# Documentation

## Initial Conditions

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.

Pass u0 as a column vector of values at the points p in the usual p, t, e mesh. See Mesh Data. You can also pass a scalar, which means the initial condition is a constant value.

 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 points in the mesh, Np.

For scalar u, specify a column vector of length Np. The value of element k corresponds to the point p(k).

For a system of N equations, specify a column vector of N*Np elements. The first Np elements contain the values of component 1, where the value of element k corresponds to point p(k). The next Np points contain the values of component 2, etc. It can be convenient to first represent the initial conditions u0 as an Np-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. Suppose that p is the usual mesh point data (see Mesh Data). Compute the initial conditions for all mesh points 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.

For the parabolic and hyperbolic solvers with scalar problems, you can also specify text expressions for the initial conditions. The initial conditions are functions of x and y alone, and, for 3-D problems, z.

For example, if you have an initial condition

$u\left(x,y\right)=\frac{xy\mathrm{cos}\left(x\right)}{1+{x}^{2}+{y}^{2}},$

then you can use this expression for the initial condition.

'x.*y.*cos(x)./(1+x.^2+y.^2)'