NAG Toolbox: nag_pde_1d_parab_fd (d03pc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

The coordinate system used:

$\mathbf{m}=0$

Indicates Cartesian coordinates.

$\mathbf{m}=1$

Indicates cylindrical polar coordinates.

$\mathbf{m}=2$

Indicates spherical polar coordinates.

Constraint: $\mathbf{m}=0$, $1$ or $2$.

2:     $\mathrm{ts}$ – double scalar

The initial value of the independent variable $t$.

Constraint: $\mathbf{ts}<\mathbf{tout}$.

3:     $\mathrm{tout}$ – double scalar

The final value of $t$ to which the integration is to be carried out.

4:     $\mathrm{pdedef}$ – function handle or string containing name of m-file

pdedef must compute the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_fd (d03pc).

[p, q, r, ires, user] = pdedef(npde, t, x, u, ux, ires, user)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

3:     $\mathrm{x}$ – double scalar

The current value of the space variable $x$.

4:     $\mathrm{u}\left(\mathbf{npde}\right)$ – double array

$\mathbf{u}\left(\mathit{i}\right)$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:     $\mathrm{ux}\left(\mathbf{npde}\right)$ – double array

$\mathbf{ux}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

6:     $\mathrm{ires}$ – int64int32nag_int scalar

Set to $-1\text{ or }1$.

7:     $\mathrm{user}$ – Any MATLAB object

pdedef is called from nag_pde_1d_parab_fd (d03pc) with the object supplied to nag_pde_1d_parab_fd (d03pc).

Output Parameters

1:     $\mathrm{p}\left(\mathbf{npde},\mathbf{npde}\right)$ – double array

$\mathbf{p}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $P_{\mathit{i},\mathit{j}}\left(x,t,U,U_x\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

2:     $\mathrm{q}\left(\mathbf{npde}\right)$ – double array

$\mathbf{q}\left(\mathit{i}\right)$ must be set to the value of $Q_{\mathit{i}}\left(x,t,U,U_x\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

3:     $\mathrm{r}\left(\mathbf{npde}\right)$ – double array

$\mathbf{r}\left(\mathit{i}\right)$ must be set to the value of $R_{\mathit{i}}\left(x,t,U,U_x\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

4:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, then nag_pde_1d_parab_fd (d03pc) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

5:     $\mathrm{user}$ – Any MATLAB object

5:     $\mathrm{bndary}$ – function handle or string containing name of m-file

bndary must compute the functions ${\beta }_i$ and ${\gamma }_i$ which define the boundary conditions as in equation (3).

[beta, gamma, ires, user] = bndary(npde, t, u, ux, ibnd, ires, user)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

3:     $\mathrm{u}\left(\mathbf{npde}\right)$ – double array

$\mathbf{u}\left(\mathit{i}\right)$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

4:     $\mathrm{ux}\left(\mathbf{npde}\right)$ – double array

$\mathbf{ux}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:     $\mathrm{ibnd}$ – int64int32nag_int scalar

Determines the position of the boundary conditions.

$\mathbf{ibnd}=0$

bndary must set up the coefficients of the left-hand boundary, $x=a$.

$\mathbf{ibnd}\ne 0$

Indicates that bndary must set up the coefficients of the right-hand boundary, $x=b$.

6:     $\mathrm{ires}$ – int64int32nag_int scalar

Set to $-1\text{ or }1$.

7:     $\mathrm{user}$ – Any MATLAB object

bndary is called from nag_pde_1d_parab_fd (d03pc) with the object supplied to nag_pde_1d_parab_fd (d03pc).

Output Parameters

1:     $\mathrm{beta}\left(\mathbf{npde}\right)$ – double array

$\mathbf{beta}\left(\mathit{i}\right)$ must be set to the value of ${\beta }_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

2:     $\mathrm{gamma}\left(\mathbf{npde}\right)$ – double array

$\mathbf{gamma}\left(\mathit{i}\right)$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,U_x\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

3:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, then nag_pde_1d_parab_fd (d03pc) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

4:     $\mathrm{user}$ – Any MATLAB object

6:     $\mathrm{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – double array

The initial values of $U\left(x,t\right)$ at $t=\mathbf{ts}$ and the mesh points $\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

7:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

The mesh points in the spatial direction. $\mathbf{x}\left(1\right)$ must specify the left-hand boundary, $a$, and $\mathbf{x}\left(\mathbf{npts}\right)$ must specify the right-hand boundary, $b$.

Constraint: $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{npts}\right)$.

8:     $\mathrm{acc}$ – double scalar

A positive quantity for controlling the local error estimate in the time integration. If $E\left(i,j\right)$ is the estimated error for $U_i$ at the $j$th mesh point, the error test is:

$$\left|E\left(i,j\right)\right|=\mathbf{acc}\times \left(1.0+\left|\mathbf{u}\left(i,j\right)\right|\right)\text{.}$$

Constraint: $\mathbf{acc}>0.0$.

9:     $\mathrm{rsave}\left(\mathit{lrsave}\right)$ – double array

lrsave, the dimension of the array, must satisfy the constraint $$\begin{gathered} \mathit{lrsave}\ge \left(6\times \mathbf{npde}+10\right)\times \mathbf{npde}\times \mathbf{npts}+\left(3\times \mathbf{npde}+21\right)\times \mathbf{npde}+ \\ 7\times \mathbf{npts}+54 \end{gathered}$$.

If $\mathbf{ind}=0$, rsave need not be set on entry.

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

10:   $\mathrm{isave}\left(\mathit{lisave}\right)$ – int64int32nag_int array

lisave, the dimension of the array, must satisfy the constraint $\mathit{lisave}\ge \mathbf{npde}\times \mathbf{npts}+24$.

If $\mathbf{ind}=0$, isave need not be set on entry.

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$\mathbf{isave}\left(1\right)$

Contains the number of steps taken in time.

$\mathbf{isave}\left(2\right)$

Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.

$\mathbf{isave}\left(3\right)$

Contains the number of Jacobian evaluations performed by the time integrator.

$\mathbf{isave}\left(4\right)$

Contains the order of the last backward differentiation formula method used.

$\mathbf{isave}\left(5\right)$

Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.

11:   $\mathrm{itask}$ – int64int32nag_int scalar

Specifies the task to be performed by the ODE integrator.

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$.

$\mathbf{itask}=2$

One step and return.

$\mathbf{itask}=3$

Stop at first internal integration point at or beyond $t=\mathbf{tout}$.

Constraint: $\mathbf{itask}=1$, $2$ or $3$.

12:   $\mathrm{itrace}$ – int64int32nag_int scalar

The level of trace information required from nag_pde_1d_parab_fd (d03pc) and the underlying ODE solver. itrace may take the value $-1$, $0$, $1$, $2$ or $3$.

$\mathbf{itrace}=-1$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed on the current error message unit (see nag_file_set_unit_error (x04aa)).

$\mathbf{itrace}>0$

Output from the underlying ODE solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

If $\mathbf{itrace}<-1$, then $-1$ is assumed and similarly if $\mathbf{itrace}>3$, then $3$ is assumed.

The advisory messages are given in greater detail as itrace increases. You are advised to set $\mathbf{itrace}=0$, unless you are experienced with Sub-chapter D02M–N.

13:   $\mathrm{ind}$ – int64int32nag_int scalar

Indicates whether this is a continuation call or a new integration.

$\mathbf{ind}=0$

Starts or restarts the integration in time.

$\mathbf{ind}=1$

Continues the integration after an earlier exit from the function. In this case, only the arguments tout and ifail should be reset between calls to nag_pde_1d_parab_fd (d03pc).

Constraint: $\mathbf{ind}=0$ or $1$.

14:   $\mathrm{cwsav}\left(10\right)$ – cell array of strings

If $\mathbf{ind}=0$, cwsav need not be set on entry.

If $\mathbf{ind}=1$, cwsav must be unchanged from the previous call to the function.

15:   $\mathrm{lwsav}\left(100\right)$ – logical array

If $\mathbf{ind}=0$, lwsav need not be set on entry.

If $\mathbf{ind}=1$, lwsav must be unchanged from the previous call to the function.

16:   $\mathrm{iwsav}\left(505\right)$ – int64int32nag_int array

If $\mathbf{ind}=0$, iwsav need not be set on entry.

If $\mathbf{ind}=1$, iwsav must be unchanged from the previous call to the function.

17:   $\mathrm{rwsav}\left(1100\right)$ – double array

If $\mathbf{ind}=0$, rwsav need not be set on entry.

If $\mathbf{ind}=1$, rwsav must be unchanged from the previous call to the function.

Optional Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

Default: the first dimension of the array u.

The number of PDEs in the system to be solved.

Constraint: $\mathbf{npde}\ge 1$.

2:     $\mathrm{npts}$ – int64int32nag_int scalar

Default: the dimension of the array x and the second dimension of the array u. (An error is raised if these dimensions are not equal.)

The number of mesh points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{npts}\ge 3$.

3:     $\mathrm{user}$ – Any MATLAB object

user is not used by nag_pde_1d_parab_fd (d03pc), but is passed to pdedef and bndary. 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:     $\mathrm{ts}$ – double scalar

The value of $t$ corresponding to the solution values in u. Normally $\mathbf{ts}=\mathbf{tout}$.

2:     $\mathrm{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – double array

$\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ will contain the computed solution at $t=\mathbf{ts}$.

3:     $\mathrm{rsave}\left(\mathit{lrsave}\right)$ – double array

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

4:     $\mathrm{isave}\left(\mathit{lisave}\right)$ – int64int32nag_int array

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$\mathbf{isave}\left(1\right)$

Contains the number of steps taken in time.

$\mathbf{isave}\left(2\right)$

Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.

$\mathbf{isave}\left(3\right)$

Contains the number of Jacobian evaluations performed by the time integrator.

$\mathbf{isave}\left(4\right)$

Contains the order of the last backward differentiation formula method used.

$\mathbf{isave}\left(5\right)$

Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.

5:     $\mathrm{ind}$ – int64int32nag_int scalar

$\mathbf{ind}=1$.

6:     $\mathrm{user}$ – Any MATLAB object

7:     $\mathrm{cwsav}\left(10\right)$ – cell array of strings

If $\mathbf{ind}=1$, cwsav must be unchanged from the previous call to the function.

8:     $\mathrm{lwsav}\left(100\right)$ – logical array

If $\mathbf{ind}=1$, lwsav must be unchanged from the previous call to the function.

9:     $\mathrm{iwsav}\left(505\right)$ – int64int32nag_int array

If $\mathbf{ind}=1$, iwsav must be unchanged from the previous call to the function.

10:   $\mathrm{rwsav}\left(1100\right)$ – double array

If $\mathbf{ind}=1$, rwsav must be unchanged from the previous call to the function.

11:   $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{tout}\le \mathbf{ts}$,
or	$\mathbf{tout}-\mathbf{ts}$ is too small,
or	$\mathbf{itask}\ne 1$, $2$ or $3$,
or	$\mathbf{m}\ne 0$, $1$ or $2$,
or	$\mathbf{m}>0$ and $\mathbf{x}\left(1\right)<0.0$,
or	$\text{the mesh points }\mathbf{x}\left(i\right)$ are not ordered,
or	$\mathbf{npts}<3$,
or	$\mathbf{npde}<1$,
or	$\mathbf{acc}\le 0.0$,
or	$\mathbf{ind}\ne 0$ or $1$,
or	lrsave is too small,
or	lisave is too small.

W  $\mathbf{ifail}=2$

The underlying ODE solver cannot make any further progress across the integration range from the current point $t=\mathbf{ts}$ with the supplied value of acc. The components of u contain the computed values at the current point $t=\mathbf{ts}$.

W  $\mathbf{ifail}=3$

In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or acc is too small for the integration to continue. Integration was successful as far as $t=\mathbf{ts}$.

   $\mathbf{ifail}=4$

In setting up the ODE system, the internal initialization function was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to $3$ in at least pdedef or bndary, when the residual in the underlying ODE solver was being evaluated.

   $\mathbf{ifail}=5$

In solving the ODE system, a singular Jacobian has been encountered. You should check your problem formulation.

W  $\mathbf{ifail}=6$

When evaluating the residual in solving the ODE system, ires was set to $2$ in at least pdedef or bndary. Integration was successful as far as $t=\mathbf{ts}$.

   $\mathbf{ifail}=7$

The value of acc is so small that the function is unable to start the integration in time.

   $\mathbf{ifail}=8$

In one of pdedef or bndary, ires was set to an invalid value.

   $\mathbf{ifail}=9$ (nag_ode_ivp_stiff_imp_revcom (d02nn))

A serious error has occurred in an internal call to the specified function. Check the problem specification and all arguments and array dimensions. Setting $\mathbf{itrace}=1$ may provide more information. If the problem persists, contact NAG.

W  $\mathbf{ifail}=10$

The required task has been completed, but it is estimated that a small change in acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $\mathbf{itask}\ne 2$.)

   $\mathbf{ifail}=11$

An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit).

   $\mathbf{ifail}=12$

Not applicable.

   $\mathbf{ifail}=13$

Not applicable.

   $\mathbf{ifail}=14$

The flux function $R_i$ was detected as depending on time derivatives, which is not permissible.

   $\mathbf{ifail}=-99$

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: d03pc_example

function  d03pc_example


fprintf('d03pc example results\n\n');

%  Solution of an elliptic-parabolic pair of PDEs.
global alpha;

% Set values for problem parameters.
npde = 2;

% Number of points on calculation mesh, and on interpolated mesh.
npts = 20;
intpts = 6;

itype  = int64(1);
neqn   = npde*npts;
lisave = neqn + 24;
nwk    = (10 + 6*npde)*neqn;
lrsave = nwk + (21 + 3*npde)*npde + 7*npts + 54;

% Define some arrays.
rsave  = zeros(lrsave, 1);
u      = zeros(npde, npts);
uinterp = zeros(npde, intpts, itype);
x     = zeros(npts, 1);
isave = zeros(lisave, 1, 'int64');
cwsav = {''; ''; ''; ''; ''; ''; ''; ''; ''; ''};
lwsav = false(100, 1);
iwsav = zeros(505, 1, 'int64');
rwsav = zeros(1100, 1);

% Set up the points on the interpolation grid.
xinterp = [0.0 0.4 0.6 0.8 0.9 1.0];

acc = 1.0e-3;
alpha = 1;

itrace = int64(0);
itask  = int64(1);
% Use cylindrical polar coordinates.
m = int64(1);

% We run through the calculation twice; once to output the interpolated
% results, and once to store the results for plotting.
niter = [5, 28];

% Prepare to store plotting results.
tsav = zeros(niter(2), 1);
usav = zeros(2, niter(2), npts);
isav = 0;

for icalc = 1:2

  % Set spatial mesh points.
  piby2 = pi/2;
  hx = piby2/(npts-1);
  x = sin(0:hx:piby2);

  % Set initial conditions.
  ts   = 0;
  tout = 1e-5;

  % Set the initial values.
  [u] = uinit(x, npts);

  % Start the integration in time.
  ind = int64(0);

  % Counter for saved results.
  isav = 0;

  % Loop over endpoints for the integration.  
  % Set itask = 1: normal computation of output values at t = tout.
  for iter = 1:niter(icalc)

    %Set the endpoint.
    if icalc == 1
      tout = 10.0*tout;
    else
      if iter < 10
        tout = 2*tout;
      else
        if iter == 10
          tout = 0.01;
        else
          if iter < 20
            tout = tout + 0.01;
          else
            tout = tout + 0.1;
          end
        end
      end
    end

    % The first time this is called, ind is 0, which (re)starts the
    % integration in time.  On exit, ind is set to 1; using this value
    % on a subsequent call continues the integration.  This means that
    % only tout and ifail should be reset between calls.
    [ts, u, rsave, isave, ind, user, cwsav, lwsav, iwsav, rwsav, ifail] = ...
    d03pc(...
           m, ts, tout, @pdedef, @bndary, u, x, acc, rsave, isave, itask, ...
          itrace, ind, cwsav, lwsav, iwsav, rwsav);

    if icalc == 1
      % Output interpolation points first time through.
      if iter == 1
        fprintf([' accuracy requirement = %12.5e\n', ...
                 ' parameter alpha =  %12.3e\n'], acc, alpha);
        fprintf(' t / x       ');
        fprintf('%8.4f', xinterp(1:intpts));
        fprintf('\n\n');
      end

      % Call d03pz to do interpolation of results onto coarser grid.
      [uinterp, ifail] = d03pz( ...
                                m, u, x, xinterp, itype);

      % Output interpolated results for this time step.
      fprintf('%7.4f  u(1)', tout);
      fprintf('%8.4f', uinterp(1,1:intpts,1));
      fprintf('\n%13s','u(2)');
      fprintf('%8.4f', uinterp(2,1:intpts,1));
      fprintf('\n\n');
    else
      % Save this timestep, and this set of results.
      isav = isav+1;
      tsav(isav) = ts;
      usav(1:2,isav,1:npts) = u(1:2,1:npts);
    end
  end

  if icalc == 1
    % Output some statistics.
    fprintf([' Number of integration steps in time = %6d\n', ...
             ' Number of function evaluations      = %6d\n', ...
             ' Number of Jacobian evaluations      = %6d\n', ...
             ' Number of iterations                = %6d\n'], ...
            isave(1), isave(2), isave(3), isave(5));
  else
    % Plot results.
    fig1 = figure;
    plot_results(x, tsav, squeeze(usav(1,:,:)), 'u_1');
    fig2 = figure;
    plot_results(x, tsav, squeeze(usav(2,:,:)), 'u_2');
  end
end



function [p, q, r, ires, user] = pdedef(npde, t, x, u, ux, ires, user)
  % Evaluate Pij, Qi and Ri which define the system of PDEs.
  global alpha;

  p = zeros(npde,npde);
  q = zeros(npde,1);
  r = zeros(npde,1);
  q(1) = 4*alpha*(u(2) + x*ux(2));
  r(1) = x*ux(1);
  r(2) = ux(2) - u(1)*u(2);
  p(2,2) = 1 - x*x;


function [beta, gamma, ires, user] = bndary(npde, t, u, ux, ibnd, ires, user)
  % Evaluate beta and gamma to define the boundary conditions.

  if (ibnd == 0)
  beta(1) = 0;
  beta(2) = 1;
  gamma(1) = u(1);
  gamma(2) = -u(1)*u(2);
  else
  beta(1) = 1;
  beta(2) = 0;
  gamma(1) = -u(1);
  gamma(2) = u(2);
  end


function [u] = uinit(x, npts)
  % Set initial values for solution.
  global alpha;

  for i = 1:npts
    u(1,i) = 2*alpha*x(i);
    u(2,i) = 1;
  end


function plot_results(x, t, u, ident)

  % Plot array as a mesh.
  mesh(x, t, u);
  set(gca, 'YScale', 'log');
  set(gca, 'YTick', [0.00001 0.0001 0.001 0.01 0.1 1]);
  set(gca, 'YMinorGrid', 'off');
  set(gca, 'YMinorTick', 'off');

  % Label the axes, and set the title.
  xlabel('x');
  ylabel('t');
  zlabel([ident,'(x,t)']);
  title({['Solution ',ident,' of elliptic-parabolic pair'], ...
         'using method of lines and BDF'});
  title(['Solution ',ident,' of elliptic-parabolic pair']);

  % Set the axes limits tight to the x and y range.
  axis([x(1) x(end) t(1) t(end)]);

  % Set the view to something nice (determined empirically).
  view(-125, 30);

d03pc example results

 accuracy requirement =  1.00000e-03
 parameter alpha =     1.000e+00
 t / x         0.0000  0.4000  0.6000  0.8000  0.9000  1.0000

 0.0001  u(1)  0.0000  0.8008  1.1988  1.5990  1.7958  1.8485
         u(2)  0.9997  0.9995  0.9994  0.9988  0.9663 -0.0000

 0.0010  u(1)  0.0000  0.7982  1.1940  1.5841  1.7179  1.6734
         u(2)  0.9969  0.9952  0.9937  0.9484  0.6385 -0.0000

 0.0100  u(1)  0.0000  0.7676  1.1239  1.3547  1.3635  1.2830
         u(2)  0.9627  0.9495  0.8754  0.5537  0.2908 -0.0000

 0.1000  u(1)  0.0000  0.3908  0.5007  0.5297  0.5120  0.4744
         u(2)  0.5468  0.4299  0.2995  0.1479  0.0724 -0.0000

 1.0000  u(1)  0.0000  0.0007  0.0008  0.0008  0.0008  0.0007
         u(2)  0.0010  0.0007  0.0005  0.0002  0.0001 -0.0000

 Number of integration steps in time =     78
 Number of function evaluations      =    378
 Number of Jacobian evaluations      =     25
 Number of iterations                =    190