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These lecture slides are delivered at The LNM Institute of Information Technology by Dr. Sham Thakur for subject of Mathematical Modeling and Simulation. Its main points are: Numerical, Derivatives, Differentiation, Integration, Methods, Forward, Difference, Backward, Errors
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2
b
a
i i x
i i
0
y
x
f(xi)
f(xi+ x)
x
y
x
f(xi)
f(xi+ x)
x
y
x
f(xi)
f(xi+ x)
x
xi
f(xi)
x
Finally, we can
see that the
derivative forms
i i x
i i
0
neighborhood of xo when the derivatives of all order are known.
if the values of function are known at a number of discrete points, { xi } where, i = 1, 2,... N.
h is a small value.
following:
( ) 2
( ) ( ) ( )
2 o o o f x o
h f x h f x hf x
solve for first derivative we obtain an approximation:
h) and f(xo). By defining the forward differences on the right side is expressed by
( ) ( ) 1 ( (^) o ) f xo h f xo h
f x
f ( xo ) f ( xo h ) f ( xo )
o
o o o x
f x f x h f x h
( ) ( ) ( ) 1
( 0. 3 ) ( 0. 1 ) 2
1
f x f x dx h
df x
= 18.5 gm/cm
( 0. 4 ) ( 0. 2 ) 2
1
f x f x dx h
df x
2 ( 0. 1 )
( 7. 2 4. 3 )
dx x
df = 14.5gm/cm
Let us first calculate the derivative at x = 0.2,
( 0. 1 ) 2 ( 0. 2 ) ( 0. 3 )
2
2
2
f x f x f x dx h
d f
x
2. 1 2. 0 4. 3 5. 8 (^70). ( 0. 1 )
2
2
2 dx x
d f
( 0. 2 ) 2 ( 0. 3 ) ( 0. 4 ) 10.
1 2
2
2
f x f x f x dx h
d f
x
and the second derivative at x = 0.2 is given as:
Then the second derivative at x= 0.3 is approximately equal to:
These estimates can also be found using forward and backward difference forms.
f(x) 2 x^5 2 x^4 5 x^3 10 x 1
( ) 10 8 15 10 4 3 2 f ^ x x x x
( yi 1 yi)/(xi 1 xi) i 1 , 2 ,
.