D02 Chapter Contents
D02 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentD02TKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

D02TKF solves a general two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations.

## 2  Specification

 SUBROUTINE D02TKF ( FFUN, FJAC, GAFUN, GBFUN, GAJAC, GBJAC, GUESS, WORK, IWORK, IFAIL)
 INTEGER IWORK(*), IFAIL REAL (KIND=nag_wp) WORK(*) EXTERNAL FFUN, FJAC, GAFUN, GBFUN, GAJAC, GBJAC, GUESS

## 3  Description

D02TKF and its associated routines (D02TVF, D02TXF, D02TYF and D02TZF) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations
 $y1m1 x = f1 x,y1,y11,…,y1m1-1,y2,…,ynmn-1 y2m2 x = f2 x,y1,y11,…,y1m1-1,y2,…,ynmn-1 ⋮ ynmn x = fn x,y1,y11,…,y1m1-1,y2,…,ynmn-1$
over an interval $\left[a,b\right]$ subject to $p$ ($\text{}>0$) nonlinear boundary conditions at $a$ and $q$ ($\text{}>0$) nonlinear boundary conditions at $b$, where $p+q=\sum _{i=1}^{n}{m}_{i}$. Note that ${y}_{i}^{\left(m\right)}\left(x\right)$ is the $m$th derivative of the $i$th solution component. Hence ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$. The left boundary conditions at $a$ are defined as
 $gizya=0, i=1,2,…,p,$
and the right boundary conditions at $b$ as
 $g-jzyb=0, j=1,2,…,q,$
where $y=\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$ and
 $zyx = y1x, y11 x ,…, y1m1-1 x ,y2x,…, ynmn-1 x .$
First, D02TVF must be called to specify the initial mesh, error requirements and other details. Note that the error requirements apply only to the solution components ${y}_{1},{y}_{2},\dots ,{y}_{n}$ and that no error control is applied to derivatives of solution components. (If error control is required on derivatives then the system must be reduced in order by introducing the derivatives whose error is to be controlled as new variables. See Section 8 in D02TVF.) Then, D02TKF can be used to solve the boundary value problem. After successful computation, D02TZF can be used to ascertain details about the final mesh and other details of the solution procedure, and D02TYF can be used to compute the approximate solution anywhere on the interval $\left[a,b\right]$.
A description of the numerical technique used in D02TKF is given in Section 3 in D02TVF.
D02TKF can also be used in the solution of a series of problems, for example in performing continuation, when the mesh used to compute the solution of one problem is to be used as the initial mesh for the solution of the next related problem. D02TXF should be used in between calls to D02TKF in this context.
See Section 8 in D02TVF for details of how to solve boundary value problems of a more general nature.
The routines are based on modified versions of the codes COLSYS and COLNEW (see Ascher et al. (1979) and Ascher and Bader (1987)). A comprehensive treatment of the numerical solution of boundary value problems can be found in Ascher et al. (1988) and Keller (1992).

## 4  References

Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput. 8 483–500
Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput. 33 659–679
Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York

## 5  Parameters

1:     FFUN – SUBROUTINE, supplied by the user.External Procedure
FFUN must evaluate the functions ${f}_{i}$ for given values $x,z\left(y\left(x\right)\right)$.
The specification of FFUN is:
 SUBROUTINE FFUN ( X, Y, NEQ, M, F)
 INTEGER NEQ, M(NEQ) REAL (KIND=nag_wp) X, Y(NEQ,0:*), F(NEQ)
1:     X – REAL (KIND=nag_wp)Input
On entry: $x$, the independent variable.
2:     Y(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3:     NEQ – INTEGERInput
On entry: the number of differential equations.
4:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
5:     F(NEQ) – REAL (KIND=nag_wp) arrayOutput
On exit: the values of ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
FFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
2:     FJAC – SUBROUTINE, supplied by the user.External Procedure
FJAC must evaluate the partial derivatives of ${f}_{i}$ with respect to the elements of
$z\left(y\left(x\right)\right)$ ($\text{}=\left({y}_{1}\left(x\right),{y}_{1}^{1}\left(x\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(x\right),{y}_{2}\left(x\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(x\right)\right)$).
The specification of FJAC is:
 SUBROUTINE FJAC ( X, Y, NEQ, M, DFDY)
 INTEGER NEQ, M(NEQ) REAL (KIND=nag_wp) X, Y(NEQ,0:*), DFDY(NEQ,NEQ,0:*)
1:     X – REAL (KIND=nag_wp)Input
On entry: $x$, the independent variable.
2:     Y(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3:     NEQ – INTEGERInput
On entry: the number of differential equations.
4:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
5:     DFDY(NEQ,NEQ,$0:*$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{DFDY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${f}_{\mathit{i}}$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$, $\mathit{j}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{M}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
FJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
3:     GAFUN – SUBROUTINE, supplied by the user.External Procedure
GAFUN must evaluate the boundary conditions at the left-hand end of the range, that is functions ${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ for given values of $z\left(y\left(a\right)\right)$.
The specification of GAFUN is:
 SUBROUTINE GAFUN ( YA, NEQ, M, NLBC, GA)
 INTEGER NEQ, M(NEQ), NLBC REAL (KIND=nag_wp) YA(NEQ,0:*), GA(NLBC)
1:     YA(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{YA}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2:     NEQ – INTEGERInput
On entry: the number of differential equations.
3:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
4:     NLBC – INTEGERInput
On entry: the number of boundary conditions at $a$.
5:     GA(NLBC) – REAL (KIND=nag_wp) arrayOutput
On exit: the values of ${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NLBC}}$.
GAFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
4:     GBFUN – SUBROUTINE, supplied by the user.External Procedure
GBFUN must evaluate the boundary conditions at the right-hand end of the range, that is functions ${\stackrel{-}{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ for given values of $z\left(y\left(b\right)\right)$.
The specification of GBFUN is:
 SUBROUTINE GBFUN ( YB, NEQ, M, NRBC, GB)
 INTEGER NEQ, M(NEQ), NRBC REAL (KIND=nag_wp) YB(NEQ,0:*), GB(NRBC)
1:     YB(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{YB}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2:     NEQ – INTEGERInput
On entry: the number of differential equations.
3:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
4:     NRBC – INTEGERInput
On entry: the number of boundary conditions at $b$.
5:     GB(NRBC) – REAL (KIND=nag_wp) arrayOutput
On exit: the values of ${\stackrel{-}{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NRBC}}$.
GBFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
5:     GAJAC – SUBROUTINE, supplied by the user.External Procedure
GAJAC must evaluate the partial derivatives of ${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ with respect to the elements of $z\left(y\left(a\right)\right)$ ($\text{}=\left({y}_{1}\left(a\right),{y}_{1}^{1}\left(a\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(a\right),{y}_{2}\left(a\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(a\right)\right)$).
The specification of GAJAC is:
 SUBROUTINE GAJAC ( YA, NEQ, M, NLBC, DGADY)
 INTEGER NEQ, M(NEQ), NLBC REAL (KIND=nag_wp) YA(NEQ,0:*), DGADY(NLBC,NEQ,0:*)
1:     YA(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{YA}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2:     NEQ – INTEGERInput
On entry: the number of differential equations.
3:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
4:     NLBC – INTEGERInput
On entry: the number of boundary conditions at $a$.
5:     DGADY(NLBC,NEQ,$0:*$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{DGADY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NLBC}}$, $\mathit{j}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{M}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
GAJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
6:     GBJAC – SUBROUTINE, supplied by the user.External Procedure
GBJAC must evaluate the partial derivatives of ${\stackrel{-}{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ with respect to the elements of $z\left(y\left(b\right)\right)$ ($\text{}=\left({y}_{1}\left(b\right),{y}_{1}^{1}\left(b\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(b\right),{y}_{2}\left(b\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(b\right)\right)$).
The specification of GBJAC is:
 SUBROUTINE GBJAC ( YB, NEQ, M, NRBC, DGBDY)
 INTEGER NEQ, M(NEQ), NRBC REAL (KIND=nag_wp) YB(NEQ,0:*), DGBDY(NRBC,NEQ,0:*)
1:     YB(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{YB}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2:     NEQ – INTEGERInput
On entry: the number of differential equations.
3:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
4:     NRBC – INTEGERInput
On entry: the number of boundary conditions at $a$.
5:     DGBDY(NRBC,NEQ,$0:*$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{DGBDY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${\stackrel{-}{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NRBC}}$, $\mathit{j}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{M}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
GBJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
7:     GUESS – SUBROUTINE, supplied by the user.External Procedure
GUESS must return initial approximations for the solution components ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}$ and the derivatives ${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$. Try to compute each derivative ${y}_{i}^{\left({m}_{i}\right)}$ such that it corresponds to your approximations to ${y}_{i}^{\left(\mathit{j}\right)}$, for $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(i\right)-1$. You should not call FFUN to compute ${y}_{i}^{\left({m}_{i}\right)}$.
If D02TKF is being used in conjunction with D02TXF as part of a continuation process, then GUESS is not called by D02TKF after the call to D02TXF.
The specification of GUESS is:
 SUBROUTINE GUESS ( X, NEQ, M, Y, DYM)
 INTEGER NEQ, M(NEQ) REAL (KIND=nag_wp) X, Y(NEQ,0:*), DYM(NEQ)
1:     X – REAL (KIND=nag_wp)Input
On entry: $x$, the independent variable; $x\in \left[a,b\right]$.
2:     NEQ – INTEGERInput
On entry: the number of differential equations.
3:     M(NEQ) – INTEGER arrayInput
On entry: ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
4:     Y(NEQ,$0:*$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ must contain ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{M}}\left(\mathit{i}\right)-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
5:     DYM(NEQ) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{DYM}}\left(\mathit{i}\right)$ must contain ${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NEQ}}$.
GUESS must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.
8:     WORK($*$) – REAL (KIND=nag_wp) arrayCommunication Array
Note: the dimension of the array WORK must be at least ${\mathbf{LRWORK}}$ (see D02TVF).
On entry: this must be the same array as supplied to D02TVF and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated routines.
9:     IWORK($*$) – INTEGER arrayCommunication Array
Note: the dimension of the array IWORK must be at least ${\mathbf{LIWORK}}$ (see D02TVF).
On entry: this must be the same array as supplied to D02TVF and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated routines.
10:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, because for this routine the values of the output parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: D02TKF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, an invalid call was made to D02TKF, for example, without a previous call to the setup routine D02TVF.
${\mathbf{IFAIL}}=2$
Numerical singularity has been detected in the Jacobian used in the underlying Newton iteration. No meaningful results have been computed. You should check carefully how you have coded FJAC, GAJAC and GBJAC. If the user-supplied routines have been coded correctly then supplying a different initial approximation to the solution in GUESS might be appropriate. See also Section 8.
${\mathbf{IFAIL}}=3$
The nonlinear iteration has failed to converge. At no time during the computation was convergence obtained and no meaningful results have been computed. You should check carefully how you have coded procedures FJAC, GAJAC and GBJAC. If the procedures have been coded correctly then supplying a better initial approximation to the solution in GUESS might be appropriate. See also Section 8.
${\mathbf{IFAIL}}=4$
The nonlinear iteration has failed to converge. At some earlier time during the computation convergence was obtained and the corresponding results have been returned for diagnostic purposes and may be inspected by a call to D02TZF. Nothing can be said regarding the suitability of these results for use in any subsequent computation for the same problem. You should try to provide a better mesh and initial approximation to the solution in GUESS. See also Section 8.
${\mathbf{IFAIL}}=5$
The expected number of sub-intervals required exceeds the maximum number specified by the argument MXMESH in the setup routine D02TVF. Results for the last mesh on which convergence was obtained have been returned. Nothing can be said regarding the suitability of these results for use in any subsequent computation for the same problem. An indication of the error in the solution on the last mesh where convergence was obtained can be obtained by calling D02TZF. The error requirements may need to be relaxed and/or the maximum number of mesh points may need to be increased. See also Section 8.

## 7  Accuracy

The accuracy of the solution is determined by the parameter TOLS in the prior call to D02TVF (see Sections 3 and 8 in D02TVF for details and advice). Note that error control is applied only to solution components (variables) and not to any derivatives of the solution. An estimate of the maximum error in the computed solution is available by calling D02TZF.

If D02TKF returns with ${\mathbf{IFAIL}}={\mathbf{2}}$, ${\mathbf{3}}$, ${\mathbf{4}}$ or ${\mathbf{5}}$ and the call to D02TKF was a part of some continuation procedure for which successful calls to D02TKF have already been made, then it is possible that the adjustment(s) to the continuation parameter(s) between calls to D02TKF is (are) too large for the problem under consideration. More conservative adjustment(s) to the continuation parameter(s) might be appropriate.

## 9  Example

The following example is used to illustrate the treatment of a high-order system, control of the error in a derivative of a component of the original system, and the use of continuation. See also D02TVF, D02TXF, D02TYF and D02TZF, for the illustration of other facilities.
Consider the steady flow of an incompressible viscous fluid between two infinite coaxial rotating discs. See Ascher et al. (1979) and the references therein. The governing equations are
 $1R f′′′+ff′′′+gg′ = 0 1R g′′+fg′-f′g = 0$
subject to the boundary conditions
 $f0=f′0= 0, g0=Ω0, f1=f′1= 0, g1=Ω1,$
where $R$ is the Reynolds number and ${\Omega }_{0},{\Omega }_{1}$ are the angular velocities of the disks.
We consider the case of counter-rotation and a symmetric solution, that is ${\Omega }_{0}=1,{\Omega }_{1}=-1$. This problem is more difficult to solve, the larger the value of $R$. For illustration, we use simple continuation to compute the solution for three different values of $R$ ($={10}^{6},{10}^{8},{10}^{10}$). However, this problem can be addressed directly for the largest value of $R$ considered here. Instead of the values suggested in Section 5 in D02TXF for NMESH, IPMESH and MESH in the call to D02TXF prior to a continuation call, we use every point of the final mesh for the solution of the first value of $R$, that is we must modify the contents of IPMESH. For illustrative purposes we wish to control the computed error in ${f}^{\prime }$ and so recast the equations as
 $y1′ = y2 y2′′′ = -Ry1y2′′+y3y3′ y3′′ = Ry2y3-y1y3′$
subject to the boundary conditions
 $y10=y20= 0, y30=Ω, y11=y21= 0, y31=-Ω, Ω=1.$
For the symmetric boundary conditions considered, there exists an odd solution about $x=0.5$. Hence, to satisfy the boundary conditions, we use the following initial approximations to the solution in GUESS:
 $y1x = -x2x-12 x-1 2 y2x = -xx-15⁢x2-5x+1 y3x = -8Ω x-12 3.$

### 9.1  Program Text

Program Text (d02tkfe.f90)

### 9.2  Program Data

Program Data (d02tkfe.d)

### 9.3  Program Results

Program Results (d02tkfe.r)