1. Finite Deference Methodby
Dr. Samah Mohamed Mabrouk

2. Finite-difference method

3. Different Difference formulasB C x
f(x)
f(x+x) i
i+1
Different Difference formulas
Taylor series expansion
First forward difference approximation
Solve for
order of x
First forward difference approximation

4. First backward difference approximation
f(x)
f(x-x) i
i-1
First backward difference approximation
Taylor series expansion
Solve for
order of x
First backward difference approximation

5. Central difference approximation
Subtracting (2) from (1)
Central difference approximation B C x
f(x)
f(x+x)
f(x-x) A
i-1 i
i+1

6. Second derivative Multiply (1) by 2 and subtracting it from (2)
Solve for
forward difference approximation

7. backward difference approximation
Central difference approximation

8. backward difference approximation
forward difference approximation
backward difference approximation
Central difference approximation

9. BVP IVP ODE the conditions given are specified at the same value of x,
Initial Value Problems
IVP
Boundary Value Problems
BVP
the conditions given are specified at the same value of x,
the two conditions are specified at different values of x.

10. Dirichlet BC

11. y0 = y y y y4 =0
x x x x x4
We then substitute in the difference equation for the inner nodes (unknowns) as follows:
Rearrange the equations and substitute by the boundary values y0 =0, y4 =0, we get the following 3x3 system

12. Writing this system in matrix-vector form as

13. Solving this system in Matlab
>> Y=A\b
Y =
>> format long
>> Y=inv(A)*b

14. In case of Neumann BC:
we apply FD for the first derivative, for example:
This means that the value un at the last node is unknown and will need an extra equation at that node
In this equation we replace un+1 from the approximation of Neumann BC

15. Example: Solve
Using FD method. Let h=0.25
Solution
du/dx=1
u0 = u u u u4
x x x x x4
We then substitute in the difference equation for the inner nodes (unknowns) and the Neumann boundary node as follows:

16. x x x x x4
u0 = u u u u4
du/dx=1 u5
We substitute by u5 in the last equation from the Neumann BC as

17. This will lead to a linear system of equations of four unknowns
In matrix form Au=b

18. Heat equation
The conservation of heat can be used to develop a heat balance for a long, thin rod. If the rod is not insulated along its length and the system is at steady state. The equation that results is: T1 T2 Ta

19. For L= 10 m rod with T(0) = 40, T(L) = 200, Ta = 20 and h = 0.01
Solve
Solution

20. Divide the rod into a grid by  x = 2m40
200 i=

21. Writing this system in matrix-vector form as

22. Exercise

23. Exercises