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Numerical Analysis - Some Formulae Post-Graduate, Biomedical Engineering Partial Derivatives u u i 1, j  u i , j  x x u u i 1, j  u i 1, j  x 2x u u i , j  u i 1, j  x x

 2u u i 1, j  2u i , j  u i 1, j  x 2 2x

(Forward)

You can deduce similar expressions for

(Centered)

u  2u . and y y 2

(Backward)

Ordinary Differential Equations Runge-Kutta 4th – order:

Heun's Method:

1  k 1  2k 2  2k 3  k 4  h 6 k 1  f (x i , y i )

y i01  y i  f ( x i , y i )h

y i 1  y i 

y i 1  y i 

f ( x i , y i )  f ( x i 1 , y i01 ) h 2

1 1 k 2  f ( x i  h , y i  k 1h ) 2 2 1 1 k 3  f (x i  h , y i  k 2h ) 2 2 k 4  f ( x i  h , y i  k 3h )

Numerical Integration Gauss Quadrature : N

Weights

Function Arguments

1.0000000

-0.577350269

1.0000000

-0.577350269

0.5555556

-0.774596669

0.8888889

0.0

0.5555556

0.774596669

2

3


Simpson's Rule:

I 

 h  f (x 0 )  4  f (x i )  2  f (x j )  f (x n )  , 3 i 1,3,5,... j 2,4,6,... 

(b  a )5 Ea   f 180 n 4

Trapezoidal Rule:

I 

n 1 h  f ( x )  2 f (x i )  f (x n )  ,  0  2 i 1 

Ea  

(b  a )3 f 12 n 2

(2)

Numerical Differentiation Forward Difference: First Derivative f ( x i 1 )  f ( x i ) f ' (x )  O (h ) h f ( x i  2 )  4f ( x i 1 )  3f ( x i ) f ' (x )  O (h 2 ) 2h Second Derivative f '' ( x ) 

f ( x i  2 )  2f ( x i 1 )  f ( x i ) O (h ) h2

Centered Difference:

First Derivative f ( x i 1 )  f ( x i 1 ) f ' (x )  O (h 2 ) 2h Second Derivative f ( x i 1 )  2f ( x i )  f ( x i 1 ) f '' ( x )  O (h 2 ) 2 h Backward Difference: First Derivative f ( x i )  f ( x i 1 ) f ' (x )  O (h ) h 3f ( x i )  4f ( x i 1 )  f ( x i 2 ) f ' (x )  O (h 2 ) 2h Second Derivative f '' ( x ) 

f ( x i )  2f ( x i 1 )  f ( x i 2 ) O (h ) h2

(4)

Formulae - Numerical Analysis  

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