Page 468 - Applied Numerical Methods Using MATLAB

PROBLEMS 457
the mesh is nonadaptively refined every time you click ‘Refine Mesh’
in the Mesh pull-down menu. You can restore the previous mesh by
clicking ‘Undo Mesh Change’ in the Mesh pull-down menu.
(b) Consider the PDE
2
2
∂ u(x, y) ∂ u(x, y)
+ = 0 for 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 (P9.8.4)
∂x 2 ∂y 2
with the Dirichlet/Neumann boundary conditions
y
y
4
u(0,y) = e − cos y, ∂u(x, y)/∂x| x=4 =−e sin 4 − e cos y (P9.8.5)
x
4
∂u(x, y)/∂y| y=0 =cos x, ∂u(x, y)/∂y| y=4 =e cos x + e sin 4 (P9.8.6)
Noting that the true analytical solution is
x
y
u(x, y) = e cos x − e cos y (P9.8.7)
use the PDEtool to solve this PDE and fill in Table P9.8.2 with the
maximum absolute error and the number of nodes together with those of
Problem 9.3(g) for comparison.
(c) Consider the PDE
2
∂ u(x, t) ∂u(x, t)
2 = for 0 ≤ x ≤ π, 0 ≤ t ≤ 0.2 (P9.8.8)
∂x 2 ∂t
with the initial/boundary conditions

u(x, 0) = sin(2x), u(0,t) = 0, u(π, t) = 0 (P9.8.9)

Noting that the true analytical solution is
u(x, t) = sin(2x)e −8t (P9.8.10)

Table P9.8.2 The Maximum Absolute Error and the Number of Nodes
The Maximum The Number
Absolute Error of Nodes

poisson() 0.2005 21 × 21
PDEtool with Initialize Mesh 0.5702 177
PDEtool with Refine Mesh
PDEtool with second Refine Mesh
PDEtool with Adaptive Mesh
PDEtool with second Adaptive Mesh