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ScheduleWeek Date Topic Classification of Topic

1 9 Feb. 2010 Introduction toNumerical Methods

and Type of Errors

Measuring errors, Binary representation,

Propagation of errors and Taylor series

2 14 Feb. 2010 Nonlinear Equations Bisection Method

3 21 Feb. 2010 Newton-Raphson Method

4 28 Feb. 2010 Interpolation Lagrange Interpolation

5 7 March 2010 Newton's Divided Difference Method

6 14 March 2010 Differentiation Newton's Forward and BackwardDivided Difference

7 21 March 2010 Regression Least squares

8 28 March 2010 Systems of LinearEquations

Gaussian Jordan

9 11 April 2010 Gaussian Seidel

10 18 April 2010 Integration Composite Trapezoidal and SimpsonRules

11 25 April 2010 Ordinary DifferentialEquations

Euler's Method

12 2 May 2010

Runge-Kutta 2nd and4th order Method

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2

Exact solution to Separable andLinear differential equations

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Ordinary differential equations of order one and degree one

1. Separable equations

Example 1

Solve the equation2

dy x

dx y given that (0) 1y

solution

By separating the variables of the equation we get

2y dy xdx

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By integration

2y dy x dx then

3 2

3 2y x C The general solution of the equation

Now (0) 1y y=1 at x=0

then1 0

3 2C

1

3C

the solution of the equation is

3 2 1

3 2 3

y x

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2. Linear equations

Linear equations are on the form

( ) ( )

dy

p x y q xdx

Example

212

dy y xdx x

2

1( )

2( )

p x

x

q x x

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6

How to solve Linear equations

Linear equations are solved by using the following method

Step 1: Calculate the following integrating factor

( )p x dx

e

Step 2: The solution is then given by

( )y q x dx

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Example 2

Solve the equation such that y(0)=244 ydx

dy

solution

This equation could be written on the form

44 ydx

dy

4)(

4)(

xq

xp

Thenxdxdxxp eee 4

4)(

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The solution will be such that

( )y q x dx cedxeye xxx

444 4

Let x=0 , y=2 c=3

Thenxey 431

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Euler Method

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Eulers Method

Step size, h

x

y

x0,y0

True value

y1, Predicted

value

00,, yyyxfdx

dy

Slope

Run

Rise

01

01

xx

yy

00 ,yxf

010001 , xxyxfyy

hyxfy 000 ,Figure 1 Graphical interpretation of the first step of Eulers method

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Eulers Method

Step size

h

True Value

yi+1, Predicted value

yi

x

y

xi xi+1

Figure 2. General graphical interpretation of Eulers method

hyxfyy iiii ,1

ii xxh 1

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How to write Ordinary Differential

Equation

Example

50,3.12 yeydx

dy x

is rewritten as 50,23.1 yye

dx

dy x

In this case

yeyxf x 23.1,

How does one write a first order differential equation in the form of

yxfdx

dy,

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ExampleA ball at 1200K is allowed to cool down in air at an ambient temperature of

300K. Assuming heat is lost only due to radiation, the differential equation for

the temperature of the ball is given by

Kdtd 12000,1081102067.2 8412

Find the temperature at 480t seconds using Eulersmethod. Assume a step size of

240h seconds.

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Solution

K

f

htf

htf iiii

09.106

2405579.4120024010811200102067.21200

2401200,01200

,

,

8412

0001

1

Step 1:

1is the approximate temperature at 240240001 httt

K09.106240 1

8412 1081102067.2 dt

d

8412

1081102067.2,

tf

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Solution ContFor 09.106,240,1 11 ti

K

f

htf

32.110

240017595.009.106

240108109.106102067.209.10624009.106,24009.106

,

8412

1112

Step 2:

2 is the approximate temperatureat 48024024012 httt

K32.110480 2

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Comparison of Exact and

Numerical Solutions

Figure 3. Comparing exact and Eulers method

0

200

400

600

800

1000

1200

1400

0 100 200 300 400 500

Time, t(sec)

Temperature,

h=240

Exact Solution

(K

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Step, h (480) Et |t|%

480

240

120

60

30

987.81

110.32

546.77

614.97

632.77

1635.4

537.26

100.80

32.607

14.806

252.54

82.964

15.566

5.0352

2.2864

18

Effect of step sizeTable 1. Temperature at 480 seconds as a function of step size, h

K57.647)480( (exact)

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Comparison with exact results

-1500

-1000

-500

0

500

1000

1500

0 100 200 300 400 500

Time , t (sec)Temperature,

Exact solution

h=120h=240

h=480

(K

Figure 4. Comparison of Eulers method with exact solution for different step sizes

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Effects of step size on Eulers

Method

-1200

-800

-400

0

400

800

0 100 200 300 400 500

Step size, h (s)Temperature,

(K

Figure 5. Effect of step size in Eulers method.

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