part 7 ordinary differential equations odes. ordinary differential equations part 7 equations which...
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PART 7PART 7Ordinary Differential Equations Ordinary Differential Equations
ODEsODEs
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Ordinary Differential EquationsPart 7
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
vm
cg
dt
dv
v - dependent variable
t - independent variable
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Ordinary Differential Equations
• When a function involves one dependent
variable, the equation is called an ordinary
differential equation (ODE).
•A partial differential equation (PDE) involves
two or more independent variables.
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Ordinary Differential Equations
Differential equations are also classified as to
their order:1. A first order equation includes a first derivative as
its highest derivative.
- - Linear 1Linear 1stst order ODE order ODE
- Non-Linear 1- Non-Linear 1stst order ODE order ODE
Where f(x,y) is nonlinearWhere f(x,y) is nonlinear
)(xfydx
dy
),( yxfdx
dy
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Ordinary Differential Equations
)(),(),( xfyyxQdx
dyyxp
dx
yd2
2
2. A second order equation includes a second
derivative.
- - Linear 2Linear 2ndnd order ODE order ODE
- Non-Linear 2nd order ODE- Non-Linear 2nd order ODE
• Higher order equations can be reduced to a system
of first order equations, by redefining a variable.
)(xfQydx
dyp
dx
yd2
2
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Ordinary Differential Equations
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Runge-Kutta Methods This chapter is devoted to solving ODE of the form:
• Euler’s Method
solution
),( yxfdx
dy
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Runge-Kutta Methods
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Euler’s Method: Example
58x20x12x2dx
dy 23 .
Obtain a solution between x = 0 to x = 4
with a step size of 0.5 for:
Initial conditions are: x = 0 to y = 1
Solution:
1255505801200112012255
50875501f01y02y
8755505850205012502255
5025550f50y01y
25550x5801
5010f0y50y
23
23
.).).(.).().().((.
).).(.,.().().(
.).).(.).().().((.
).).(.,.().().(
....
).).(,()().(
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Euler’s Method: Example• Although the computation captures the general trend
solution, the error is considerable.• This error can be reduced by using a smaller step size.
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Improvements of Euler’s method• A fundamental source of error in Euler’s
method is that the derivative at the beginning of the interval is assumed to apply across the entire interval.
• Simple modifications are available:– Heun’s Method– The Midpoint Method– Ralston’s Method
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Runge-Kutta Methods• Runge-Kutta methods achieve the accuracy of a Taylor series
approach without requiring the calculation of higher derivatives.
1
1 1 2 2
1
2 1 11 1
3 3 21 1 22 2
1 1 1 1,2 2 1, 1 1
( , , )
' constants
( , )
( , )
( , )
( , )
' and ' are constants
i i i i
n n
i i
i i
i i
n i n i n n n n n
y y x y h h
a k a k a k
a s
k f x y
k f x p h y q k h
k f x p h y q k h q k h
k f x p h y q k h q k h q k h
p s q s
Increment function (representative slope over the interval)
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Runge-Kutta Methods• Various types of RK methods can be devised by employing
different number of terms in the increment function as specified by n.
1. First order RK method with n=1 is Euler’s method.– Error is proportional to O(h)
2. Second order RK methods:- Error is proportional to O(h2)
• Values of a1, a2, p1, and q11 are evaluated by setting the second order equation to Taylor series expansion to the second order term.
),(
),(
)(
11112
1
22111
hkqyhpxfk
yxfk
hkakayy
ii
ii
ii
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Runge-Kutta Methods
2
12
1
1
112
12
21
qa
pa
aa
• Three equations to evaluate the four unknown constants are derived:
A value is assumed for one of the unknowns to solve for the other three.
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Runge-Kutta Methods
2
1,
2
1,1 1121221 qapaaa
• We can choose an infinite number of values for a2,there are an infinite number of second-order RK methods.
• Every version would yield exactly the same results if the solution to ODE were quadratic, linear, or a constant.
• However, they yield different results if the solution is more complicated (typically the case).
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Runge-Kutta MethodsThree of the most commonly used methods are:
• Huen Method with a Single Corrector (a2=1/2)
• The Midpoint Method (a2= 1)
• Raltson’s Method (a2= 2/3)
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Runge-Kutta Methods• Heun’s Method: Involves the determination
of two derivatives for the interval at the initial point and the end point.
y
f(xi,yi)
xi xi+hx
f(xi+h,yi+k1h)
y
a
xi xi+hx
Slope: 0.5(k1+k2)
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Runge-Kutta Methods• Midpoint Method:Uses Euler’s method to predict a value of y at the midpoint of the interval:
f(xi+h/2,yi+k1h/2)
y
a
xi xi+hx
Slope: k2
y
f(xi,yi)
xi xi+h/2x
Chapter 25
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Runge-Kutta Methods• Ralston’s Method:
x
f(xi+ 3/4 h, yi+3/4k1h)
xi+h x
y
f(xi,yi)xi xi+3/4h
y
a
xi
Slope: (1/3k1+2/3k2)
Chapter 25
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Chapter 25 22
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Runge-Kutta Methods
3. Third order RK methods
Chapter 25
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Runge-Kutta Methods4. Fourth order RK methods
hkkkkyy ii )22(6
143211
),(
)2
1,
2
1(
)2
1,
2
1(
),(
33
23
12
1
hkyhxfk
hkyhxfk
hkyhxfk
yxfk
where
ii
ii
ii
ii
Chapter 25
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Comparison of Runge-Kutta Methods
Use first to fourth order RK methods to solve the equation from x = 0 to x = 4
Initial condition y(0) = 2, exact answer of y(4) = 75.33896
yeyxf x 5.04),( 8.0
Chapter 25