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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_ode_bvp_coll_nlin_solve (d02tl)

## Purpose

nag_ode_bvp_coll_nlin_solve (d02tl) solves a general two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations.

## Syntax

[rcomm, icomm, user, ifail] = d02tl(ffun, fjac, gafun, gbfun, gajac, gbjac, guess, rcomm, icomm, 'user', user)
[rcomm, icomm, user, ifail] = nag_ode_bvp_coll_nlin_solve(ffun, fjac, gafun, gbfun, gajac, gbjac, guess, rcomm, icomm, 'user', user)

## Description

nag_ode_bvp_coll_nlin_solve (d02tl) and its associated functions (nag_ode_bvp_coll_nlin_setup (d02tv), nag_ode_bvp_coll_nlin_contin (d02tx), nag_ode_bvp_coll_nlin_interp (d02ty) and nag_ode_bvp_coll_nlin_diag (d02tz)) 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, nag_ode_bvp_coll_nlin_setup (d02tv) 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 Further Comments in nag_ode_bvp_coll_nlin_setup (d02tv).) Then, nag_ode_bvp_coll_nlin_solve (d02tl) can be used to solve the boundary value problem. After successful computation, nag_ode_bvp_coll_nlin_diag (d02tz) can be used to ascertain details about the final mesh and other details of the solution procedure, and nag_ode_bvp_coll_nlin_interp (d02ty) can be used to compute the approximate solution anywhere on the interval $\left[a,b\right]$.
A description of the numerical technique used in nag_ode_bvp_coll_nlin_solve (d02tl) is given in Description in nag_ode_bvp_coll_nlin_setup (d02tv).
nag_ode_bvp_coll_nlin_solve (d02tl) 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. nag_ode_bvp_coll_nlin_contin (d02tx) should be used in between calls to nag_ode_bvp_coll_nlin_solve (d02tl) in this context.
See Further Comments in nag_ode_bvp_coll_nlin_setup (d02tv) for details of how to solve boundary value problems of a more general nature.
The functions 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).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ffun}$ – function handle or string containing name of m-file
ffun must evaluate the functions ${f}_{i}$ for given values $x,z\left(y\left(x\right)\right)$.
[f, user] = ffun(x, y, neq, m, user)

Input Parameters

1:     $\mathrm{x}$ – double scalar
$x$, the independent variable.
2:     $\mathrm{y}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
4:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
5:     $\mathrm{user}$ – Any MATLAB object
ffun is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{f}\left({\mathbf{neq}}\right)$ – double array
${\mathbf{f}}\left(\mathit{i}\right)$ must contain ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
2:     $\mathrm{user}$ – Any MATLAB object
2:     $\mathrm{fjac}$ – function handle or string containing name of m-file
fjac must evaluate the partial derivatives of ${f}_{i}$ with respect to the elements of
$z\left(y\left(x\right)\right)=\left({y}_{1}\left(x\right),{y}_{1}^{1}\left(x\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(x\right),{y}_{2}\left(x\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(x\right)\right)$.
[dfdy, user] = fjac(x, y, neq, m, dfdy, user)

Input Parameters

1:     $\mathrm{x}$ – double scalar
$x$, the independent variable.
2:     $\mathrm{y}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
4:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
5:     $\mathrm{dfdy}\left({\mathbf{neq}},{\mathbf{neq}},0:*\right)$ – double array
Set to zero.
6:     $\mathrm{user}$ – Any MATLAB object
fjac is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{dfdy}\left({\mathbf{neq}},{\mathbf{neq}},0:*\right)$ – double array
${\mathbf{dfdy}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ 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.
2:     $\mathrm{user}$ – Any MATLAB object
3:     $\mathrm{gafun}$ – function handle or string containing name of m-file
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 $z\left(y\left(a\right)\right)$.
[ga, user] = gafun(ya, neq, m, nlbc, user)

Input Parameters

1:     $\mathrm{ya}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{ya}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
3:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     $\mathrm{nlbc}$int64int32nag_int scalar
The number of boundary conditions at $a$.
5:     $\mathrm{user}$ – Any MATLAB object
gafun is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{ga}\left({\mathbf{nlbc}}\right)$ – double array
${\mathbf{ga}}\left(\mathit{i}\right)$ must contain ${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nlbc}}$.
2:     $\mathrm{user}$ – Any MATLAB object
4:     $\mathrm{gbfun}$ – function handle or string containing name of m-file
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 $z\left(y\left(b\right)\right)$.
[gb, user] = gbfun(yb, neq, m, nrbc, user)

Input Parameters

1:     $\mathrm{yb}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{yb}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
3:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     $\mathrm{nrbc}$int64int32nag_int scalar
The number of boundary conditions at $b$.
5:     $\mathrm{user}$ – Any MATLAB object
gbfun is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{gb}\left({\mathbf{nrbc}}\right)$ – double array
${\mathbf{gb}}\left(\mathit{i}\right)$ must contain ${\stackrel{-}{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nrbc}}$.
2:     $\mathrm{user}$ – Any MATLAB object
5:     $\mathrm{gajac}$ – function handle or string containing name of m-file
gajac must evaluate the partial derivatives of ${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ with respect to the elements of $z\left(y\left(a\right)\right)=\left({y}_{1}\left(a\right),{y}_{1}^{1}\left(a\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(a\right),{y}_{2}\left(a\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(a\right)\right)$.
[dgady, user] = gajac(ya, neq, m, nlbc, dgady, user)

Input Parameters

1:     $\mathrm{ya}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{ya}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
3:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     $\mathrm{nlbc}$int64int32nag_int scalar
The number of boundary conditions at $a$.
5:     $\mathrm{dgady}\left({\mathbf{nlbc}},{\mathbf{neq}},0:*\right)$ – double array
Set to zero.
6:     $\mathrm{user}$ – Any MATLAB object
gajac is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{dgady}\left({\mathbf{nlbc}},{\mathbf{neq}},0:*\right)$ – double array
${\mathbf{dgady}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ 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.
2:     $\mathrm{user}$ – Any MATLAB object
6:     $\mathrm{gbjac}$ – function handle or string containing name of m-file
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 $z\left(y\left(b\right)\right)=\left({y}_{1}\left(b\right),{y}_{1}^{1}\left(b\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(b\right),{y}_{2}\left(b\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(b\right)\right)$.
[dgbdy, user] = gbjac(yb, neq, m, nrbc, dgbdy, user)

Input Parameters

1:     $\mathrm{yb}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{yb}}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
3:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     $\mathrm{nrbc}$int64int32nag_int scalar
The number of boundary conditions at $b$.
5:     $\mathrm{dgbdy}\left({\mathbf{nrbc}},{\mathbf{neq}},0:*\right)$ – double array
Set to zero.
6:     $\mathrm{user}$ – Any MATLAB object
gbjac is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{dgbdy}\left({\mathbf{nrbc}},{\mathbf{neq}},0:*\right)$ – double array
${\mathbf{dgbdy}}\left(\mathit{i},\mathit{j},\mathit{k}+1\right)$ 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.
2:     $\mathrm{user}$ – Any MATLAB object
7:     $\mathrm{guess}$ – function handle or string containing name of m-file
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 nag_ode_bvp_coll_nlin_solve (d02tl) is being used in conjunction with nag_ode_bvp_coll_nlin_contin (d02tx) as part of a continuation process, then guess is not called by nag_ode_bvp_coll_nlin_solve (d02tl) after the call to nag_ode_bvp_coll_nlin_contin (d02tx).
[y, dym, user] = guess(x, neq, m, user)

Input Parameters

1:     $\mathrm{x}$ – double scalar
$x$, the independent variable; $x\in \left[a,b\right]$.
2:     $\mathrm{neq}$int64int32nag_int scalar
The number of differential equations.
3:     $\mathrm{m}\left({\mathbf{neq}}\right)$int64int32nag_int array
${\mathbf{m}}\left(\mathit{i}\right)$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     $\mathrm{user}$ – Any MATLAB object
guess is called from nag_ode_bvp_coll_nlin_solve (d02tl) with the object supplied to nag_ode_bvp_coll_nlin_solve (d02tl).

Output Parameters

1:     $\mathrm{y}\left({\mathbf{neq}},0:*\right)$ – double array
${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ 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:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
2:     $\mathrm{dym}\left({\mathbf{neq}}\right)$ – double array
${\mathbf{dym}}\left(\mathit{i}\right)$ must contain ${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
3:     $\mathrm{user}$ – Any MATLAB object
8:     $\mathrm{rcomm}\left(*\right)$ – double array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument rcomm in the previous call to nag_ode_bvp_coll_nlin_setup (d02tv).
This must be the same array as supplied to nag_ode_bvp_coll_nlin_setup (d02tv) and must remain unchanged between calls.
9:     $\mathrm{icomm}\left(*\right)$int64int32nag_int array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument icomm in the previous call to nag_ode_bvp_coll_nlin_setup (d02tv).
This must be the same array as supplied to nag_ode_bvp_coll_nlin_setup (d02tv) and must remain unchanged between calls.

### Optional Input Parameters

1:     $\mathrm{user}$ – Any MATLAB object
user is not used by nag_ode_bvp_coll_nlin_solve (d02tl), but is passed to ffun, fjac, gafun, gbfun, gajac, gbjac and guess. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

### Output Parameters

1:     $\mathrm{rcomm}\left(*\right)$ – double array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument rcomm in the previous call to nag_ode_bvp_coll_nlin_setup (d02tv).
Contains information about the solution for use on subsequent calls to associated functions.
2:     $\mathrm{icomm}\left(*\right)$int64int32nag_int array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument icomm in the previous call to nag_ode_bvp_coll_nlin_setup (d02tv).
Contains information about the solution for use on subsequent calls to associated functions.
3:     $\mathrm{user}$ – Any MATLAB object
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_ode_bvp_coll_nlin_solve (d02tl) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Either the setup function has not been called or the communication arrays have become corrupted. No solution will be computed.
${\mathbf{ifail}}=2$
Numerical singularity has been detected in the Jacobian used in the Newton iteration.
No results have been generated. Check the coding of the functions for calculating the Jacobians of system and boundary conditions.
${\mathbf{ifail}}=3$
All Newton iterations that have been attempted have failed to converge.
No results have been generated. Check the coding of the functions for calculating the Jacobians of system and boundary conditions.
Try to provide a better initial solution approximation.
${\mathbf{ifail}}=4$
A Newton iteration has failed to converge. The computation has not succeeded but results have been returned for an intermediate mesh on which convergence was achieved.
These results should be treated with extreme caution.
${\mathbf{ifail}}=5$
The expected number of sub-intervals required to continue the computation exceeds the maximum specified.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy of the solution is determined by the argument tols in the prior call to nag_ode_bvp_coll_nlin_setup (d02tv) (see Description and Further Comments in nag_ode_bvp_coll_nlin_setup (d02tv) 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 nag_ode_bvp_coll_nlin_diag (d02tz).

If nag_ode_bvp_coll_nlin_solve (d02tl) returns with ${\mathbf{ifail}}={\mathbf{2}}$, ${\mathbf{3}}$, ${\mathbf{4}}$ or ${\mathbf{5}}$ and the call to nag_ode_bvp_coll_nlin_solve (d02tl) was a part of some continuation procedure for which successful calls to nag_ode_bvp_coll_nlin_solve (d02tl) have already been made, then it is possible that the adjustment(s) to the continuation parameter(s) between calls to nag_ode_bvp_coll_nlin_solve (d02tl) is (are) too large for the problem under consideration. More conservative adjustment(s) to the continuation parameter(s) might be appropriate.

## 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 nag_ode_bvp_coll_nlin_setup (d02tv), nag_ode_bvp_coll_nlin_contin (d02tx), nag_ode_bvp_coll_nlin_interp (d02ty) and nag_ode_bvp_coll_nlin_diag (d02tz), 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 Arguments in nag_ode_bvp_coll_nlin_contin (d02tx) for nmesh, ipmesh and mesh in the call to nag_ode_bvp_coll_nlin_contin (d02tx) 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:
 $y1x = -x2x-12 x-1 2 y2x = -xx-15⁢x2-5x+1 y3x = -8Ω x-12 3.$
```function d02tl_example

fprintf('d02tl example results\n\n');

global omega sqrofr

% Initialize variables and arrays.
neq  = int64(3);
nlbc = neq;
nrbc = neq;
ncol = int64(7);
mmax = int64(3);
m    = int64([1; 3; 2]);
tols = [1; 1; 1]/10^4;

% Set values for problem-specific physical parameters.
omega = 1;
r     = 10^6;

% Set up the mesh.
nmesh  = int64(11);
mxmesh = int64(51);
ipmesh = zeros(mxmesh, 1, 'int64');
mesh   = zeros(mxmesh, 1);

mesh(1:nmesh)     = [0:0.1:1];
ipmesh(1)         = 1;
ipmesh(2:nmesh-1) = 2;
ipmesh(nmesh)     = 1;

% d02tv is a setup routine to be called prior to d02tl.
[rcomm, icomm, ifail] = d02tv(...
m, nlbc, nrbc, ncol, tols, nmesh, mesh, ipmesh);

% Set number of continuation steps.
ncont = 3;

% We run through the calculation three times with different parameter sets.
for jcont = 1:ncont
sqrofr = sqrt(r);
fprintf('\n Tolerance = %8.1e  R = %10.3e\n\n', tols(1), r);

% Call d02tk to solve BVP for this set of parameters.
[rcomm, icomm, ifail] = ...
d02tl(...
@ffun, @fjac, @gafun, @gbfun, @gajac, @gbjac, @guess, rcomm, icomm);

% Call d02tz to extract mesh from solution.
[nmesh, mesh, ipmesh, ermx, iermx, ijermx, ifail] = ...
d02tz( ...
mxmesh, rcomm, icomm);

% Output mesh results.
fprintf(' Mesh size = %d\n', nmesh);
fprintf(' Maximum error = %10.2e in interval %d for component %d\n\n', ...
ermx, iermx, ijermx);
fprintf('Mesh points:\n');
for imesh = 1:nmesh
fprintf( '%4d(%d) %6.4f', imesh, ipmesh(imesh), mesh(imesh));
if mod(imesh, 4) == 0
fprintf('\n');
end
end

% Output solution, and store it for plotting.
xarray = zeros(nmesh, 1);
yarray = zeros(nmesh, 3);
fprintf('\n\n    x        f         f''        g\n');
for imesh = 1:nmesh
% Call d02ty to perform interpolation on the solution.
[y, rcomm, ifail] = d02ty(...
mesh(imesh), neq, mmax, rcomm, icomm);
fprintf(' %6.3f  %8.4f  %8.4f  %8.4f\n', mesh(imesh), y(:,1));
xarray(imesh) = mesh(imesh);
yarray(imesh, 1:3) = y(1:3,1);
end

% Plot results for this parameter set.
if jcont==1
fig1 = figure;
else
if jcont==2
fig2 = figure;
else
fig3 = figure;
end
end
display_plot(xarray, yarray, r);

% Select mesh for next calculation.
if jcont < ncont
r = 100*r;
ipmesh(2:nmesh-1) = 2;

% d02tx allows the current solution to be used for continuation
[rcomm, icomm, ifail] = d02tx(...
nmesh, mesh, ipmesh, rcomm, icomm);
end
end

function [f, user] = ffun(x, y, neq, m, user)
% Evaluate derivative functions (rhs of system of ODEs).

global omega sqrofr; % For communication with main routine.
f = zeros(neq, 1);
f(1,1) =   y(2,1);
f(2,1) = -(y(1,1)*y(2,3) + y(3,1)*y(3,2))*sqrofr;
f(3,1) =  (y(2,1)*y(3,1) - y(1,1)*y(3,2))*sqrofr;

function [dfdy, user] = fjac(x, y, neq, m, dfdy, user)
% Evaluate Jacobians (partial derivatives of f).

global omega sqrofr
dfdy = zeros(neq, neq, 3);
dfdy(1,2,1) =  1.0;
dfdy(2,1,1) = -y(2,3)*sqrofr;
dfdy(2,2,3) = -y(1,1)*sqrofr;
dfdy(2,3,1) = -y(3,2)*sqrofr;
dfdy(2,3,2) = -y(3,1)*sqrofr;
dfdy(3,1,1) = -y(3,2)*sqrofr;
dfdy(3,2,1) =  y(3,1)*sqrofr;
dfdy(3,3,1) =  y(2,1)*sqrofr;
dfdy(3,3,2) = -y(1,1)*sqrofr;

function [ga, user] = gafun(ya, neq, m, nlbc, user)
% Evaluate boundary conditions at left-hand end of range.

global omega sqrofr
ga = zeros(nlbc, 1);
ga(1) = ya(1);
ga(2) = ya(2);
ga(3) = ya(3) - omega;

function [dgady, user] = gajac(ya, neq, m, nlbc, dgady, user)
% Evaluate Jacobians (partial derivatives of ga).

dgady = zeros(nlbc, neq, 3);

function [gb, user] = gbfun(yb, neq, m, nrbc, user)
% Evaluate boundary conditions at right-hand end of range.

global omega sqrofr
gb = zeros(nrbc, 1);
gb(1) = yb(1);
gb(2) = yb(2);
gb(3) = yb(3) + omega;

function [dgbdy, user] = gbjac(yb, neq, m, nrbc, dgbdy, user)
% Evaluate Jacobians (partial derivatives of gb).

dgbdy = zeros(nrbc, neq, 3);
dgbdy(1,1,1) = 1;
dgbdy(2,2,1) = 1;
dgbdy(3,3,1) = 1;

function [y, dym, user] = guess(x, neq, m, user)
% Evaluate initial approximations to solution components and derivatives.

global omega sqrofr
y = zeros(neq, 3);
dym = zeros(neq, 1);
y(1,1) = -x^2*(x - 0.5)*(x - 1)^2;
y(2,1) = -x*(x - 1)*(5*x^2 - 5*x + 1);
y(3,1) = -8*omega*(x - 0.5)^3;
y(2,2) = -(20*x^3 - 30*x^2 + 12*x - 1);
y(2,3) = -(60*x^2 - 60*x + 12*x);
y(3,2) = -24*omega*(x - 0.5)^2;

dym(1) = y(2,1);
dym(2) = -(120*x - 60);
dym(3) = -56*omega*(x - 0.5);

function display_plot(x, y, r)
% Plot two of the curves, then add the other one.
[haxes, hline1, hline2] = plotyy(x, y(:,2), x, y(:,3));
% We want the third curve to be plotted on the left-hand y-axis.
hold(haxes(1), 'on');
hline3 = plot(x, y(:,1));
% Set the axis limits and the tick specifications to beautify the plot.
set(haxes(1), 'YLim', [-0.1 0.4]);
set(haxes(2), 'YLim', [-1 1]);
set(haxes(1), 'XMinorTick', 'on', 'YMinorTick', 'on');
set(haxes(2), 'YMinorTick', 'on');
set(haxes(1), 'YTick', [-0.1:0.1:0.4]);
set(haxes(2), 'YTick', [-1:0.5:1]);
for iaxis = 1:2
% These properties must be the same for both sets of axes.
set(haxes(iaxis), 'XLim', [0 1]);
set(haxes(iaxis), 'XTick', [0:0.2:1]);
end
set(gca, 'box', 'off');
ord = log10(r);
title(['Flow between Discs, Re = 10^n, n = ', num2str(ord)]);
% Label the axes.
xlabel('x');
ylabel(haxes(1), 'f and f''');
ylabel(haxes(2), 'g');
% Add a legend.
legend('f''','f','g','Location','Best')
% Set some features of the three lines.
set(hline1, 'Linewidth', 0.25, 'Marker', '+', 'LineStyle', '-', ...
'Color', 'Magenta');
set(hline2, 'Linewidth', 0.25, 'Marker', 'x', 'LineStyle', '--');
set(hline3, 'Linewidth', 0.25, 'Marker', '*', 'LineStyle', ':');
```
```d02tl example results

Tolerance =  1.0e-04  R =  1.000e+06

Mesh size = 21
Maximum error =   6.16e-10 in interval 20 for component 3

Mesh points:
1(1) 0.0000   2(3) 0.0500   3(2) 0.1000   4(3) 0.1500
5(2) 0.2000   6(3) 0.2500   7(2) 0.3000   8(3) 0.3500
9(2) 0.4000  10(3) 0.4500  11(2) 0.5000  12(3) 0.5500
13(2) 0.6000  14(3) 0.6500  15(2) 0.7000  16(3) 0.7500
17(2) 0.8000  18(3) 0.8500  19(2) 0.9000  20(3) 0.9500
21(1) 1.0000

x        f         f'        g
0.000    0.0000    0.0000    1.0000
0.050    0.0070    0.1805    0.4416
0.100    0.0141    0.0977    0.1886
0.150    0.0171    0.0252    0.0952
0.200    0.0172   -0.0165    0.0595
0.250    0.0157   -0.0400    0.0427
0.300    0.0133   -0.0540    0.0322
0.350    0.0104   -0.0628    0.0236
0.400    0.0071   -0.0683    0.0156
0.450    0.0036   -0.0714    0.0078
0.500    0.0000   -0.0724    0.0000
0.550   -0.0036   -0.0714   -0.0078
0.600   -0.0071   -0.0683   -0.0156
0.650   -0.0104   -0.0628   -0.0236
0.700   -0.0133   -0.0540   -0.0322
0.750   -0.0157   -0.0400   -0.0427
0.800   -0.0172   -0.0165   -0.0595
0.850   -0.0171    0.0252   -0.0952
0.900   -0.0141    0.0977   -0.1886
0.950   -0.0070    0.1805   -0.4416
1.000   -0.0000   -0.0000   -1.0000

Tolerance =  1.0e-04  R =  1.000e+08

Mesh size = 21
Maximum error =   4.49e-09 in interval 6 for component 3

Mesh points:
1(1) 0.0000   2(3) 0.0176   3(2) 0.0351   4(3) 0.0520
5(2) 0.0689   6(3) 0.0859   7(2) 0.1030   8(3) 0.1351
9(2) 0.1672  10(3) 0.2306  11(2) 0.2939  12(3) 0.4713
13(2) 0.6486  14(3) 0.7455  15(2) 0.8423  16(3) 0.8824
17(2) 0.9225  18(3) 0.9449  19(2) 0.9673  20(3) 0.9836
21(1) 1.0000

x        f         f'        g
0.000    0.0000    0.0000    1.0000
0.018    0.0025    0.1713    0.3923
0.035    0.0047    0.0824    0.1381
0.052    0.0056    0.0267    0.0521
0.069    0.0058    0.0025    0.0213
0.086    0.0057   -0.0073    0.0097
0.103    0.0056   -0.0113    0.0053
0.135    0.0052   -0.0135    0.0027
0.167    0.0047   -0.0140    0.0020
0.231    0.0038   -0.0142    0.0015
0.294    0.0029   -0.0142    0.0012
0.471    0.0004   -0.0143    0.0002
0.649   -0.0021   -0.0143   -0.0008
0.745   -0.0035   -0.0142   -0.0014
0.842   -0.0049   -0.0139   -0.0022
0.882   -0.0054   -0.0127   -0.0036
0.922   -0.0058   -0.0036   -0.0141
0.945   -0.0057    0.0205   -0.0439
0.967   -0.0045    0.0937   -0.1592
0.984   -0.0023    0.1753   -0.4208
1.000   -0.0000    0.0000   -1.0000

Tolerance =  1.0e-04  R =  1.000e+10

Mesh size = 21
Maximum error =   3.13e-06 in interval 7 for component 3

Mesh points:
1(1) 0.0000   2(3) 0.0063   3(2) 0.0125   4(3) 0.0185
5(2) 0.0245   6(3) 0.0308   7(2) 0.0370   8(3) 0.0500
9(2) 0.0629  10(3) 0.0942  11(2) 0.1256  12(3) 0.4190
13(2) 0.7125  14(3) 0.8246  15(2) 0.9368  16(3) 0.9544
17(2) 0.9719  18(3) 0.9803  19(2) 0.9886  20(3) 0.9943
21(1) 1.0000

x        f         f'        g
0.000    0.0000    0.0000    1.0000
0.006    0.0009    0.1623    0.3422
0.013    0.0016    0.0665    0.1021
0.019    0.0018    0.0204    0.0318
0.025    0.0019    0.0041    0.0099
0.031    0.0019   -0.0014    0.0028
0.037    0.0019   -0.0031    0.0007
0.050    0.0019   -0.0038   -0.0002
0.063    0.0018   -0.0038   -0.0003
0.094    0.0017   -0.0039   -0.0003
0.126    0.0016   -0.0039   -0.0002
0.419    0.0004   -0.0041   -0.0001
0.712   -0.0008   -0.0042    0.0001
0.825   -0.0013   -0.0043    0.0002
0.937   -0.0018   -0.0043    0.0003
0.954   -0.0019   -0.0042    0.0001
0.972   -0.0019   -0.0003   -0.0049
0.980   -0.0019    0.0152   -0.0252
0.989   -0.0015    0.0809   -0.1279
0.994   -0.0008    0.1699   -0.3814
1.000    0.0000    0.0000   -1.0000
```

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