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8/2/2019 Runge Kutta Method. 1
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8/2/2019 Runge Kutta Method. 1
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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
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