Page 456 - Applied Numerical Methods Using MATLAB

PROBLEMS 445
Divide the solution region into M x × M y = 40 × 40 sections.

2
2
∂ u(x, y) ∂ u(x, y) 2 2
(c) + + 4π(x + y )u(x, y)
∂x 2 ∂y 2
2
2
= 4π cos(π(x + y )) for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (P9.1.6)
with the boundary conditions
2 2
u(0,y) = sin(πy ), u(1,y) = sin(π(y + 1)) (P9.1.7)
2 2
u(x, 0) = sin(πx ), u(x, 1) = sin(π(x + 1)) (P9.1.8)
Divide the solution region into M x × M y = 40 × 40 sections.

2
2
∂ u(x, y) ∂ u(x, y) 2x+y
(d) + = 10e for 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 (P9.1.9)
∂x 2 ∂y 2
with the boundary conditions

y
u(0,y) = 2e , u(1,y) = 2e 2x+y ,
2x
u(x, 0) = 2e , u(x, 2) = 2e 2x+2 (P9.1.10)
Divide the solution region into M x × M y = 20 × 40 sections.

2
2
∂ u(x, y) ∂ u(x, y)
(e) + = 0 for 0 ≤ x ≤ 1, 0 ≤ y ≤ π/2 (P9.1.11)
∂x 2 ∂y 2
with the boundary conditions

u(0,y) = 4cos(3y), u(1,y) = 4e −3 cos(3y),
u(x, 0) = 4e −3x , u(x, π/2) = 0 (P9.1.12)

Divide the solution region into M x × M y = 20 × 20 sections.
9.2 More General PDE Having Nonunity Coefficients

Consider the following PDE having nonunity coefficients.
2
2
2
∂ u(x, y) ∂ u(x, y) ∂ u(x, y)
A + B + C + g(x, y)u(x, y) = f (x, y)
∂x 2 ∂x∂y ∂y 2
(P9.2.1)
Modify the routine “poisson()” so that it can solve this kind of PDEs and
declare it as
function [u,x,y] = poisson_abc(ABC,f,g,bx0,bxf,by0,...,Mx,My,tol,imax)