ORDINARY DIFFERENTIAL EQUATIONS

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ORDINARY DIFFERENTIAL EQUATIONS

• ENGR 351
• Numerical Methods for Engineers
• Southern Illinois University Carbondale
• College of Engineering
• Dr. L.R. Chevalier
• Dr. B.A. DeVantier

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Ordinary Differential Equationswhere to use them
The dissolution (solubilization) of a contaminant
into groundwater is governed by the
equation where kl is a lumped mass transfer
coefficient and Cs is the maximum solubility of
the contaminant into the water (a constant).
Given C(0)2 mg/L, Cs 500 mg/L and kl 0.1
day-1, estimate C(0.5) and C(1.0) using a
numerical method for ODEs.
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Ordinary Differential Equationswhere to use them
A mass balance for a chemical in a completely
mixed reactor can be written as where V is
the volume (10 m3), c is concentration (g/m3), F
is the feed rate (200 g/min), Q is the flow rate
(1 m3/min), and k is reaction rate (0.1
m3/g/min). If c(0)0, solve the ODE for c(0.5)
and c(1.0)
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Ordinary Differential Equationswhere to use them
Before coming to an exam Friday afternoon, Mr.
Bringer forgot to place 24 cans of a refreshing
beverage in the refrigerator. His guest are
arriving in 5 minutes. So, of course he puts the
beverage in the refrigerator immediately. The
cans are initially at 75?, and the refrigerator
is at a constant temperature of 40?.
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Ordinary Differential Equationswhere to use them
The rate of cooling is proportional to the
difference in the temperature between the
beverage and the surrounding air, as expressed by
the following equation with k 0.1/min. Use a
numerical method to determine the temperature of
the beverage after 5 minutes and 10 minutes.
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Ordinary Differential Equations

• A differential equation defines a relationship
  between an unknown function and one or more of
  its derivatives
• Physical problems using differential equations
• electrical circuits
• heat transfer
• motion

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Ordinary Differential Equations

• The derivatives are of the dependent variable
  with respect to the independent variable
• First order differential equation with y as the
  dependent variable and x as the independent
  variable would be
• dy/dx f(x,y)

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Ordinary Differential Equations

• A second order differential equation would have
  the form

does not necessarily have to include all of these
variables
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Ordinary Differential Equations

• An ordinary differential equation is one with a
  single independent variable.
• Thus, the previous two equations are ordinary
  differential equations
• The following is not

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Ordinary Differential Equations

• The analytical solution of ordinary differential
  equation as well as partial differential
  equations is called the closed form solution
• This solution requires that the constants of
  integration be evaluated using prescribed values
  of the independent variable(s).

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Ordinary Differential Equations

• An ordinary differential equation of order n
  requires that n conditions be specified.
• Boundary conditions
• Initial conditions

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Ordinary Differential Equations

• An ordinary differential equation of order n
  requires that n conditions be specified.
• Boundary conditions
• Initial conditions

consider this beam where the deflection is zero
at the boundaries x 0 and x L These are
boundary conditions
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consider this beam where the deflection is zero
at the boundaries x 0 and x L These are
boundary conditions
P
a
yo
In some cases, the specific behavior of a
system(s) is known at a particular time.
Consider how the deflection of a beam at x a is
shown at time t 0 to be equal to yo. Being
interested in the response for t gt 0, this is
called the initial condition.
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Ordinary Differential Equations

• At best, only a few differential equations can be
  solved analytically in a closed form.
• Solutions of most practical engineering problems
  involving differential equations require the use
  of numerical methods.

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Scope of Lectures on ODE

• One Step Methods
• Eulers Method
• Heuns Method
• Improved Polygon
• Runge Kutta
• Systems of ODE
• Adaptive step size control

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Scope of Lectures on ODE

• Boundary value problems
• Case studies

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Specific Study Objectives

• Understand the visual representation of Eulers,
  Heuns and the improved polygon methods.
• Understand the difference between local and
  global truncation errors
• Know the general form of the Runge-Kutta methods.
  
• Understand the derivation of the second-order RK
  method and how it relates to the Taylor series
  expansion.

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Specific Study Objectives

• Realize that there are an infinite number of
  possible versions for second- and higher-order RK
  methods
• Know how to apply any of the RK methods to
  systems of equations
• Understand the difference between initial value
  and boundary value problems

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Review of Analytical Solution
At this point lets consider initial
conditions. y(0)1 and y(0)2
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What we see are different values of C for the
two different initial conditions. The resulting
equations are
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One Step Methods

• Focus is on solving ODE in the form

h
y
yi1
slope f
yi
x
This is the same as saying new value old value
slope x step size
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Eulers Method

• The first derivative provides a direct estimate
  of the slope at xi
• The equation is applied iteratively, or one step
  at a time, over small distance in order to reduce
  the error
• Hence this is often referred to as Eulers
  One-Step Method

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Example
For the initial condition y(1)1, determine y for
h 0.1 analytically and using Eulers method
given
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Error Analysis of Eulers Method

• Truncation error - caused by the nature of the
  techniques employed to approximate values of y
• local truncation error (from Taylor Series)
• propagated truncation error
• sum of the two global truncation error
• Round off error - caused by the limited number of
  significant digits that can be retained by a
  computer or calculator

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....end of example
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Higher Order Taylor Series Methods

• This is simple enough to implement with
  polynomials
• Not so trivial with more complicated ODE
• In particular, ODE that are functions of both
  dependent and independent variables require
  chain-rule differentiation
• Alternative one-step methods are needed

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Modification of Eulers Methods

• A fundamental error in Eulers method is that the
  derivative at the beginning of the interval is
  assumed to apply across the entire interval
• Two simple modifications will be demonstrated
• These modification actually belong to a larger
  class of solution techniques called Runge-Kutta
  which we will explore later.

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Heuns Method

• Determine the derivative for the interval
• the initial point
• end point
• Use the average to obtain an improved estimate of
  the slope for the entire interval

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y
Use this average slope to predict yi1
xi xi1

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y
y
xi xi1
x
xi xi1
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y
x
xi xi1
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Improved Polygon Method

• Another modification of Eulers Method
• Uses Eulers to predict a value of y at the
  midpoint of the interval
• This predicted value is used to estimate the
  slope at the midpoint

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Improved Polygon Method

• We then assume that this slope represents a valid
  approximation of the average slope for the entire
  interval
• Use this slope to extrapolate linearly from xi to
  xi1 using Eulers algorithm

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Runge-Kutta Methods
Both Heuns and the Improved Polygon Method have
been introduced graphically. However, the
algorithms used are not as straight forward as
they can be. Lets review the Runge-Kutta
Methods. Choices in values of variable will give
us these methods and more. It is recommend that
you use this algorithm on your homework and/or
programming assignments.
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Runge-Kutta Methods

• RK methods achieve the accuracy of a Taylor
  series approach without requiring the calculation
  of a higher derivative
• Many variations exist but all can be cast in the
  generalized form

f is called the incremental function
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, Incremental Functioncan be interpreted as a
representative slope over the interval
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NOTE ks are recurrence relationships, that is
k1 appears in the equation for k2 which appears
in the equation for k3 This recurrence makes RK
methods efficient for computer calculations
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Second Order RK Methods
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Second Order RK Methods

• We have to determine values for the constants a1,
  a2, p1 and q11
• To do this consider the Taylor series in terms of
  yi1 and f(xi,yi)

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Now, f(xi , yi ) must be determined by the chain
rule for differentiation
The basic strategy underlying Runge-Kutta
methods is to use algebraic manipulations to
solve for values of a1, a2, p1 and q11
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By setting these two equations equal to each
other and recalling
we derive three equations to evaluate the four
unknown constants
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Because we have three equations with four
unknowns, we must assume a value of one of the
unknowns. Suppose we specify a value for
a2. What would the equations be?
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Because we can choose an infinite number of
values for a2 there are an infinite number of
second order RK methods. Every solution would
yield exactly the same result if the solution to
the ODE were quadratic, linear or a
constant. Lets review three of the most
commonly used and preferred versions.
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Consider the following Case 1 a2 1/2 Case
2 a2 1 These two methods have been
previously studied. What are they?
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Case 1 a2 1/2 This is Heuns Method with a
single corrector. Note that k1 is the slope
at the beginning of the interval and k2 is the
slope at the end of the interval.
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Case 2 a2 1 This is the Improved Polygon
Method.
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Ralstons Method Ralston (1962) and Ralston and
Rabinowitiz (1978) determined that choosing a2
2/3 provides a minimum bound on the truncation
error for the second order RK algorithms. This
results in a1 1/3 and p1 q11 3/4
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Example
Evaluate the following ODE using Heuns Methods
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Third Order Runge-Kutta Methods

• Derivation is similar to the one for the
  second-order
• Results in six equations and eight unknowns.
• One common version results in the following

Note the third term
NOTE if the derivative is a function of x only,
this reduces to Simpsons 1/3 Rule
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Fourth Order Runge Kutta

• The most popular
• The following is sometimes called the classical
  fourth-order RK method

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• Note that for ODE that are a function of x alone
  that this is also the equivalent of Simpsons
  1/3 Rule

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Example
Use 4th Order RK to solve the following
differential equation
using an interval of h 0.1
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Higher Order RK Methods

• When more accurate results are required, Buchers
  (1964) fifth order RK method is recommended
• There is a similarity to Booles Rule
• The gain in accuracy is offset by added
  computational effort and complexity

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Systems of Equations

• Many practical problems in engineering and
  science require the solution of a system of
  simultaneous differential equations

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• Solution requires n initial conditions
• All the methods for single equations can be used
• The procedure involves applying the one-step
  technique for every equation at each step before
  proceeding to the next step

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Boundary Value Problems

• Recall that the solution to an nth order ODE
  requires n conditions
• If all the conditions are specified at the same
  value of the independent variable, then we are
  dealing with an initial value problem
• Problems so far have been devoted to this type of
  problem

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Boundary Value Problems

• In contrast, we may also have conditions a
  different value of the independent variable.
• These are often specified at the extreme point or
  boundaries of as system and customarily referred
  to as boundary value problems
• To approaches to the solution
• shooting method
• finite difference approach

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General Methods for Boundary Value Problems
The conservation of heat can be used to develop a
heat balance for a long, thin rod. If the rod
is not insulated along its length and the system
is at steady state. The equation that results is
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Clearly this second order ODE needs 2
conditions. This can be satisfied by knowing the
temperature at the boundaries, i.e. T1 and T2
T(0) T1 T(L) T2
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Use these conditions to solve the equation
analytically. For a 10 m rod with Ta 20 T(0)
40 T(10) 200 h 0.01
T(0) T1 T(L) T2
Now that we have an analytical solution, lets
evaluate our two proposed numerical methods.
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Shooting Method
Given
We need an initial value of z. For the shooting
method, guess an initial value. Guessing z(0)
10
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Guessing z(0) 10
Using a fourth-order RK method with a step
size of 2, T(10) 168.38 This differs from the
BC T(10) 200 Making another guess, z(0) 20
T(10) 285.90 Because the original ODE is
linear, the estimates of z(0) are linearly
related.
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Using a linear interpolation formula between the
values of z(0), determine a new value of
z(0) Recall first estimate z(0) 10
T(20) 168.38 second estimate z(0)20
T(20) 285.90 What is z(0) that would
give us T(20)200?
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We can now use this to solve the first order ODE
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For nonlinear boundary value problems, linear
interpolation will not necessarily result in an
accurate estimation. One alternative is to apply
three applications of the shooting method and use
quadratic interpolation..
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Finite Difference Methods
The finite divided difference approximation
for the 2nd derivative can be substituted into
the governing equation.
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Collect terms
We can now apply this equation to each interior
node on the rod. Divide the rod into a grid, and
consider a node to be at each division. i.e..
D x 2m
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D x 2 m
T(0)
T(10)
L 10 m
Consider the previous problem L 10 m Ta
20 T(0) 40 T(10) 200 h 0.01
We need to solve for the temperature at the
interior nodes (4 unknowns). Apply the
governing equation at these nodes
(4 equations). What is the matrix?
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
Notice the labeling for numbering Dx and i
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
40
200
Note also that the dependent values are known at
the boundaries (hence the term boundary value
problem)
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
40
200
Apply the governing equation at node 1
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
40
200
Apply the equation at node 2
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
40
200
We get a similar equation at node 3
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x0 2 4 6
8 10
T(0)
T(10)
i0 1 2 3
4 5
At node 4, we consider the boundary at the right.
40
200
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For the four interior nodes, we get the
following 4 x 4 matrix
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Example
Consider the previous example, but with Dx1.
What is the matrix?
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Specific Study Objectives

• Understand the visual representation of Eulers,
  Heuns and the improved polygon methods.
• Understand the difference between local and
  global truncation errors
• Know the general form of the Runge-Kutta methods.
  
• Understand the derivation of the second-order RK
  method and how it relates to the Taylor series
  expansion.

80
Specific Study Objectives

• Realize that there are an infinite number of
  possible versions for second- and higher-order RK
  methods
• Know how to apply any of the RK methods to
  systems of equations
• Understand the difference between initial value
  and boundary value problems

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end of lecture on ODE