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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_ode_bvp_coll_nlin (d02tk)

## Purpose

nag_ode_bvp_coll_nlin (d02tk) solves a general two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations.

## Syntax

[work, iwork, ifail] = d02tk(ffun, fjac, gafun, gbfun, gajac, gbjac, guess, work, iwork)
[work, iwork, ifail] = nag_ode_bvp_coll_nlin(ffun, fjac, gafun, gbfun, gajac, gbjac, guess, work, iwork)

## Description

nag_ode_bvp_coll_nlin (d02tk) 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
 y1(m1) (x) = f1 (x,y1,y1(1), … ,y1(m1 − 1),y2, … ,yn(mn − 1)) y2(m2) (x) = f2 (x,y1,y1(1), … ,y1(m1 − 1),y2, … ,yn(mn − 1)) ⋮ yn(mn) (x) = fn (x,y1,y1(1), … ,y1(m1 − 1),y2, … ,yn(mn − 1))
$y1(m1) (x) = f1 (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1)) y2(m2) (x) = f2 (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1)) ⋮ yn(mn) (x) = fn (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1))$
over an interval [a,b]$\left[a,b\right]$ subject to p$p$ ( > 0$\text{}>0$) nonlinear boundary conditions at a$a$ and q$q$ ( > 0$\text{}>0$) nonlinear boundary conditions at b$b$, where p + q = i = 1n mi $p+q=\sum _{i=1}^{n}{m}_{i}$. Note that yi(m)(x)${y}_{i}^{\left(m\right)}\left(x\right)$ is the m$m$th derivative of the i$i$th solution component. Hence yi(0)(x) = yi(x)${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$. The left boundary conditions at a$a$ are defined as
 gi(z(y(a))) = 0,  i = 1,2, … ,p, $gi(z(y(a)))=0, i=1,2,…,p,$
and the right boundary conditions at b$b$ as
 gj(z(y(b))) = 0,  j = 1,2, … ,q, $g-j(z(y(b)))=0, j=1,2,…,q,$
where y = (y1,y2,,yn)$y=\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$ and
 z(y(x)) = (y1(x), y1(1) (x) , … , y1(m1 − 1) (x) ,y2(x), … , yn(mn − 1) (x) ) . $z(y(x)) = (y1(x), y1(1) (x) ,…, y1(m1-1) (x) ,y2(x),…, yn(mn-1) (x) ) .$
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 y1,y2,,yn${y}_{1},{y}_{2},\dots ,{y}_{n}$ 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 [Further Comments] in (d02tv).) Then, nag_ode_bvp_coll_nlin (d02tk) 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 [a,b]$\left[a,b\right]$.
A description of the numerical technique used in nag_ode_bvp_coll_nlin (d02tk) is given in Section [Description] in (d02tv).
nag_ode_bvp_coll_nlin (d02tk) 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 (d02tk) in this context.
See Section [Further Comments] in (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:     ffun – function handle or string containing name of m-file
ffun must evaluate the functions fi${f}_{i}$ for given values x,z(y(x))$x,z\left(y\left(x\right)\right)$.
[f] = ffun(x, y, neq, m)

Input Parameters

1:     x – double scalar
x$x$, the independent variable.
2:     y(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
y(i,j)${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(x)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(x) = yi(x)${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3:     neq – int64int32nag_int scalar
The number of differential equations.
4:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.

Output Parameters

1:     f(neq) – double array
The values of fi${f}_{\mathit{i}}$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
2:     fjac – function handle or string containing name of m-file
fjac must evaluate the partial derivatives of fi${f}_{i}$ with respect to the elements of
z(y(x))$z\left(y\left(x\right)\right)$ ( = (y1(x),y11(x),,y1(m11)(x),y2(x),,yn(mn1)(x))$\text{}=\left({y}_{1}\left(x\right),{y}_{1}^{1}\left(x\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(x\right),{y}_{2}\left(x\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(x\right)\right)$).
[dfdy] = fjac(x, y, neq, m)

Input Parameters

1:     x – double scalar
x$x$, the independent variable.
2:     y(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
y(i,j)${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(x)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(x) = yi(x)${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3:     neq – int64int32nag_int scalar
The number of differential equations.
4:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.

Output Parameters

1:     dfdy(neq,neq, : $:$) – double array
dfdy(i,j,k + 1)${\mathbf{dfdy}}\left(\mathit{i},\mathit{j},\mathit{k}+1\right)$ must contain the partial derivative of fi${f}_{\mathit{i}}$ with respect to yj(k)${y}_{\mathit{j}}^{\left(\mathit{k}\right)}$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$, j = 1,2,,neq$\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and k = 0,1,,m(j)1$\mathit{k}=0,1,\dots ,{\mathbf{m}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
3:     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 gi(z(y(a)))${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ for given values of z(y(a))$z\left(y\left(a\right)\right)$.
[ga] = gafun(ya, neq, m, nlbc)

Input Parameters

1:     ya(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
ya(i,j)${\mathbf{ya}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(a)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(a) = yi(a)${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2:     neq – int64int32nag_int scalar
The number of differential equations.
3:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     nlbc – int64int32nag_int scalar
The number of boundary conditions at a$a$.

Output Parameters

1:     ga(nlbc) – double array
The values of gi(z(y(a)))${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$, for i = 1,2,,nlbc$\mathit{i}=1,2,\dots ,{\mathbf{nlbc}}$.
4:     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 gi(z(y(b)))${\stackrel{-}{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ for given values of z(y(b))$z\left(y\left(b\right)\right)$.
[gb] = gbfun(yb, neq, m, nrbc)

Input Parameters

1:     yb(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
yb(i,j)${\mathbf{yb}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(b)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(b) = yi(b)${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2:     neq – int64int32nag_int scalar
The number of differential equations.
3:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     nrbc – int64int32nag_int scalar
The number of boundary conditions at b$b$.

Output Parameters

1:     gb(nrbc) – double array
The values of gi(z(y(b)))${\stackrel{-}{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$, for i = 1,2,,nrbc$\mathit{i}=1,2,\dots ,{\mathbf{nrbc}}$.
5:     gajac – function handle or string containing name of m-file
gajac must evaluate the partial derivatives of gi(z(y(a)))${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ with respect to the elements of z(y(a))$z\left(y\left(a\right)\right)$ ( = (y1(a),y11(a),,y1(m11)(a),y2(a),,yn(mn1)(a))$\text{}=\left({y}_{1}\left(a\right),{y}_{1}^{1}\left(a\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(a\right),{y}_{2}\left(a\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(a\right)\right)$).
[dgady] = gajac(ya, neq, m, nlbc)

Input Parameters

1:     ya(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
ya(i,j)${\mathbf{ya}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(a)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(a) = yi(a)${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2:     neq – int64int32nag_int scalar
The number of differential equations.
3:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     nlbc – int64int32nag_int scalar
The number of boundary conditions at a$a$.

Output Parameters

1:     dgady(nlbc,neq, : $:$) – double array
dgady(i,j,k + 1)${\mathbf{dgady}}\left(\mathit{i},\mathit{j},\mathit{k}+1\right)$ must contain the partial derivative of gi(z(y(a)))${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$ with respect to yj(k)(a)${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(a\right)$, for i = 1,2,,nlbc$\mathit{i}=1,2,\dots ,{\mathbf{nlbc}}$, j = 1,2,,neq$\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and k = 0,1,,m(j)1$\mathit{k}=0,1,\dots ,{\mathbf{m}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
6:     gbjac – function handle or string containing name of m-file
gbjac must evaluate the partial derivatives of gi(z(y(b)))${\stackrel{-}{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ with respect to the elements of z(y(b))$z\left(y\left(b\right)\right)$ ( = (y1(b),y11(b),,y1(m11)(b),y2(b),,yn(mn1)(b))$\text{}=\left({y}_{1}\left(b\right),{y}_{1}^{1}\left(b\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(b\right),{y}_{2}\left(b\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(b\right)\right)$).
[dgbdy] = gbjac(yb, neq, m, nrbc)

Input Parameters

1:     yb(neq, : $:$) – double array
The second dimension of the array must be at least maxdeg$\mathit{maxdeg}$
yb(i,j)${\mathbf{yb}}\left(\mathit{i},\mathit{j}\right)$ contains yi(j)(b)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(b) = yi(b)${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2:     neq – int64int32nag_int scalar
The number of differential equations.
3:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4:     nrbc – int64int32nag_int scalar
The number of boundary conditions at a$a$.

Output Parameters

1:     dgbdy(nrbc,neq, : $:$) – double array
dgbdy(i,j,k)${\mathbf{dgbdy}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of gi(z(y(b)))${\stackrel{-}{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$ with respect to yj(k)(b)${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(b\right)$, for i = 1,2,,nrbc$\mathit{i}=1,2,\dots ,{\mathbf{nrbc}}$, j = 1,2,,neq$\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and k = 0,1,,m(j)1$\mathit{k}=0,1,\dots ,{\mathbf{m}}\left(\mathit{j}\right)-1$. Only nonzero partial derivatives need be set.
7:     guess – function handle or string containing name of m-file
guess must return initial approximations for the solution components yi(j)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}$ and the derivatives yi(mi)${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$. Try to compute each derivative yi(mi)${y}_{i}^{\left({m}_{i}\right)}$ such that it corresponds to your approximations to yi(j) ${y}_{i}^{\left(\mathit{j}\right)}$, for j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(i\right)-1$. You should not call ffun to compute yi(mi)${y}_{i}^{\left({m}_{i}\right)}$.
If nag_ode_bvp_coll_nlin (d02tk) 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 (d02tk) after the call to nag_ode_bvp_coll_nlin_contin (d02tx).
[y, dym] = guess(x, neq, m)

Input Parameters

1:     x – double scalar
x$x$, the independent variable; x[a,b]$x\in \left[a,b\right]$.
2:     neq – int64int32nag_int scalar
The number of differential equations.
3:     m(neq) – int64int32nag_int array
mi${m}_{\mathit{i}}$, the order of the i$\mathit{i}$th differential equation, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.

Output Parameters

1:     y(neq, : $:$) – double array
The second dimension of the array will be maxdeg$\mathit{maxdeg}$
y(i,j)${\mathbf{y}}\left(\mathit{i},\mathit{j}\right)$ must contain yi(j)(x)${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and j = 0,1,,m(i)1$\mathit{j}=0,1,\dots ,{\mathbf{m}}\left(\mathit{i}\right)-1$.
Note:  yi(0)(x) = yi(x)${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
2:     dym(neq) – double array
dym(i)${\mathbf{dym}}\left(\mathit{i}\right)$ must contain yi(mi)(x)${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}\left(x\right)$, for i = 1,2,,neq$\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
8:     work( : $:$) – double array
Note: the dimension of the array work must be at least lrwork${\mathbf{lrwork}}$ (see 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:     iwork( : $:$) – int64int32nag_int array
Note: the dimension of the array iwork must be at least liwork${\mathbf{liwork}}$ (see 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.

None.

None.

### Output Parameters

1:     work( : $:$) – double array
Note: the dimension of the array work must be at least lrwork${\mathbf{lrwork}}$ (see nag_ode_bvp_coll_nlin_setup (d02tv)).
Contains information about the solution for use on subsequent calls to associated functions.
2:     iwork( : $:$) – int64int32nag_int array
Note: the dimension of the array iwork must be at least liwork${\mathbf{liwork}}$ (see nag_ode_bvp_coll_nlin_setup (d02tv)).
Contains information about the solution for use on subsequent calls to associated functions.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Note: nag_ode_bvp_coll_nlin (d02tk) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, an invalid call was made to nag_ode_bvp_coll_nlin (d02tk), for example, without a previous call to the setup function nag_ode_bvp_coll_nlin_setup (d02tv).
ifail = 2${\mathbf{ifail}}=2$
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 functions have been coded correctly then supplying a different initial approximation to the solution in guess might be appropriate. See also Section [Further Comments].
ifail = 3${\mathbf{ifail}}=3$
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 [Further Comments].
ifail = 4${\mathbf{ifail}}=4$
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 nag_ode_bvp_coll_nlin_diag (d02tz). 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 [Further Comments].
ifail = 5${\mathbf{ifail}}=5$
The expected number of sub-intervals required exceeds the maximum number specified by the argument mxmesh in the setup function nag_ode_bvp_coll_nlin_setup (d02tv). 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 nag_ode_bvp_coll_nlin_diag (d02tz). The error requirements may need to be relaxed and/or the maximum number of mesh points may need to be increased. See also Section [Further Comments].

## Accuracy

The accuracy of the solution is determined by the parameter tols in the prior call to nag_ode_bvp_coll_nlin_setup (d02tv) (see Sections [Description] and [Further Comments] in (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 (d02tk) returns with ${\mathbf{ifail}}={\mathbf{2}}$, 3${\mathbf{3}}$, 4${\mathbf{4}}$ or 5${\mathbf{5}}$ and the call to nag_ode_bvp_coll_nlin (d02tk) was a part of some continuation procedure for which successful calls to nag_ode_bvp_coll_nlin (d02tk) 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 (d02tk) 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
 1/(sqrt(R)) f ′ ′ ′ + ff ′ ′ ′ + gg′ = 0 1/(sqrt(R))g ′ ′ + fg′ − f′g = 0
$1R f′′′+ff′′′+gg′ = 0 1R g′′+fg′-f′g = 0$
subject to the boundary conditions
 f(0) = f′(0) = 0,   g(0) = Ω0,   f(1) = f′(1) = 0,   g(1) = Ω1, $f(0)=f′(0)= 0, g(0)=Ω0, f(1)=f′(1)= 0, g(1)=Ω1,$
where R$R$ is the Reynolds number and Ω0,Ω1${\Omega }_{0},{\Omega }_{1}$ are the angular velocities of the disks.
We consider the case of counter-rotation and a symmetric solution, that is Ω0 = 1, Ω1 = 1${\Omega }_{0}=1,{\Omega }_{1}=-1$. This problem is more difficult to solve, the larger the value of R$R$. For illustration, we use simple continuation to compute the solution for three different values of R$R$ ( = 106,108,1010$={10}^{6},{10}^{8},{10}^{10}$). However, this problem can be addressed directly for the largest value of R$R$ considered here. Instead of the values suggested in Section [Parameters] in (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$R$, that is we must modify the contents of ipmesh. For illustrative purposes we wish to control the computed error in f${f}^{\prime }$ and so recast the equations as
 y1 ′ = y2 y2 ′ ′ ′ = − sqrt(R)(y1y2 ′ ′ + y3y3 ′ ) y3 ′ ′ = sqrt(R)(y2y3 − y1y3 ′ )
$y1′ = y2 y2′′′ = -R(y1y2′′+y3y3′) y3′′ = R(y2y3-y1y3′)$
subject to the boundary conditions
 y1(0) = y2(0) = 0,   y3(0) = Ω,   y1(1) = y2(1) = 0,   y3(1) = − Ω,   Ω = 1. $y1(0)=y2(0)= 0, y3(0)=Ω, y1(1)=y2(1)= 0, y3(1)=-Ω, Ω=1.$
For the symmetric boundary conditions considered, there exists an odd solution about x = 0.5$x=0.5$. Hence, to satisfy the boundary conditions, we use the following initial approximations to the solution in guess:
 y1(x) = − x2(x − (1/2))(x − 1)2 y2(x) = − x(x − 1)(5x2 − 5x + 1) y3(x) = − 8Ω(x − (1/2))3.
$y1(x) = -x2(x-12) (x-1) 2 y2(x) = -x(x-1)(5⁢x2-5x+1) y3(x) = -8Ω (x-12) 3.$
```function nag_ode_bvp_coll_nlin_example
global omega sqrofr; % For communication with local functions

% Initialize variables and arrays.
neq = 3;
nlbc = 3;
nrbc = 3;
ncol = 7;
mmax = 3;
m = [1; 3; 2];
tols = [1.0e-04; 1.0e-04; 1.0e-04];

% Set values for problem-specific physical parameters.
omega = 1.0;
r = 1.0e+6;

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

% Set location of mesh boundaries, then calculate initial spacing.
mesh(1) = 0.0;
mesh(nmesh) = 1.0;
mstep = (mesh(nmesh) - mesh(1))/double(nmesh-1);
for i = 2:nmesh-1
mesh(i) = mstep*double(i-1);
ipmesh(i) = 2;
end

% Specify mesh end points as fixed.
ipmesh(1)     = 1;
ipmesh(nmesh) = 1;

fprintf('nag_ode_bvp_coll_nlin example program results\n\n');

% nag_ode_bvp_coll_nlin_setup is a setup routine to be called prior to nag_ode_bvp_coll_nlin.
[work, iwork, ifail] = nag_ode_bvp_coll_nlin_setup(int64(m), int64(nlbc), int64(nrbc), ...
int64(ncol), tols, int64(nmesh), mesh, int64(ipmesh));
if ifail ~= 0
% Unsuccessful call.  Print message and exit.
error('Warning: nag_ode_bvp_coll_nlin_setup returned with ifail = %1d ',ifail);
end

% 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 nag_ode_bvp_coll_nlin to solve BVP for this set of parameters.
[work, iwork, ifail] = nag_ode_bvp_coll_nlin(@ffun, @fjac, ...
@gafun, @gbfun, @gajac, @gbjac,...
@guess, work, iwork);
if ifail ~= 0
% Unsuccessful call.  Print message and exit.
error('Warning: nag_ode_bvp_coll_nlin returned with ifail = %1d ',ifail);
end

% Call nag_ode_bvp_coll_nlin_diag to extract mesh from solution.
[nmesh, mesh, ipmesh, ermx, iermx, ijermx, ifail] = nag_ode_bvp_coll_nlin_diag( ...
int64(mxmesh), work, iwork);

% Output mesh results.
fprintf(' Used a mesh of %d points\n', nmesh);
fprintf([' Maximum error = %10.2e in interval %d for component %d\n\n', ...
' Mesh points:\n'], ermx, iermx, ijermx);
for imesh = 1:int32(nmesh) % can't use int64 in loop range.
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:int32(nmesh) % can't use int64 in loop range.
% Call nag_ode_bvp_coll_nlin_interp to perform interpolation on the solution.
[y, work, ifail] = nag_ode_bvp_coll_nlin_interp(mesh(imesh), int64(neq), int64(mmax), ...
work, iwork);
fprintf(' %6.3f  %8.4f  %8.4f  %8.4f\n', mesh(imesh), ...
y(1,1), y(2,1), y(3,1));
xarray(imesh) = mesh(imesh);
for jcomp = 1:3
yarray(imesh, jcomp) = y(jcomp,1);
end
end

% Plot results for this parameter set.
fig = figure('Number', 'off');
display_plot(xarray, yarray, r)

% Select mesh for next calculation.
if jcont < ncont
r = r*1.0e+02;
for i = 2:int32(nmesh)-1 % can't use int64 in loop range.
ipmesh(i) = 2;
end

% nag_ode_bvp_coll_nlin_contin allows the current solution to be used as an initial
% approximation to the solution of a related problem.
[work, iwork, ifail] = nag_ode_bvp_coll_nlin_contin(nmesh, mesh, ipmesh, work, iwork);
end
end

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

global omega sqrofr; % For communication with main routine.
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] = gafun(ya, neq, m, nlbc)
% Evaluate boundary conditions at left-hand end of range.

global omega sqrofr; % For communication with main routine.
ga = zeros(nlbc, 1);
ga(1) = ya(1);
ga(2) = ya(2);
ga(3) = ya(3) - omega;
function [dgady] = gajac(ya, neq, m, nlbc)
% Evaluate Jacobians (partial derivatives of ga).

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

global omega sqrofr; % For communication with main routine.
gb = zeros(nrbc, 1);
gb(1) = yb(1);
gb(2) = yb(2);
gb(3) = yb(3) + omega;
function [dgbdy] = gbjac(yb, neq, m, nrbc)
% Evaluate Jacobians (partial derivatives of gb).

dgbdy = zeros(nrbc, neq, 1);
dgbdy(1,1,1) = 1.0;
dgbdy(2,2,1) = 1.0;
dgbdy(3,3,1) = 1.0;
function [y, dym] = guess(x, neq, m)
% Evaluate initial approximations to solution components and derivatives.

global omega sqrofr; % For communication with main routine.
y = zeros(neq, 3);
dym = zeros(neq, 1);
y(1,1) = -x^2*(x - 0.5)*(x - 1.0)^2;
y(2,1) = -x*(x - 1.0)*(5.0*x^2 - 5.0*x + 1.0);
y(3,1) = -8.0*omega*(x - 0.5)^3;
y(2,2) = -(20.0*x^3 - 30.0*x^2 + 12.0*x - 1.0);
y(2,3) = -(60.0*x^2 - 60.0*x + 12.0*x);
y(3,2) = -24.0*omega*(x - 0.5)^2;

dym(1) = y(2,1);
dym(2) = -(120.0*x - 60.0);
dym(3) = -56.0*omega*(x - 0.5);
function display_plot(x, y, r)
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% 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(1), 'XMinorTick', 'on', 'YMinorTick', 'on');
set(haxes(1), 'YTick', [-0.1 0.0 0.1 0.2 0.3 0.4]);
set(haxes(2), 'YLim', [-1 1]);
set(haxes(2), 'YMinorTick', 'on');
set(haxes(2), 'YTick', [-1 -0.5 0 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 0.4 0.6 0.8 1]);
set(haxes(iaxis), 'FontSize', 13);
end
set(gca, 'box', 'off'); % so ticks aren't shown on opposite axes.
title(['Incompressible Fluid Flow between Discs. ', ...
'Solutions for Re = ', num2str(r)], titleFmt{:});
% Label the axes.
xlabel('x', labFmt{:});
ylabel(haxes(1), 'f and f''', labFmt{:});
ylabel(haxes(2), 'g', labFmt{:});
legend('f''','f','g','Location','Best')
% Set some features of the three lines.
set(hline1, 'Linewidth', 0.25, 'Marker', '+', 'Line', '-');
set(hline2, 'Linewidth', 0.25, 'Marker', 'x', 'Line', '--');
set(hline3, 'Linewidth', 0.25, 'Marker', '*', 'Line', ':');
```
```
nag_ode_bvp_coll_nlin example program results

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

Used a mesh of 21 points
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

Used a mesh of 21 points
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

Used a mesh of 21 points
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

```
```function d02tk_example
global omega sqrofr; % For communication with local functions

% Initialize variables and arrays.
neq = 3;
nlbc = 3;
nrbc = 3;
ncol = 7;
mmax = 3;
m = [1; 3; 2];
tols = [1.0e-04; 1.0e-04; 1.0e-04];

% Set values for problem-specific physical parameters.
omega = 1.0;
r = 1.0e+6;

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

% Set location of mesh boundaries, then calculate initial spacing.
mesh(1) = 0.0;
mesh(nmesh) = 1.0;
mstep = (mesh(nmesh) - mesh(1))/double(nmesh-1);
for i = 2:nmesh-1
mesh(i) = mstep*double(i-1);
ipmesh(i) = 2;
end

% Specify mesh end points as fixed.
ipmesh(1)     = 1;
ipmesh(nmesh) = 1;

fprintf('d02tk example program results\n\n');

% d02tv is a setup routine to be called prior to d02tk.
[work, iwork, ifail] = d02tv(int64(m), int64(nlbc), int64(nrbc), ...
int64(ncol), tols, int64(nmesh), mesh, int64(ipmesh));
if ifail ~= 0
% Unsuccessful call.  Print message and exit.
error('Warning: d02tv returned with ifail = %1d ',ifail);
end

% 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.
[work, iwork, ifail] = d02tk(@ffun, @fjac, ...
@gafun, @gbfun, @gajac, @gbjac,...
@guess, work, iwork);
if ifail ~= 0
% Unsuccessful call.  Print message and exit.
error('Warning: d02tk returned with ifail = %1d ',ifail);
end

% Call d02tz to extract mesh from solution.
[nmesh, mesh, ipmesh, ermx, iermx, ijermx, ifail] = d02tz( ...
int64(mxmesh), work, iwork);

% Output mesh results.
fprintf(' Used a mesh of %d points\n', nmesh);
fprintf([' Maximum error = %10.2e in interval %d for component %d\n\n', ...
' Mesh points:\n'], ermx, iermx, ijermx);
for imesh = 1:int32(nmesh) % can't use int64 in loop range.
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:int32(nmesh) % can't use int64 in loop range.
% Call d02ty to perform interpolation on the solution.
[y, work, ifail] = d02ty(mesh(imesh), int64(neq), int64(mmax), ...
work, iwork);
fprintf(' %6.3f  %8.4f  %8.4f  %8.4f\n', mesh(imesh), ...
y(1,1), y(2,1), y(3,1));
xarray(imesh) = mesh(imesh);
for jcomp = 1:3
yarray(imesh, jcomp) = y(jcomp,1);
end
end

% Plot results for this parameter set.
fig = figure('Number', 'off');
display_plot(xarray, yarray, r)

% Select mesh for next calculation.
if jcont < ncont
r = r*1.0e+02;
for i = 2:int32(nmesh)-1 % can't use int64 in loop range.
ipmesh(i) = 2;
end

% d02tx allows the current solution to be used as an initial
% approximation to the solution of a related problem.
[work, iwork, ifail] = d02tx(nmesh, mesh, ipmesh, work, iwork);
end
end

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

global omega sqrofr; % For communication with main routine.
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] = gafun(ya, neq, m, nlbc)
% Evaluate boundary conditions at left-hand end of range.

global omega sqrofr; % For communication with main routine.
ga = zeros(nlbc, 1);
ga(1) = ya(1);
ga(2) = ya(2);
ga(3) = ya(3) - omega;
function [dgady] = gajac(ya, neq, m, nlbc)
% Evaluate Jacobians (partial derivatives of ga).

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

global omega sqrofr; % For communication with main routine.
gb = zeros(nrbc, 1);
gb(1) = yb(1);
gb(2) = yb(2);
gb(3) = yb(3) + omega;
function [dgbdy] = gbjac(yb, neq, m, nrbc)
% Evaluate Jacobians (partial derivatives of gb).

dgbdy = zeros(nrbc, neq, 1);
dgbdy(1,1,1) = 1.0;
dgbdy(2,2,1) = 1.0;
dgbdy(3,3,1) = 1.0;
function [y, dym] = guess(x, neq, m)
% Evaluate initial approximations to solution components and derivatives.

global omega sqrofr; % For communication with main routine.
y = zeros(neq, 3);
dym = zeros(neq, 1);
y(1,1) = -x^2*(x - 0.5)*(x - 1.0)^2;
y(2,1) = -x*(x - 1.0)*(5.0*x^2 - 5.0*x + 1.0);
y(3,1) = -8.0*omega*(x - 0.5)^3;
y(2,2) = -(20.0*x^3 - 30.0*x^2 + 12.0*x - 1.0);
y(2,3) = -(60.0*x^2 - 60.0*x + 12.0*x);
y(3,2) = -24.0*omega*(x - 0.5)^2;

dym(1) = y(2,1);
dym(2) = -(120.0*x - 60.0);
dym(3) = -56.0*omega*(x - 0.5);
function display_plot(x, y, r)
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% 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(1), 'XMinorTick', 'on', 'YMinorTick', 'on');
set(haxes(1), 'YTick', [-0.1 0.0 0.1 0.2 0.3 0.4]);
set(haxes(2), 'YLim', [-1 1]);
set(haxes(2), 'YMinorTick', 'on');
set(haxes(2), 'YTick', [-1 -0.5 0 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 0.4 0.6 0.8 1]);
set(haxes(iaxis), 'FontSize', 13);
end
set(gca, 'box', 'off'); % so ticks aren't shown on opposite axes.
title(['Incompressible Fluid Flow between Discs. ', ...
'Solutions for Re = ', num2str(r)], titleFmt{:});
% Label the axes.
xlabel('x', labFmt{:});
ylabel(haxes(1), 'f and f''', labFmt{:});
ylabel(haxes(2), 'g', labFmt{:});
legend('f''','f','g','Location','Best')
% Set some features of the three lines.
set(hline1, 'Linewidth', 0.25, 'Marker', '+', 'Line', '-');
set(hline2, 'Linewidth', 0.25, 'Marker', 'x', 'Line', '--');
set(hline3, 'Linewidth', 0.25, 'Marker', '*', 'Line', ':');
```
```
d02tk example program results

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

Used a mesh of 21 points
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

Used a mesh of 21 points
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

Used a mesh of 21 points
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

```