Two Dimensional PDEs

Problem #1: Alternating Direction Implicit (ADI) Method
Research and write a brief explanation of the ADI Method and solve:
Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point
Charge
! ! !
?2u(x, y) = f (x, y) for -1 = x = 1, -1 = y = 1
boundary conditions: u(x,y) = 0 for all boundaries
f (0.5, 0.5) = -1
f (-0.5,-0.5) = 1
elsewhere : f (x, y) = 0
Two Dimensional Temperature Diffusion:
! ! !
10-4 ?2u(x, y, t )
?x2 +
?2u(x, y, t )
?y2
?
? ?
?
? ?
=
?u(x, y, t )
?t
for 0 = x = 4, 0 = y = 4 0 = t = 5000
u(x, y,0) = 0
u(x, y, t ) = ey cos x – ex cos y for x = 0, x = 4, y = 0, y = 4
! ! Present results for t= 5000
Problem #2: Crank-Nicolson Problem
Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method.
Elliptic PDE
ENGR516 Assignment #5: Two Dimensional PDEs
Problem #1: Alternating Direction Implicit (ADI) Method
Research and write a brief explanation of the ADI Method and solve:
Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point
Charge
! ! !
?2u(x, y) = f (x, y) for -1 = x = 1, -1 = y = 1
boundary conditions: u(x,y) = 0 for all boundaries
f (0.5, 0.5) = -1
f (-0.5,-0.5) = 1
elsewhere : f (x, y) = 0
Two Dimensional Temperature Diffusion:
! ! !
10-4 ?2u(x, y, t )
?x2 +
?2u(x, y, t )
?y2
?
? ?
?
? ?
=
?u(x, y, t )
?t
for 0 = x = 4, 0 = y = 4 0 = t = 5000
u(x, y,0) = 0
u(x, y, t ) = ey cos x – ex cos y for x = 0, x = 4, y = 0, y = 4
! ! Present results for t= 5000
Problem #2: Crank-Nicolson Problem
Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method.
Assignment #8
Expand the explicit method for hyperbolic equation to two dimensions and solve:
Two Dimensional Wave Vibration Over Square Membrane
! ! !
0.25
?2u(x, y,t )
?x2 +
?2u(x, y,t )
?y2
?
? ?
?
? ?
=
?2u(x, y,t )
?t 2
for 0 = x = 2, 0 = y = 2 and 0 = t = 2
u(0, y,t ) = 0, u(2, y,t ) = 0
u(x,0,t ) = 0, u(x,2,t ) = 0
u(x, y,0) = 0.1sin(p x)sin(p y / 2),
?u(x, y,0)
?t
= 0
! ! Present Results for t = 0.1 and t = 1.8