What is numerical differentiation? What is finite difference? How to apply that to boundary value problems? #WikiCourses #Num001 https://wikicourses.wikispaces.com/Topic+Boundary+Value+Problems+-+Finite+Difference

1. Boundary Value Problems – Finite Difference
Mohammad Tawfik
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Boundary Value Problems
Mohammad Tawfik

2. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Objectives
• Applying finite difference as a numerical
method for differentiation
• Solve a Boundary Value Problem using
finite difference method
• Applying weighted residual methods for
the solution of BVP’s

3. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Numerical Differentiation

4. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Numerical Differentiation
• In most of the practical applications, the
differentiation of the function does not present a
great challenge!
• In some cases, the differentiation becomes very
expensive in terms of the computational
requirements
• If you are using the computer, you need to teach
it how to differentiate!!!
• If the function is given in the form of a table,
what are you going to do?!

5. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Numerical Differentiation
• In the previous lectures, we learned that the
slope of the function may be approximated by a
difference expression
• Recall, the parachutist:
• Note that this evaluates the derivative at the
starting point in terms of what is going to happen
in the future!
12
12
tt
vv
t
v
dt
dv







6. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Forward Difference
• When you evaluate the slope at some
point in terms of the value of the function
at higher values of time/space, then we
call this forward difference
t
vv
tt
vv
dt
dv
tt 






12
12
12
1

7. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
How about the second derivative?
• The second derivative of a function is
defined as the rate of change of slope.











 121
1
2
2
tttttt
dt
dv
dt
dv
tdt
vd

8. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substitute for the 1st derivative











 121
1
2
2
tttttt
dt
dv
dt
dv
tdt
vd














t
vv
t
vv
tdt
vd
tt
1223
2
2
1
1
2
123
2
2
2
1
t
vvv
dt
vd
tt





9. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Backward Difference
• When you evaluate the slope at some point in
terms of the value of the function at lower
values of time/space, then we call this backward
difference
• Note that this is the same formula for the forward
difference!
t
vv
tt
vv
dt
dv
tt 






21
21
21
2

10. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
How about the second derivative?
• The second derivative of a function is
defined as the rate of change of slope.











 212
1
2
2
tttttt
dt
dv
dt
dv
tdt
vd

11. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substitute for the 1st derivative











 212
1
2
2
tttttt
dt
dv
dt
dv
tdt
vd














t
vv
t
vv
tdt
vd
tt
2110
2
2
1
2
2
210
2
2
2
2
t
vvv
dt
vd
tt





12. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Central Difference
• If you know some information about the
previous and the upcoming points, why
don’t you make use of it?
t
vv
tt
vv
dt
dv
tt 





 2
02
02
02
1
2
012
2
2
2
1
t
vvv
dt
vd
tt





13. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Applications
Finite Difference

14. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Finite Difference
• The finite difference is a direct application
of the numerical differentiation.
• Substitute each derivative in a DE by its
FD equivalent to transform the differential
equation into difference equation

15. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example: The Parachutist
• The DE is:
• Applying the FD
relation:
• You get the
Euler formula!
m
cvmg
v


m
cv
g
t
vv
dt
dv



 12







m
cv
gtvv 1
12

16. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example: Heat transfer problem
dx
q1=-kAdT/dx
q2=q1+(dq1/dx)dx
q3=hPdx(T-Ta)
Tin Tout
0 L x
Ta
d2T/dx2 - b2 T = -b2Ta
Area A
Perim. P
(b2 = hP/kA)
We need 2 conditions: T(0) = Tin; T(L) = Tout
Conditions at 2 boundaries: Boundary value problem

17. Boundary Value Problems – Finite Difference
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Finite difference methods
Discretize into n sub-domains
0 x1 x2 xn=Lxn-1
(equidistant for simplicity)
Approximate derivatives using neighboring points
d2T/dx2|i  (Ti-1-2Ti+Ti+1)/h2
h
Apply equation at each internal point:
(Ti-1-2Ti+Ti+1)/h2 - b2 Ti = -b2 Ta Solve n-1 equations in n-1 unknowns