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Introduction: Numerical methods for differential equations are of great

importance to the engineer because practical problemsoften lead to differential equations that cannot be solvedaccurately to get exact results.

These are a group of implicit and explicit methods used forapproximation of solutions of Ordinary Differentialequations.

The methods are classified as :

ExplicitEmbeddedImplicit

We shall mainly talk about the explicit method used to findthe solution of ODE.

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Explicit method: Forward Euler method

Generic second-order method

Kuttas third-order method

Classic fourth-order method

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Engineering Applications Control Systems

Laplace transforms

Electrical equations Chemical engineering

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Formulae

34

23

12

1

4321

,

2

1,

2

1

2

1,

2

1

,

226

1

kyhxfhk

kyhxfhk

kyhxfhk

yxfhk

kkkky

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Problem :1) Using Runge Kutta fourth order method, find y(0.2) for theequation dy/dx= (y-x)/(y+x), y(0)=1 taking h=0.2

>> By data f(x,y) = (y-x)/(y+x), x0=0, y0=1, h=0.2We shall first compute k1, k2, k3, k4

k1 = h f(x0,y0) = (0.2) f(0,1) = (0.2) [(1-0)/(1+0)] = 0.2

k2 = h f(x0 + h/2 , y0 +k1/2) = (0.2) f(0.1, 1.1) =(0.2) [(1.1-0.1)/ (1.1+0.1)] = 0.1667

k3 = h f(x0 +h/2 , y0 +k2/2) = (0.2) f(0.1, 1.0835) =(0.2) [(1.0835-0.1)/ (1.0835+0.1)] = 0.1662

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k4 = h f(x0 + h , y0 +k3) = (0.2) f(0.2, 1.1662) =

(0.2) [(1.1662-0.2)/ (1.1662+0.2)] = 0.1414

We have, y(x0 + h) = y0 + 1/6 (k1 +k2 + k3 + k4 )

Therefore,

y(0.2) = 1 + 1/6 [(0.2 + 2(0.1667) + 2(0.1662) + 0.1414]

Thus y(0.2) = 1.1679

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2) Use Runge Kutta fourth order method to find y at x=0.1 given

that dy/dx= 3ex + 2y, y(0)=0 taking h=0.1

>> By data f(x, y) = 3ex + 2y, x0=0, y0=0, h=0.1

We shall first compute k1, k2, k3, k4

k1 = h f(x0,y0) = (0.1) f(0,0) = (0.1) [3e0

+ 2 0] = 0.3k2 = h f(x0 + h/2 , y0 +k1/2) = (0.1) f(0.05, 0.15)

= (0.1) [3e0.05 + 2(0.15)] = 0.3454

k3 = h f(x0 +h/2 , y0 +k2/2) = (0.1) f(0.05, 0.1727)

= (0.1) [3e0.05 + 2(0.1727)] = 0.3499

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k4 = h f(x0 + h , y0 +k3) = (0.1) f(0.1, 0.3499)

= (0.1) [3e0.1 + 2(0.3499)] = 0.4015

We have, y(x0 + h) = y0 + 1/6 (k1 +2k2 +2 k3 + k4 )

Therefore,

y(0.1) = 0 + 1/6 [(0.3 + 2(0.3454) + 2(0.3499) + 0.4015]

Thus y(0.2) = 0.3487

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Engineering Example:A 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 thetemperature of the ball is given by

Find the temperature at t= 480 seconds seconds using Runge

Kutta 4th order method. Assume a step size of h= 240 seconds.

Kdt

d12000,1081102067.2

8412

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Step 1:

8412 1081102067.2 dt

d

84121081102067.2, tf

hkkkkii 43211 226

1

1200)0(,0,000

ti

5579.410811200102067.21200,0, 841201

ftfk o

38347.0108105.653102067.205.653,120

2405579.42

11200,240

2

10

2

1,

2

1

8412

1002

f

fhkhtfk

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Continued..

8954.310810.1154102067.20.1154,120

24038347.02

11200,240

2

10

2

1,

2

1

8412

2003

f

fhkhtfk

0069750.0108110.265102067.210.265,240240984.31200,2400,

8412

3004

f

fhkhtfk

K

hkkkk

65.675

2401848.26

11200

240069750.08954.3238347.025579.46

11200

2261 432101

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1 is the approximatetemperature at t1= t0+ h= 0+ 240=240

( 240) = 1 = 675.65K

Step 2:Kti 65.675,240,1

11

44199.0108165.675102067.265.675,240, 8412111

ftfk

31372.0108161.622102067.261.622,360

24044199.0

2

165.675,240

2

1240

2

1,

2

1

8412

1112

f

fhkhtfk

34775.0108100.638102067.200.638,360

24031372.02

165.675,240

2

1240

2

1,

2

1

8412

2113

f

fhkhtfk

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2 is the approximate temperature at t2= t1 +h =240+240=480

(480) = 2 = 594.91K

25351.0108119.592102067.219.592,48024034775.065.675,240240,

8412

3114

f

fhkhtfk

K

hkkkk

91.594

2400184.26

165.675

24025351.034775.0231372.0244199.06

165.675

2261

432112

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Comparison with other methods

0

200

400

600

800

1000

1200

1400

0 100 200 300 400 500

Time, t(sec)

Temperature,

(K)

Exact

4th order

Heun

Euler

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Conclusion The solutions obtained from range kutta method is

almost accurate when compared with that of othermethods.

The graph in the previous slide also depicts that rangeKutta methods solution is almost near exact solution.

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Bibliography Engineering mathematics by Dr K.S.C

Wikipedia

http://numericalmethods.eng.usf.edu/topics/runge_kutta_4th_method.html

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Queries ???

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Thank you