nag_inteq_abel1_weak (d05bec) (PDF version)
d05 Chapter Contents
d05 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_inteq_abel1_weak (d05bec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_inteq_abel1_weak (d05bec) 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.

2  Specification

#include <nag.h>
#include <nagd05.h>
void  nag_inteq_abel1_weak (
double (*ck)(double t, Nag_Comm *comm),
double (*cf)(double t, Nag_Comm *comm),
double (*cg)(double s, double y, Nag_Comm *comm),
Nag_WeightMode wtmode, Integer iorder, double tlim, double tolnl, Integer nmesh, double yn[], double rwsav[], Integer lrwsav, Nag_Comm *comm, NagError *fail)

3  Description

nag_inteq_abel1_weak (d05bec) computes the numerical solution of the weakly singular convolution Volterra–Abel integral equation of the first kind
ft+1π0tkt-s t-s gs,ysds=0,  0tT. (1)
Note the constant 1π  in (1). It is assumed that the functions involved in (1) are sufficiently smooth and if
ft=tβwt  with  β>-12​ and ​wt​ smooth, (2)
then the solution yt is unique and has the form yt=tβ-1/2zt, (see Lubich (1987)). It is evident from (1) that f0=0. You are required to provide the value of yt at t=0. If y0 is unknown, Section 8 gives a description of how an approximate value can be obtained.
The function uses a fractional BDF linear multi-step method selected by you to generate a family of quadrature rules (see nag_inteq_abel_weak_weights (d05byc)). The BDF methods available in nag_inteq_abel1_weak (d05bec) are of orders 4, 5 and 6 (=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 yt 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 function 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 nag_inteq_abel1_weak (d05bec) is to be called several times (with the same value of N) within the same program with different functions.

4  References

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. Vietnam 16 211–262
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. Anal 7 97–106

5  Arguments

1:     ckfunction, supplied by the userExternal Function
ck must evaluate the kernel kt of the integral equation (1).
The specification of ck is:
double  ck (double t, Nag_Comm *comm)
1:     tdoubleInput
On entry: t, the value of the independent variable.
2:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to ck.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_inteq_abel1_weak (d05bec) you may allocate memory and initialize these pointers with various quantities for use by ck when called from nag_inteq_abel1_weak (d05bec) (see Section 3.2.1 in the Essential Introduction).
2:     cffunction, supplied by the userExternal Function
cf must evaluate the function ft in (1).
The specification of cf is:
double  cf (double t, Nag_Comm *comm)
1:     tdoubleInput
On entry: t, the value of the independent variable.
2:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to cf.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_inteq_abel1_weak (d05bec) you may allocate memory and initialize these pointers with various quantities for use by cf when called from nag_inteq_abel1_weak (d05bec) (see Section 3.2.1 in the Essential Introduction).
3:     cgfunction, supplied by the userExternal Function
cg must evaluate the function gs,ys in (1).
The specification of cg is:
double  cg (double s, double y, Nag_Comm *comm)
1:     sdoubleInput
On entry: s, the value of the independent variable.
2:     ydoubleInput
On entry: the value of the solution y at the point s.
3:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to cg.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_inteq_abel1_weak (d05bec) you may allocate memory and initialize these pointers with various quantities for use by cg when called from nag_inteq_abel1_weak (d05bec) (see Section 3.2.1 in the Essential Introduction).
4:     wtmodeNag_WeightModeInput
On entry: if the fractional weights required by the method need to be calculated by the function then set wtmode=Nag_InitWeights.
If wtmode=Nag_ReuseWeights, the function assumes the fractional weights have been computed by a previous call and are stored in rwsav.
Constraint: wtmode=Nag_InitWeights or Nag_ReuseWeights.
Note: when nag_inteq_abel1_weak (d05bec) is re-entered with a value of wtmode=Nag_ReuseWeights, the values of nmesh, iorder and the contents of rwsav MUST NOT be changed
5:     iorderIntegerInput
On entry: p, the order of the BDF method to be used.
Suggested value: iorder=4.
Constraint: 4iorder6.
6:     tlimdoubleInput
On entry: the final point of the integration interval, T.
Constraint: tlim>10×machine precision.
7:     tolnldoubleInput
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 8).
Suggested value: tolnl=ε where ε is the machine precision.
Constraint: tolnl>10×machine precision.
8:     nmeshIntegerInput
On entry: N, the number of equispaced points at which the solution is sought.
Constraint: nmesh=2m+2×iorder-1, where m1.
9:     yn[nmesh]doubleInput/Output
On entry: yn[0] must contain the value of yt at t=0 (see Section 8).
On exit: yn[i-1] contains the approximate value of the true solution yt at the point t=i-1×h, for i=1,2,,nmesh, where h=tlim/nmesh-1.
10:   rwsav[lrwsav]doubleCommunication Array
On entry: if wtmode=Nag_ReuseWeights, rwsav must contain fractional weights computed by a previous call of nag_inteq_abel1_weak (d05bec) (see description of wtmode).
On exit: contains fractional weights which may be used by a subsequent call of nag_inteq_abel1_weak (d05bec).
11:   lrwsavIntegerInput
On entry: the dimension of the array rwsav.
Constraint: lrwsav2×iorder+6×nmesh+8×iorder2-16×iorder+1.
12:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
13:   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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_FAILED_START
An error occurred when trying to compute the starting values.
NE_FAILED_STEP
An error occurred when trying to compute the solution at a specific step.
NE_INT
On entry, iorder=value.
Constraint: 4iorder6.
NE_INT_2
On entry, lrwsav=value.
Constraint: lrwsav2×iorder+6×nmesh+8×iorder2-16×iorder+1; that is, value.
On entry, nmesh=value and iorder=value.
Constraint: nmesh=2m+2×iorder-1, for some m.
On entry, nmesh=value and iorder=value.
Constraint: nmesh2×iorder+1.
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.
NE_REAL
On entry, tlim=value.
Constraint: tlim>10×machine precision 
On entry, tolnl=value.
Constraint: tolnl>10×machine precision.

7  Accuracy

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 ε, where ε is the machine precision, should be sufficient.

8  Further Comments

Also when solving (1) the initial value y0 is required. This value may be computed from the limit relation (see Gorenflo and Pfeiffer (1991))
-2π k0 g 0,y0 = lim t0 ft t . (3)
If the value of the above limit is known then by solving the nonlinear equation (3) an approximation to y0 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 pth-order approximation can be used:
lim t0 ft t fhphp/2 . (4)
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 lim t0 ft t  by analytical methods.
Also when solving (1), initially, nag_inteq_abel1_weak (d05bec) computes the solution of a system of nonlinear equation for obtaining the 2p-2 starting values. nag_zero_nonlin_eqns_rcomm (c05qdc) is used for this purpose. If a failure with fail.code= NE_FAILED_START occurs (corresponding to an error exit from nag_zero_nonlin_eqns_rcomm (c05qdc)), 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, (5)
is required at each step of computation, where Ψn and α are constants. nag_inteq_abel1_weak (d05bec) calls nag_zero_cont_func_cntin_rcomm (c05axc) to find the root of this equation.
When a failure with fail.code= NE_FAILED_STEP occurs (which corresponds to an error exit from nag_zero_cont_func_cntin_rcomm (c05axc)), 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 fail.code= NE_FAILED_START or NE_FAILED_STEP 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.

9  Example

We solve the following integral equations.
Example 1
The density of the probability that a Brownian motion crosses a one-sided moving boundary at before time t, satisfies the integral equation (see Hairer et al. (1988))
-1t exp 12-at2/t+0texp -12at-as2/t-s t-s ysds=0,  0t7.
In the case of a straight line at=1+t, the exact solution is known to be
yt=12πt3 exp- 1+t 2/2t
Example 2
In this example we consider the equation
-2log1+t+t 1+t +0t ys t-s ds= 0,   0t 5.
The solution is given by yt=11+t .
In the above examples, the fourth-order BDF is used, and nmesh is set to 26+7.

9.1  Program Text

Program Text (d05bece.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (d05bece.r)


nag_inteq_abel1_weak (d05bec) (PDF version)
d05 Chapter Contents
d05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012