Differential Equations - Personal Web PagesITCS 4133/5133: Numerical Comp. Methods/Analysis 2 Ordinary Differential Equations An equation that deﬁnes a relationship between an unknown

Differential Equations

� Definitions

� Finite Differences

� Taylor Series based Methods: Euler Method

� Runge-Kutta Methods

⇒ Improved Euler, Midpoint methods⇒ Runge Kutta (2nd, 4th order) methods

� Predictor-Corrector Methods

⇒ Euler-Trapezoidal, Milne Simpson Methods

ITCS 4133/5133: Numerical Comp. Methods/Analysis 1 Ordinary Differential Equations

Motivation

� Engineering problems require estimates of derivatives of functionsfor analysis

� Approaches:

1. Use function differences between neighboring points, divided bydistance between the points,

2. Fit a function to the relationship between the independent anddependent variable (say an nth order polynomial) and use itsderivative


Differential Equations

An equation that defines a relationship between an unknown function and oneor more derivatives

dy

dx= f (x, y)

d2y

dx2= f

(x, y,

dy

dx

)Definitions:

� Order: is the order of the highest derivative

� f (.) may be a function any combination of x, y, and (in case of sec-ond order) dy/dx

� Ordinary Diff. Eq.: f (.) is a function of a single variable


Definitions(contd)

� Linearity: determined by whether f (.) is linear in x, y, dy/dx,

� Solution: is a function of the independent variable.

� Boundary Conditions: constraints placed on the solution space.


Example Applications

1. Electrical Circuit: Relationship between current and time

Ldi

dt+ Ri = E, i = 0 at t = 0

2. 1D Heat Flow:

H = KAdT

dr

where K is the coeff. of thermal conductivity, H is the quantity ofheat, A is the area perpendicular to heat flow, T is the temperature.


Example Applications


Reminder:Finite Differences

Forward Difference

df (x)

dx=f (x + ∆x)− f (x)

∆x

Backward Difference

df (x)

dx=f (x)− f (x−∆x)

∆x

Two Step Method

df (x)

dx=f (x + ∆x)− f (x−∆x)

2∆x


Taylor Series Based Methods:Euler’s Method

Given

y′ = f (x, y)

treat f (x, y) as a constant and the derivative as a tangent (quotient)

y′ =y1 − y0

x1 − x0

Thus,

y1 − y0 = f (x0, y0)(x1 − x0)

y1 = y0 + hf (x0, y0)

where h = (b− a)/n, n is the number of values of x.


Taylor Series Based Methods:Euler’s Method

y1 = y0 + hf (x0, y0)


Euler’s method: From Taylor Series

� Taylor’s series, truncated to the first term is given by

y(x + h) = y(x) + hy′(x) +h2

2y′′(η)

� Euler’s method follows, since y(x + h) = yi+1, y(x) = yi, y′(x) =

f (xi, yi), x < η < x + h


Euler’s Method: Example 1

◦ y′ = x + y, 0 ≤ x ≤ 1, y(0) = 2


Euler’s Method: Example 2

◦ y′ =

{y(−2x + 1/x), x 6= 0

1 x = 0

where y(0) = 0.0, 0 ≤ x ≤ 2.0.


Euler’s Method: Algorithm


Euler’s Method: Notes, Errors

� dydx

is evaluated at beginning of interval

� Error e increases with the width of (x − x0), as higher order termsbecome more important

� Also known as one-step Euler method

� Local Error: Range over a single step size; measure difference be-tween numerical solution at end of step (starting with exact solutionat beginning of the step) and the exact solution at end of step.

� Global Error: Accumulates over the range of the solution; measuredas the difference between numerical and exact solutions.

� Errors using Euler’s method can be approximated using the secondorder term of the Taylor series:

ε =h2

2y′′(η)


Runge-Kutta Methods

� To improve on Euler’s method, we can use additional terms of theTaylor series.

� Problem: Need to compute additional higher order derivatives, whichcan be problematic for complex functions.

� Runge-Kutta methods determine the y value (dependent variable)based on the value at the beginning of the interval, step size andsome representative slope over the interval

� Euler’s and the mnodified Euler’s methods are special cases of thesetechniques

� Runge-Kutta methods are classified based on their order; fourth or-der is the most commonly used.

� Higher order derivates are not required of these methods


Second Order Runge-Kutta Methods

General Form:

k1 = hf (xn, yn)//slope at beginning of intervalk2 = hf (xn + c2h, yn + a21k1)//slope at end of interval

Iteration:

yn+1 = yn + w1k1 + w2k2

Example Methods:

◦ Improved Euler: c2 = 1, a21 = 1, w1 = w2 = 0.5

◦ Midpoint: c2 = 1, a21 = 2/3, w1 = 0, w2 = 1

◦ Heun: c2 = 2/3, a21 = 2/3, w1 = 1/4, w2 = 3/4


Second Order Runge Kutta Methods: MidpointMethod

� Idea: Approximate the value of y at (x + h/2) by summing currentvalue of y and one-half the change in y from Euler’s method:

k1 = hf (xi, yi)// change in y : Euler’s methodk2 = hf (xi + h/2, yi + k1/2)// change in y: slope est. at midpoint

Iteration:

yi+1 = yi + k2


Runge Kutta Methods:Improved Euler’sMethod

� A second order Runge-Kutta method.

� Estimates of y′ at start and midpoint of interval are averaged to pro-duce a revised estimate of y at end of interval.

� Procedure:

k1 = hf (xn, yn)

k2 = hf (xn + h, yn + k1)

yn+1 = yn +1

2k1 +

1

2k2


Improved Euler’s Method: Procedure

⇒ Evaluate y′ at start of interval

⇒ Estimate y at end of interval using Euler’s method

⇒ Evaluate y′ at end of interval

⇒ Compute average slope

⇒ Compute a revised y at end of interval using average slope


Example : Midpoint and Improved Euler’sMethods


Classic Runge-Kutta Method: Fourth Order

A commonly used class of Runge-Kutta methods.

k1 = hf (xi, yi)

k2 = hf (xi + 0.5h, yi + 0.5k1)

k3 = hf (xi + 0.5h, yi + 0.5k2)

k4 = hf (xi + 0.5h, yi + k3)

Iteration:

yi+1 = yi +1

6(k1 + 2k2 + 2k3 + k4)


Classic Runge-Kutta Method: Algorithm


Classic Runge-Kutta Method: Example 1


Classic Runge-Kutta Method: Example 2


Predictor-Corrector Methods

� Euler’s method and Runge-Kutta methods generally require stepsizes to be small, else might not yield precise solutions

� Predictor-Corrector methods can be used to increase the accuracyof solutions

� These methods use solutions from previous intervals to project tothe end of the next interval, followed by iterative refinement.

� Disadvantage: Requires values from previous intervals - one-stepmethods such as the Euler’s method have to be used.

� Predictor: Gets an initial estimate at the end of the interval.

� Corrector: Improves the estimate by iteration.


Euler-Trapezoidal Method

� Uses Euler’s method as predictor and Trapezoidal rule as cor-rector

Predictor

yi+1,0 = yi,∗ + hdy

dx

∣∣∣∣i,∗

Corrector

yi+1,j = yi,∗ +h

2

[dy

dx

∣∣∣∣i,∗

+dy

dx

∣∣∣∣i+1,j−1

]


Euler-Trapezoidal Method: Example


Euler-Trapezoidal Method: Example


Milne-Simpson Method

� Uses Milne’s method as predictor and Simpson’s rule as cor-rector

Predictor

yi+1,0 = yi−3,∗ +4h

3

[2dy

dx

∣∣∣∣i,∗− dy

dx

∣∣∣∣i−1,∗

+ 2dy

dx

∣∣∣∣i−2,∗

]Corrector

yi+1,j = yi−1,∗ +h

3

[dy

dx

∣∣∣∣i+1,j−1

+ 4dy

dx

∣∣∣∣i,∗

+dy

dx

∣∣∣∣i−1,∗

]


Milne-Simpson Method: Example


Milne-Simpson: Example



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Differential Equations - Personal Web PagesITCS 4133/5133: Numerical Comp. Methods/Analysis 2 Ordinary Differential Equations An equation that deﬁnes a relationship between an unknown