Set Initial Conditions

What Are Initial Conditions?

The term initial condition has two meanings:

For time-dependent problems, the initial condition is the solution u at the initial time, and also the initial time-derivative if the m coefficient is nonzero. Set the initial condition in the model using setInitialConditions.
For nonlinear stationary problems, the initial condition is a guess or approximation of the solution u at the initial iteration of the nonlinear solver. Set the initial condition in the model using setInitialConditions.
If you do not specify the initial condition for a stationary problem, solvepde uses the zero function for the initial iteration.

Constant Initial Conditions

For a system of N equations, you can give constant initial conditions as either a scalar or as a vector with N components. For example, if the initial condition is u = 15 for all components, use the following command.

setInitialConditions(model,15);

If N = 3, and the initial condition is 15 for the first equation, 0 for the second equation, and –3 for the third equation, use the following commands.

u0 = [15,0,-3];
setInitialConditions(model,u0);

If the m coefficient is nonzero, give an initial condition for the time derivative as well. Set this initial derivative in the same form as the first initial condition. For example, if the initial derivative of the solution is [4,3,0], use the following commands.

u0 = [15,0,-3];
ut0 = [4,3,0];
setInitialConditions(model,u0,ut0);

Nonconstant Initial Conditions

If your initial conditions are not constant, set them by writing a function of the form.

function u0 = initfun(location)

solvepde computes and populates the data in the location structure array and passes this data to your function. You can define your function so that its output depends on this data. You can use any name instead of location. To use additional arguments in your function, wrap your function (that takes additional arguments) with an anonymous function that takes only the location argument. For example:

u0 = @(location) initfunWithAdditionalArgs(location,arg1,arg2...)
setInitialConditions(model,u0)

location is a structure array with fields location.x, location.y, and, for 3-D problems, location.z. Your function must return a matrix u0 of size N-by-M, where N is the number of equations in your PDE and M = length(location.x). The fields in location are row vectors.

For example, suppose you have a 2-D problem with N = 2 equations:

$\begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}} - \nabla \cdot (\nabla u) = [\begin{matrix} 3 + x \\ 4 - x - y \end{matrix}] \\ u (0) = [\begin{matrix} 4 + x^{2} + y^{2} \\ 0 \end{matrix}] \\ \frac{\partial u}{\partial t} (0) = [\begin{matrix} 0 \\ \sin (x y) \end{matrix}] \end{array}$

This problem has m = 1, c = 1, and f = $[\begin{matrix} 3 + x \\ 4 - x - y \end{matrix}]$ . Because m is nonzero, give both an initial value of u and an initial value of the derivative of u.

Write the following function files. Save them to a location on your MATLAB^® path.

function uinit = u0fun(location)

M = length(location.x);
uinit = zeros(2,M);
uinit(1,:) = 4 + location.x.^2 + location.y.^2;

function utinit = ut0fun(location)

M = length(location.x);
utinit = zeros(2,M);
utinit(2,:) = sin(location.x.*location.y);

Pass the initial conditions to your PDE model:

u0 = @u0fun;
ut0 = @ut0fun;
setInitialConditions(model,u0,ut0);

Nodal Initial Conditions

You can use results of previous analysis as nodal initial conditions for your current model. The geometry and mesh of the model you used to obtain the results and the current model must be the same. For example, solve a time-dependent PDE problem for times from t0 to t1 with a time step tstep.

results = solvepde(model,t0:tstep:t1);

If later you need to solve this PDE problem for times from t1 to t2, you can use results to set initial conditions. If you do not explicitly specify the time step, setInitialConditions uses results corresponding to the last solution time, t1.

setInitialConditions(model,results)

To use results for a particular solution time instead of the last one, specify the solution time index as a third parameter of setInitialConditions. For example, to use the solution at time t0 + 10*tstep, specify 11 as the third parameter.

setInitialConditions(model,results,11)