d05bef computes the solution of a weakly singular nonlinear convolution Volterra–Abel integral equation of the first kind using a fractional Backward Differentiation Formulae (BDF) method.
Note the constant $\frac{1}{\sqrt{\pi}}$ in (1). It is assumed that the functions involved in (1) are sufficiently smooth and if
$$f\left(t\right)={t}^{\beta}w\left(t\right)\text{\hspace{1em} with \hspace{1em}}\beta >-\frac{1}{2}\text{ and}w\left(t\right)\text{ smooth,}$$
(2)
then the solution $y\left(t\right)$ is unique and has the form $y\left(t\right)={t}^{\beta -1/2}z\left(t\right)$, (see Lubich (1987)). It is evident from (1) that $f\left(0\right)=0$. You are required to provide the value of $y\left(t\right)$ at $t=0$. If $y\left(0\right)$ is unknown, Section 9 gives a description of how an approximate value can be obtained.
The routine uses a fractional BDF linear multi-step method selected by you to generate a family of quadrature rules (see d05byf). The BDF methods available in d05bef are of orders $4$, $5$ and $6$ ($\text{}=p$ say). For a description of the theoretical and practical background related to these methods we refer to Lubich (1987) 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 $T/(N-1)$, $N$ being the number of points at which the solution is sought. These methods require $2p-2$ 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 d05bef is to be called several times (with the same value of $N$) within the same program 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
Gorenflo R and Pfeiffer A (1991) On analysis and discretization of nonlinear Abel integral equations of first kind Acta Math. Vietnam16 211–262
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 (1987) Fractional linear multistep methods for Abel–Volterra integral equations of the first kind IMA J. Numer. Anal7 97–106
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).
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 d05bef 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 d05bef. If your code inadvertently does return any NaNs or infinities, d05bef 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).
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 d05bef 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 d05bef. If your code inadvertently does return any NaNs or infinities, d05bef 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 $g(s,y\left(s\right))$ in (1).
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 d05bef 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 d05bef. If your code inadvertently does return any NaNs or infinities, d05bef 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 by a previous call and are stored in work.
Constraint:
${\mathbf{initwt}}=\text{'I'}$ or $\text{'S'}$.
Note: when d05bef is re-entered with a value of ${\mathbf{initwt}}=\text{'S'}$, the values of nmesh, iorder and the contents of workmust 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, $T$.
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.
On entry: $N$, the number of equispaced points at which the solution is sought.
Constraint:
${\mathbf{nmesh}}={2}^{m}+2\times {\mathbf{iorder}}-1$, where $m\ge 1$.
9: $\mathbf{yn}\left({\mathbf{nmesh}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{yn}}\left(1\right)$ must contain the value of $y\left(t\right)$ at $t=0$ (see Section 9).
On exit: ${\mathbf{yn}}\left(\mathit{i}\right)$ contains the approximate value of the true solution $y\left(t\right)$ at the point $t=(\mathit{i}-1)\times h$, for $\mathit{i}=1,2,\dots ,{\mathbf{nmesh}}$, where $h={\mathbf{tlim}}/({\mathbf{nmesh}}-1)$.
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 d05bef (see description of initwt).
On exit: contains fractional weights which may be used by a subsequent call of d05bef.
11: $\mathbf{lwk}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which d05bef is called.
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, ${\mathbf{initwt}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{initwt}}=\text{'I'}$ or $\text{'S'}$.
On entry, ${\mathbf{iorder}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{lwk}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lwk}}\ge (2\times {\mathbf{iorder}}+6)\times {\mathbf{nmesh}}+8\times {{\mathbf{iorder}}}^{2}-16\times {\mathbf{iorder}}+1$; that is, $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{nmesh}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{iorder}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nmesh}}={2}^{m}+2\times {\mathbf{iorder}}-1$, for some $m$.
On entry, ${\mathbf{nmesh}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{iorder}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nmesh}}\ge 2\times {\mathbf{iorder}}+1$.
On entry, ${\mathbf{tlim}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tlim}}>10\times \mathit{machineprecision}$
On entry, ${\mathbf{tolnl}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tolnl}}>10\times \mathit{machineprecision}$.
${\mathbf{ifail}}=2$
An error occurred when trying to compute the starting values.
Relaxing the value of tolnl and/or increasing the value of nmesh may overcome this problem (see Section 9 for further details).
${\mathbf{ifail}}=3$
An error occurred when trying to compute the solution at a specific step.
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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 (3) 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
d05bef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d05bef 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.
9Further Comments
Also when solving (1) the initial value $y\left(0\right)$ is required. This value may be computed from the limit relation (see Gorenflo and Pfeiffer (1991))
If the value of the above limit is known then by solving the nonlinear equation (3) an approximation to $y\left(0\right)$ can be computed. If the value of the above limit is not known, an approximation should be provided. Following the analysis presented in Gorenflo and Pfeiffer (1991), the following $p$th-order approximation can be used:
However, it must be emphasized that the approximation in (4) may result in an amplification of the rounding errors and hence you are advised (if possible) to determine
$\underset{t\to 0}{\mathrm{lim}}}\phantom{\rule{0.25em}{0ex}}\frac{f\left(t\right)}{\sqrt{t}$
by analytical methods.
Also when solving (1), initially, d05bef computes the solution of a system of nonlinear equation for obtaining the $2p-2$ starting values. c05qdf is used for this purpose. If a failure with ${\mathbf{ifail}}={\mathbf{2}}$ occurs (corresponding 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
is required at each step of computation, where ${\Psi}_{n}$ and $\alpha $ are constants. d05bef calls c05axf to find the root of this equation.
When a failure with ${\mathbf{ifail}}={\mathbf{3}}$ occurs (which corresponds to an error exit from c05axf), you are advised to either 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
We solve the following integral equations.
Example 1
The density of the probability that a Brownian motion crosses a one-sided moving boundary $a\left(t\right)$ before time $t$, satisfies the integral equation (see Hairer et al. (1988))