This section describes how to write the coefficient f
in
the equation
$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f.$$
or in similar equations. The number of rows in f
indicates N,
the number of equations, see Systems of PDEs. Give f
as any of the
following:
A column vector with N components.
For example, if N = 3, f
could
be:
f = [3;4;10];
A character array with N rows. The rows of the character array are MATLAB^{®} expressions as described in Specify Scalar PDE Coefficients in String Form, with additional options for nonlinear equations. The additional options are:
Represent the i
th component of
the solution u
using 'u(i)'
.
Similarly, represent the i
th components
of the gradients of the solution u
using 'ux(i)'
, 'uy(i)'
and 'uz(i)'
.
Pad the rows with spaces so each row has the same number
of characters (char
does this
automatically). For example, if N = 3, f
could be:
f = char('sin(x)+cos(y)','cosh(x.*y)*(1+u(1).^2)','x.*y./(1+x.^2+y.^2)')
f = sin(x)+cos(y) cosh(x.*y)*(1+u(1).^2) x.*y./(1+x.^2+y.^2)
For 2-D geometry, a function as described in Specify 2-D Scalar Coefficients in Function Form. The
function should return a matrix of size N-by-Nt,
where Nt is the number of triangles in the mesh.
The function should evaluate f
at the triangle
centroids, as in Specify 2-D Scalar Coefficients in Function Form. Give solvers the function
name as a string '
,
or as a function handle @filename
'filename
, where filename.m
is
a file on your MATLAB path. For details on writing the function,
see Calculate Coefficients in Function Form.
For example, if N = 3, f
could be:
function f = fcoeffunction(p,t,u,time) N = 3; % Number of equations % Triangle point indices it1 = t(1,:); it2 = t(2,:); it3 = t(3,:); % Find centroids of triangles xpts = (p(1,it1)+p(1,it2)+p(1,it3))/3; ypts = (p(2,it1)+p(2,it2)+p(2,it3))/3; [ux,uy] = pdegrad(p,t,u); % Approximate derivatives uintrp = pdeintrp(p,t,u); % Interpolated values at centroids nt = size(t,2); % Number of columns f = zeros(N,nt); % Allocate f % Now the particular functional form of f f(1,:) = xpts - ypts + uintrp(1,:); f(2,:) = 1 + tanh(ux(1,:)) + tanh(uy(3,:)); f(3,:) = (5+uintrp(3,:)).*sqrt(xpts.^2+ypts.^2);
Because this function depends on the solution u,
if the equation is elliptic, use the pdenonlin
solver.
The initial value can be all 0
s in the case of
Dirichlet boundary conditions:
np = size(p,2); % number of points u0 = zeros(N*np,1); % initial guess
For 3-D geometry, a function as described in Specify 3-D PDE Coefficients in Function Form.
The function should return a matrix of size N-by-Nr,
where Nr is the number of points in the region
that the solver passes. The function should evaluate f
at
these points. Give solvers the function as a function handle @filename
,
where filename.m
is a file on your MATLAB path,
or is an anonymous function.
Caution
It is not reliable to specify |