1. Finite Deference Methodby
Dr. Samah Mohamed Mabrouk

2. Difference equations for the Laplace and Poisson equations
Poisson’s equation
Where  is the Laplacian operator
Laplace’s equation
SE301_Topic 6
Al-Amer2005

3. Central difference approximation for Second derivative
j+1 j y
j-1 x y
j+1 j
j-1 i
i-1
i+1

4. For x =y = h
North
East
South
West
SE301_Topic 6
Al-Amer2005

5. Example 1
Solve the Laplace equation on a square plate of side 12 cm, using a grid of mesh 4cm and a Dirchlet B.C. as u(x,0)=u(0,y)=u(12,y)=100 and u(x,12)=0
solution
L = 12 cm , h= 4 cm
N= 12/4 = 3
u=0
-4uij+ uE+uW+uN+uS = 0 u3 u4
(1) -4u1+u u3 = 0
u=100
u=100 u1 u2
(2) -4u2+u u4 = 0
(3) -4u3+100+u4+u1+0 = 0
x =
u=100
(4) -4u4+100+u3+u2+0 = 0

6. So we can solve only for u1 and u3
In matrix form Au=b
But the problem is symmetry, u1 =u2 and u3 =u4
So we can solve only for u1 and u3
(1) -4u1+u u3 = 0
(3) -4u3+100+u4+u1+0 = 0
Then replace for u1 =u2 and u3 =u4, the system of equations is reduced to
-3u1 +u3 = -200
Which has a solution
u1 =u2 =87.5
and u3 =u4 =62.5
u1-3u3 = -100
SE301_Topic 6
Al-Amer2005

7. Solve the mixed BVP for the poisson equation (uxx + uyy =12xy)
Example 2
Solve the mixed BVP for the poisson equation (uxx + uyy =12xy)
on a rectangle plate as shown in the fig. (take x =y =0.5) u3 u4
u=3
u=0
u=3y3 u1 u2
u=0.375
x =
u=0

8. solution -4uij+ uE+uW+uN+uS = 12h2xy=3xy u5 u6
(1) -4u1+u2+u3 =3(0.5)(0.5) u3 u4
u=3
u=0
(2) -4u u1+u4 = 3(1)(0.5)
u=3y3 u1 u2
u=0.375
(3) -4u3+u4+u1+u5 = 3(0.5)(1)
(4) -4u4+3+u3+u2+u6 = 3(1)(1)
x =
u=0

9. u5 u6 u3 u4
u=3
u=0
u=3y3 u1 u2
Then replace for u5 and u6
And put the system in the matrix form
u=0.375
x =
u=0

10. Which has a solution
SE301_Topic 6
Al-Amer2005

11. Exercise
For a rectangle thin plate of dimension 4*3 units, u(x,0)=u(x,3)=10x, u(0,y)=0 and u(4,y)= y(y-3).
Solve the Poisson’s equation 2u = 5x , take x = y = 1

12. Exercises