$) starting values which are evaluated internally. The computation of the lag term arising from the discretization of (1) is performed by fast Fourier transform (FFT) techniques when $N>32+2p-1$, and directly otherwise. The routine does not provide an error estimate and you are advised to check the behaviour of the solution with a different value of $N$. An option is provided which avoids the re-evaluation of the fractional weights when d05bdf is to be called several times (with the same value of $N$) within the same program unit with different functions.

## 4References

Baker C T H and Derakhshan M S (1987) FFT techniques in the numerical solution of convolution equations J. Comput. Appl. Math. 20 5–24
Hairer E, Lubich Ch and Schlichte M (1988) Fast numerical solution of weakly singular Volterra integral equations J. Comput. Appl. Math. 23 87–98
Lubich Ch (1985) Fractional linear multistep methods for Abel–Volterra integral equations of the second kind Math. Comput. 45 463–469

## 5Arguments

1:     $\mathbf{ck}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
ck must evaluate the kernel $k\left(t\right)$ of the integral equation (1).
The specification of ck is:
Fortran Interface
 Function ck ( t)
 Real (Kind=nag_wp) :: ck Real (Kind=nag_wp), Intent (In) :: t
#include nagmk26.h
 double ck (const double *t)
1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: $t$, the value of the independent variable.
ck must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.
Note: ck should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.
2:     $\mathbf{cf}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
cf must evaluate the function $f\left(t\right)$ in (1).
The specification of cf is:
Fortran Interface
 Function cf ( t)
 Real (Kind=nag_wp) :: cf Real (Kind=nag_wp), Intent (In) :: t
#include nagmk26.h
 double cf (const double *t)
1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: $t$, the value of the independent variable.
cf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.
Note: cf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.
3:     $\mathbf{cg}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
cg must evaluate the function $g\left(s,y\left(s\right)\right)$ in (1).
The specification of cg is:
Fortran Interface
 Function cg ( s, y)
 Real (Kind=nag_wp) :: cg Real (Kind=nag_wp), Intent (In) :: s, y
#include nagmk26.h
 double cg (const double *s, const double *y)
1:     $\mathbf{s}$ – Real (Kind=nag_wp)Input
On entry: $s$, the value of the independent variable.
2:     $\mathbf{y}$ – Real (Kind=nag_wp)Input
On entry: the value of the solution $y$ at the point s.
cg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.
Note: cg should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.
4:     $\mathbf{initwt}$ – Character(1)Input
On entry: if the fractional weights required by the method need to be calculated by the routine then set ${\mathbf{initwt}}=\text{'I'}$ (Initial call).
If ${\mathbf{initwt}}=\text{'S'}$ (Subsequent call), the routine assumes the fractional weights have been computed on a previous call and are stored in work.
Constraint: ${\mathbf{initwt}}=\text{'I'}$ or $\text{'S'}$.
Note: when d05bdf is re-entered with the value of ${\mathbf{initwt}}=\text{'S'}$, the values of nmesh, iorder and the contents of work must not be changed.
5:     $\mathbf{iorder}$ – IntegerInput
On entry: $p$, the order of the BDF method to be used.
Suggested value: ${\mathbf{iorder}}=4$.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
6:     $\mathbf{tlim}$ – Real (Kind=nag_wp)Input
On entry: the final point of the integration interval, $T$.
Constraint: .
7:     $\mathbf{tolnl}$ – Real (Kind=nag_wp)Input
On entry: the accuracy required for the computation of the starting value and the solution of the nonlinear equation at each step of the computation (see Section 9).
Suggested value: ${\mathbf{tolnl}}=\sqrt{\epsilon }$ where $\epsilon$ is the machine precision.
Constraint: .
8:     $\mathbf{nmesh}$ – IntegerInput
On entry: $N$, the number of equispaced points at which the solution is sought.
Constraint: ${\mathbf{nmesh}}={2}^{m}+2×{\mathbf{iorder}}-1$, where $m\ge 1$.
9:     $\mathbf{yn}\left({\mathbf{nmesh}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{yn}}\left(\mathit{i}\right)$ contains the approximate value of the true solution $y\left(t\right)$ at the point $t=\left(\mathit{i}-1\right)×h$, for $\mathit{i}=1,2,\dots ,{\mathbf{nmesh}}$, where $h={\mathbf{tlim}}/\left({\mathbf{nmesh}}-1\right)$.
10:   $\mathbf{work}\left({\mathbf{lwk}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{initwt}}=\text{'S'}$, work must contain fractional weights computed by a previous call of d05bdf (see description of initwt).
On exit: contains fractional weights which may be used by a subsequent call of d05bdf.
11:   $\mathbf{lwk}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which d05bdf is called.
Constraint: ${\mathbf{lwk}}\ge \left(2×{\mathbf{iorder}}+6\right)×{\mathbf{nmesh}}+8×{{\mathbf{iorder}}}^{2}-16×{\mathbf{iorder}}+1$.
12:   $\mathbf{nct}\left({\mathbf{nmesh}}/32+1\right)$ – Integer arrayWorkspace
13:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{iorder}}<4$ or ${\mathbf{iorder}}>6$, or , or ${\mathbf{initwt}}\ne \text{'I'}$ or $\text{'S'}$, or ${\mathbf{initwt}}=\text{'S'}$ on the first call to d05bdf, or , or ${\mathbf{nmesh}}\ne {2}^{m}+2×{\mathbf{iorder}}-1,m\ge 1$, or ${\mathbf{lwk}}<\left(2×{\mathbf{iorder}}+6\right)×{\mathbf{nmesh}}+8×{{\mathbf{iorder}}}^{2}-16×{\mathbf{iorder}}+1$.
${\mathbf{ifail}}=2$
The routine cannot compute the $2p-1$ starting values due to an error solving the system of nonlinear equations. Relaxing the value of tolnl and/or increasing the value of nmesh may overcome this problem (see Section 9 for further details).
${\mathbf{ifail}}=3$
The routine cannot compute the solution at a specific step due to an error in the solution of the nonlinear equation (2). Relaxing the value of tolnl and/or increasing the value of nmesh may overcome this problem (see Section 9 for further details).
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The accuracy depends on nmesh and tolnl, the theoretical behaviour of the solution of the integral equation and the interval of integration. The value of tolnl controls the accuracy required for computing the starting values and the solution of (2) at each step of computation. This value can affect the accuracy of the solution. However, for most problems, the value of $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision, should be sufficient.

## 8Parallelism and Performance

d05bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d05bdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

In solving (1), initially, d05bdf computes the solution of a system of nonlinear equations for obtaining the $2p-1$ starting values. c05qdf is used for this purpose. When a failure with ${\mathbf{ifail}}={\mathbf{2}}$ occurs (which corresponds to an error exit from c05qdf), you are advised to either relax the value of tolnl or choose a smaller step size by increasing the value of nmesh. Once the starting values are computed successfully, the solution of a nonlinear equation of the form
 $Yn-αgtn,Yn-Ψn=0,$ (2)
is required at each step of computation, where ${\Psi }_{n}$ and $\alpha$ are constants. d05bdf calls c05axf to find the root of this equation.
If a failure with ${\mathbf{ifail}}={\mathbf{3}}$ occurs (which corresponds to an error exit from c05axf), you are advised to relax the value of the tolnl or choose a smaller step size by increasing the value of nmesh.
If a failure with ${\mathbf{ifail}}={\mathbf{2}}$ or ${\mathbf{3}}$ persists even after adjustments to tolnl and/or nmesh then you should consider whether there is a more fundamental difficulty. For example, the problem is ill-posed or the functions in (1) are not sufficiently smooth.

## 10Example

In this example we solve the following integral equations
 $yt = t + 38 π t2 - ∫0t 1 t-s y s 3 ds , 0≤t≤7 ,$
with the solution $y\left(t\right)=\sqrt{t}$, and
 $y t = 3-t t - ∫0t 1t-s exp s 1-s 2 - ys 2 d s , 0≤t≤5 ,$
with the solution $y\left(t\right)=\left(1-t\right)\sqrt{t}$. In the above examples, the fourth-order BDF is used, and nmesh is set to ${2}^{6}+7$.

### 10.1Program Text

Program Text (d05bdfe.f90)

None.

### 10.3Program Results

Program Results (d05bdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017