## 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 = ... @(location,state) myfunWithAdditionalArgs(location,state,arg1,arg2...) 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

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