Presentation topicWeddle’s rule

Course titleNumerical Analysis -II:

Course code: Math-402

Presented To: Ma’am Samra

Presented By: Group-5

Group Members

NAME: ROLL-NUMBER:

1. Maria Nawaz 11031509-009

2. Rida Munir 11031509-011

3. Iqra Mushtaq 11031509-029

4. Asma Jamil 11031509-037

5. Farwa Saba 11031509-059

6. Fakhra Mubeen 11031509-068

Team work Data searched by:

Rida Munir,Asma Jamil And Farwa

Data composed by:

Asma Jamil and Farwa Saba

Data presented by:

Rida Munir

Query satisfation will be done by:

Rida Munir

Data will be explained by:

Asma Jamil,Farwa Saba,Rida Munir, Maria Nawaz

Iqrq Mushtaq & Fakhra Mubeen

HISTORY: Thomas Weddle (1817 November 30

Stamfordham, Northumberland–1853 December 4 Bagshot) was a mathematician who introduced the Weddle surface. He was mathematics professor at the Royal Military College at Sandhurst.

Weddle's Rule is a method of integration, the Newton-Cotes formula with N=6

INTRODUCTION:

Numerical integration is the process of computing the value of definite integral from a set of numerical values of the integrand. The process is sometimes referred as mechanical quadrature. The approximate values of the definite integral which are either impossible to be computed analytically or for which anti derivative is too complex to be used.

The inverse process to differentiation in calculus is represented by

I =∫f (x)dx

Weedle’s rule :

Let the values of a function f(x) be tabulated at points xi equally spaced by h= xi+1- xi so the f1=f(x1),f2=f(x2),… . Then Weddle’s rule approximating the integral of f(x) is given by putting n = 6 in the Newton Cote’s formula. For Weddle’s rule the number of subinterval should be taken as multiple of 6.

Since we take n = 6 means f(x) can be approximated by a polynomial of 6th degree so that seventh and higher order differences are vanishes in the Newton Cote’s formula.

GRAPHICAL REPRESENTATION:

Illustration of Weddle’s rule

FORMULA:

.ff5ff6ff5f10

h3dxf(x)I 6543210

b

a

6

abh

Where n = 6,

DERIVATION: To find the area we consider the integral as follows

I=∫f(x)dx=∫f(x)dx=∫f(x)dx

Where xi =a+ih and h=b-a/6

Substituting n=6 in the above equation,we get

I=6h[f(a)+3∆f(a)+9/2∆^2f(a)+4∆^3f(a)+1/24×246/5∆^4f(a)+1/120×66∆^5f(a)+1/720×246/7∆^6f(a)]

I =6h/20 [ 20 f(a) + 60 {f(a+h)-f(a)} + 90 {f(a+2h)-2f(a+h)+f(a)} + 80 {f(a+3h)-3f(a+2h)+3f(a+h)-f(a)}

+ 41 { f(a+4h)-4f(a+3h)+6 f(a+2h) -4f(a+h)+f(a)} + 11 { f(a+5h)-5 f(a+4h) +10 f(a+3h) -10 f(a+2h)+ 5 f(a+h) – f(a) } + { f( a+6h) -

6f(a+5h) +15 f(a+4h)- 20 f(a+3h) +15 f(a+2h) – 6 f(a+h) + f(a) } ]

After simplifying we get Weddle’s rule as follows

I = 8h/10[ f(a) +5 f(a+h) + f(a+2h) + 6 f(a+3h) + f(a+4h) +5 f(a+5h) + f(a+6h) ]

ExampleEvaluate

.xdx2sineI0.5

0.2

x Using Weddl’s rule

Solution

05.06

0.20.5

6

abh

Calculate the values of x2sinef(x) x

x 0.2 0.25 0.3 0.35 0.4 0.45 0.5

f(x) 0.475 0.165 0.716

0.914 1.070 1.228 1.387

.ff5ff6ff5f10

h3dxf(x)I 6543210

b

a

.2760.03873.1)2285.1(50702.1)9142.0(67622.0)6156.0(54756.010

)05.03(I

APPLICATIONS

1.Weddle’s rule is very used when have to solve multiple integrals.

2.It gives more accurate solution as compare to any other formulas of Quadrature rule.

3.Weddle’s rule is very useful whenever you have a definite integral that can not be evaluated in closed form,or that is very difficult to evaluate in closed form. Since most “real world” integrals fall into one of these two categories.

4.weddles rule is one of the most useful integration techniques,especially when you have a working knowledge of programming language like C or MATLAB.

APPLICATION: 5. One application relates to power and energy. Power

is the rate of change of energy. (Some people prefer to think of it as the "flow" of energy.) It can be written as

P = dE/dt

If you multiply both sides of this equation by dt, you have

(dt)P = dE

dE = Pdt

APPLICATIONS:

Integrating both sides of the equation gives you

∫dE = ∫(P)dt

E = ∫(P)dt

Imagine you have

P = ((sint)^2 e^-t )/ln(t)   Watts

SUGESTIONS: How would you integrate

something like this? How would you integrate

something like this? Well... chances are, you would need to use a computer. How does the computer solve it? It uses a numerical technique such as Weddle's rule.

REFERENCES: John.L.Gidley, Howard F. Rase (1955):

Numerical integration:A tool for chemical engineers, J. Chem. Educ. , 32(10),535.

Golub G.H., Welsch J. H. (1969): Calculation of Gaussian quadrature rules,Mathematics of computation,23, 221-230.

Lessels, Jonathan D (1982): Numerical calculation of Mean mission duration, Reliability IEEE transaction Vol R- 31, Issue 5, 420-422