The finite difference method involves establishing nodal networks, deriving finite difference approximations for the governing equations at interior and exterior nodal points, and developing and solving a system of simultaneous algebraic nodal equations. This method is commonly used for heat transfer problems and involves discretizing the domain, deriving equations for each node, and solving the resulting system of equations numerically.

1. FINITE DIFFERENCE
In numerical analysis, two different approaches are commonly used:
The finite difference and the finite element methods. In heat transfer
problems, the finite difference method is used more often and will be
discussed here. The finite difference method involves:
Establish nodal networks
Derive finite difference approximations for the governing
equation at both interior and exterior nodal points
Develop a system of simultaneous algebraic nodal
equations
Solve the system of equations using numerical schemes

2. The Nodal Networks

3. Finite Difference Approximation

4. Finite Difference Approximation cont.

5. Finite Difference Approximation cont.

6. A System of Algebraic Equations

7. Matrix Form

8. Numerical Solutions

9. Iteration

10. Example

11. Example (cont.)

12. Example (cont.)

13. Summary of nodal finite-difference
relations for various configurations:
Case 1 Interior Node
0
4T
T
T
T
T n
m,
n
1,
m
n
1,
m
1
n
m,
1
n
m, 



 




14. Case 2
Node at an internal corner with convection
1, , 1 1, , 1 ,
2( ) ( ) 2 2(3 ) 0
m n m n m n m n m n
h x h x
T T T T T T
k k
    
 
      

15. Case 3
Node at a plane surface with convection
1, , 1 , 1 ,
(2 ) 2 2( 2) 0
m n m n m n m n
h x h x
T T T T T
k k
   
 
     

16. Case 4
Node at an external corner with convection
, 1 1, ,
( ) 2 2( 1) 0
m n m n m n
h x h x
T T T T
k k
  
 
    

17. Case 5
Node at a plane surface with uniform heat flux
0
4
'
'
2
)
2
( ,
1
,
1
,
,
1 




 

 n
m
n
m
n
m
n
m T
k
x
q
T
T
T