# Wolfram Language™

## Compute Sensitivities of PDEs over Regions

Compute the parametric dependence of the wave equation , .

Specify the wave equation .

In[1]:=
`eqn = D[u[t, x, y], t, t] == c^2 Laplacian[u[t, x, y], {x, y}];`

Specify initial conditions .

In[2]:=
```ics = {u[0, x, y] == Exp[-((a x)^2 + (b x)^2)], Derivative[1, 0, 0][u][0, x, y] == 0};```

Specify a fixed Dirichlet boundary condition.

In[3]:=
`bcs = DirichletCondition[u[t, x, y] == 0, True];`

Set up the parametric function.

In[4]:=
```pfun = ParametricNDSolveValue[{eqn, ics, bcs}, u, {t, 0, 5}, {x, y} \[Element] Disk[], {a, b, c}];```

Find the sensitivities , , and for parameters , , and .

In[5]:=
```ifda = D[pfun[a, 1, 1], a] /. {a -> 1}; ifda = D[pfun[1, b, 1], b] /. {b -> 1}; ifdc = D[pfun[1, 1, c], c] /. {c -> 1};```

Visualize the corresponding sensitivity bands by plotting the parametric function for , , and at and overlaying the solution with of the sensitivity.

In[6]:=
```Plot3D[Evaluate[(pfun[a, b, c][\[Tau], x, y] + .5 {0, 1, -1} D[pfun[a, b, c][\[Tau], x, y], a]) /. {a -> 1, b -> 1, c -> 1, \[Tau] -> 3}], {x, y} \[Element] Disk[], PlotRange -> All, Boxed -> False, Axes -> False, Mesh -> 5, PlotStyle -> {Automatic, Opacity[0.3], Opacity[0.3]}]```
Out[6]=