Initial conditions has two meanings:
For the parabolic
and hyperbolic
solvers,
the initial condition u0
is the solution u at
the initial time. You must specify the initial condition for these
solvers. Pass the initial condition in the first argument or arguments.
u = parabolic(u0,... or u = hyperbolic(u0,ut0,...
For the hyperbolic
solver, you must also
specify ut0
, which is the value of the derivative
of u with respect to time at the initial time. ut0
has
the same form as u0
.
For nonlinear elliptic problems, the initial condition u0
is
a guess or approximation of the solution u at the
initial iteration of the pdenonlin
nonlinear
solver. You pass u0
in the 'U0'
name-value
pair.
u = pdenonlin(b,p,e,t,c,a,f,'U0',u0)
If you do not specify initial conditions, pdenonlin
uses
the zero function for the initial iteration.
You can specify initial conditions as a constant by passing a scalar or string vector.
For scalar problems or systems of equations, give
a scalar as the initial condition. For example, set u0
to 5
for
an initial condition of 5 in every component.
For systems of N equations, give
a string vector initial condition with N rows.
For example, if there are N = 3 equations, you
can give initial conditions u0
= char('3','-3','0')
.
You can specify text expressions for the initial conditions.
The initial conditions are functions of x and y alone,
and, for 3-D problems, z. The text expressions
represent vectors at nodal points, so use .*
for
multiplication, ./
for division, and .^
for
exponentiation.
For example, if you have an initial condition
$$u(x,y)=\frac{xy\mathrm{cos}(x)}{1+{x}^{2}+{y}^{2}},$$
then you can use this expression for the initial condition.
'x.*y.*cos(x)./(1+x.^2+y.^2)'
For a system of N > 1 equations, use a text array with one row for each component, such as
char('x.^2 + 5*cos(x.*y)',... 'tanh(x.*y)./(1+z.^2)')
Pass u0
as a column vector of values at the
mesh nodes. The nodes are either model.Mesh.Nodes
,
or the p
data from initmesh
or meshToPet
.
See Mesh Data.
Tip For reliability, the initial conditions and boundary conditions should be consistent. |
The size of the column vector u0
depends
on the number of equations, N, and on the number
of nodes in the mesh, N_{p}
.
For scalar u, specify a column vector of
length N_{p}
. The value of
element k
corresponds to the node p(k)
.
For a system of N equations, specify a column
vector of N*N_{p}
elements.
The first N_{p}
elements contain
the values of component 1, where the value of element k
corresponds
to node p(k)
. The next N_{p}
points
contain the values of component 2, etc. It can be convenient to first
represent the initial conditions u0
as an N_{p}
-by-N
matrix,
where the first column contains entries for component 1, the second
column contains entries for component 2, etc. The final representation
of the initial conditions is u0(:)
.
For example, suppose you have a function myfun(x,y)
that
calculates the value of the initial condition u0(x,y)
as
a row vector of length N for a 2-D problem. Suppose
that p
is the usual mesh node data (see Mesh Data). Compute the initial conditions
for all mesh nodes p
.
% Assume N and p exist; N = 1 for a scalar problem np = size(p,2); % Number of mesh points u0 = zeros(np,N); % Allocate initial matrix for k = 1:np x = p(1,k); y = p(2,k); u0(k,:) = myfun(x,y); % Fill in row k end u0 = u0(:); % Convert to column form
Specify u0
as the initial condition.
hyperbolic
| parabolic
| pdenonlin