## f Coefficient for specifyCoefficients

This section describes how to write the coefficient f in the equation

$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\nabla u\right)+au=f$

or in similar equations. The question is how to write the coefficient f for inclusion in the PDE model via specifyCoefficients.

N is the number of equations, see Equations You Can Solve Using PDE Toolbox. Give f as either of the following:

• If f is constant, give a column vector with N components. For example, if N = 3, f could be:

f = [3;4;10];
• If f is not constant, give a function handle. The function must be of the form

fcoeff = fcoeffunction(location,state)

Pass the coefficient to specifyCoefficients as a function handle, such as

specifyCoefficients(model,"f",@fcoeffunction,...)

solvepde or solvepdeeig compute and populate the data in the location and state structure arrays and pass this data to your function. You can define your function so that its output depends on this data. You can use any names instead of location and state, but the function must have exactly two arguments. To use additional arguments in your function, wrap your function (that takes additional arguments) with an anonymous function that takes only the location and state arguments. For example:

fcoeff = ...
specifyCoefficients(model,"f",fcoeff,...
• location is a structure with these fields:

• location.x

• location.y

• location.z

• location.subdomain

The fields x, y, and z represent the x-, y-, and z- coordinates of points for which your function calculates coefficient values. The subdomain field represents the subdomain numbers, which currently apply only to 2-D models. The location fields are row vectors.

• state is a structure with these fields:

• state.u

• state.ux

• state.uy

• state.uz

• state.time

The state.u field represents the current value of the solution u. The state.ux, state.uy, and state.uz fields are estimates of the solution’s partial derivatives (∂u/∂x, ∂u/∂y, and ∂u/∂z) at the corresponding points of the location structure. The solution and gradient estimates are N-by-Nr matrices. The state.time field is a scalar representing time for time-dependent models.

Your function must return a matrix of size N-by-Nr, where Nr is the number of points in the location that solvepde passes. Nr is equal to the length of the location.x or any other location field. The function should evaluate f at these points.

For example, if N = 3, f could be:

function f = fcoeffunction(location,state)

N = 3; % Number of equations
nr = length(location.x); % Number of columns
f = zeros(N,nr); % Allocate f

% Now the particular functional form of f
f(1,:) = location.x - location.y + state.u(1,:);
f(2,:) = 1 + tanh(state.ux(1,:)) + tanh(state.uy(3,:));
f(3,:) = (5 + state.u(3,:)).*sqrt(location.x.^2 + location.y.^2);

This represents the coefficient function

$f=\left[\begin{array}{c}x-y+u\left(1\right)\\ 1+\mathrm{tanh}\left(\partial u\left(1\right)/\partial x\right)+\mathrm{tanh}\left(\partial u\left(3\right)/\partial y\right)\\ \left(5+u\left(3\right)\right)\sqrt{{x}^{2}+{y}^{2}}\end{array}\right]$