Page 459 - Applied Numerical Methods Using MATLAB

448 PARTIAL DIFFERENTIAL EQUATIONS
9.3 Elliptic PDEs with Neumann Boundary Condition
Consider the PDE (E9.1.1) (dealt with in Example 9.1)

2
2
∂ u(x, y) ∂ u(x, y)
+ = 0 for 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 (P9.3.1)
∂x 2 ∂y 2
with different boundary conditions of Neumann type, which was discussed
in Section 9.1. Modify the routine “poisson()” so that it can deal with the
Neumann boundary condition and declare it as

function [u,x,y] = poisson_Neuman(f,g,bx0,bxf,by0,byf,x0,xf,y0,yf,...)

where the third/fourth/fifth/sixth input arguments are supposed to carry the
functions of
u(x 0 ,y)/u(x f ,y)/u(x, y 0 )/u(x, y f )

or

∂u(x, y)/∂x| x=x 0 /∂u(x, y)/∂x| x=x f /∂u(x, y)/∂y| y=y 0 /∂u(x, y)/∂y| y=y f

and the seventh/eighth/ninth/tenth input arguments are to carry x 0 /x f /y 0 /y f
or [x 0 1]/[x f 1]/[y 0 ]/[y f 1] depending on whether each boundary condition
is of Dirichlet or Neumann type. Use it to solve the PDE with the
following boundary conditions and plot the solutions by using the MATLAB
command “mesh()”. Divide the solution region (domain) into M x × M y =
20 × 20 sections.

(cf) You may refer to the related part of the program in the previous page.
y 4
(a) ∂u(x, y)/∂x| x=0 =− cos y, u(4,y) = e cos 4 − e cos y (P9.3.2)
x
4
∂u(x, y)/∂y| y=0 = cos x, u(x, 4) = e cos x − e cos 4 (P9.3.3)
y
4
y
(b) u(0,y) = e − cos y, ∂u(x, y)/∂x| x=4 =−e sin 4 − e cos y(P9.3.4)
4
x
x
u(x, 0) = cos x − e , ∂u(x, y)/∂y| y=4 = e cos x + e sin 4 (P9.3.5)
4
y
(c) ∂u(x, y)/∂x| x=0 =− cos y, u(4,y) = e cos 4 − e cos y (P9.3.6)
4
x
x
u(x, 0) = cos x − e , ∂u(x, y)/∂y = e cos x + e sin 4 (P9.3.7)
y=4
4
y
y
(d) u(0,y) = e − cos y, ∂u(x, y)/∂x =−e sin 4 − e cos y(P9.3.8)
x=4
4 x
∂u(x, y)/∂y = cos x, u(x, 4) = e cos x − e cos 4 (P9.3.9)
y=0