NAG Toolbox: nag_ode_bvp_coll_nlin_solve (d02tl)

Purpose

Syntax

Description

References

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

   $\mathbf{ifail}=1$

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

Further Comments

Example

Open in the MATLAB editor: d02tl_example

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);
  dgady(1,1,1) = 1;
  dgady(2,2,1) = 1;
  dgady(3,3,1) = 1;

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');
  % Add title.
  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