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ORDINARY DIFFERENTIAL EQUATIONS Student Notes. ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier. Photo Credit: Mr. Jeffrey Burdick. Ordinary Differential Equations… where to use them.

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Ordinary differential equations student notes

ORDINARY DIFFERENTIAL EQUATIONSStudent Notes

ENGR 351

Numerical Methods for Engineers

Southern Illinois University Carbondale

College of Engineering

Dr. L.R. Chevalier

Dr. B.A. DeVantier


Ordinary differential equations student notes

Photo Credit: Mr. Jeffrey Burdick


Ordinary differential equations where to use them

Ordinary Differential Equations…where 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 ODE’s.


Ordinary differential equations where to use them1

Ordinary Differential Equations…where 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)


Ordinary differential equations where to use them2

Ordinary Differential Equations…where to use them

Before coming to an exam Friday afternoon, Mr. Jones forgot to place 24 cans of a refreshing beverage in the refrigerator. His guests 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.


Ordinary differential equations where to use them3

Ordinary Differential Equations…where 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.


Ordinary differential equations

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

    • contaminant transport


Ordinary differential equations1

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)


Ordinary differential equations2

Ordinary Differential Equations

  • A second order differential equation would have the form:

}

does not necessarily have to include

all of these variables


Ordinary differential equations3

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:


Ordinary differential equations4

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).


Ordinary differential equations5

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


Ordinary differential equations student notes

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 > 0, this is called the initial condition.


Ordinary differential equations6

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.


Scope of lectures on ode

Scope of Lectures on ODE

  • One Step Methods

    • Euler’s Method

    • Heun’s Method

    • Improved Polygon

    • Runge Kutta

    • Systems of ODE

  • Boundary Value Problems


Specific study objectives

Specific Study Objectives

  • Understand the visual representation of Euler’s, Heun’s 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


Specific study objectives1

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


Review of analytical solution

Review of Analytical Solution

At this point lets consider

initial conditions.

y(0)=1

and

y(0)=2


Ordinary differential equations student notes

What we see are different

values of C for the two

different initial conditions.

The resulting equations

are:


One step methods

One Step Methods

  • Focus is on solving ODE in the form

y

xi, yi

x


One step methods1

One Step Methods

  • Focus is on solving ODE in the form

y

xi, yi

x


One step methods2

One Step Methods

  • Focus is on solving ODE in the form

h

y

xi, yi

x

This is the same as saying:

new value = old value + slope x step size


One step methods3

One Step Methods

  • Focus is on solving ODE in the form

y

yi

slope = f = dy/dx

This is the same as saying:

new value = old value + (dy/dx)(h)

where h is the step size

x

h


One step methods4

One Step Methods

  • Focus is on solving ODE in the form

y

yi+1

yi

slope = f = dy/dx

This is the same as saying:

new value = old value + (dy/dx)(h)

where h is the step size

x

h


Euler s method

Euler’s 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 Euler’s One-Step Method


Example

Example

For the initial condition y(1)=1, determine y for h = 0.1 analytically and using Euler’s method given:

STRATEGY


Strategy

Strategy

  • Determine the analytical solution based on the initial conditions y(1) = 1

    • i.e. x=1, y=1

  • Determine xi+1 = xi + h

  • Recognize that f(x,y) = f = 4x2

  • yi+1 = yi + 4(xi)2h

  • Recognize the meaning of “one-step”

    • yi+1 = yi + 4(xi)2h

    • yi+2 = yi+1 + 4(xi+1)2h

    • yi+3 = yi+2 + 4(xi+2)2h…..and so on


Error analysis of euler s method

Error Analysis of Euler’s 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


Higher order taylor series methods

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


Modification of euler s methods

Modification of Euler’s Methods

  • A fundamental error in Euler’s 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 graphically in order to give insight on the different strategies that can be employed

  • These modification actually belong to a larger class of solution techniques called Runge-Kutta which we will explore and apply later


Heun s method

Heun’s 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


Ordinary differential equations student notes

y

Take the slope at xi

Project to get f(xi+1 )

based on the step size h

h

x

xixi+1


Ordinary differential equations student notes

y

Use this “average” slope

to predict yi+1

x

xixi+1

{


Ordinary differential equations student notes

y

Use this “average” slope

to predict yi+1

x

xixi+1

{


Ordinary differential equations student notes

y

y

x

xixi+1

Euler’s

x

xi xi+1

Heun’s


Ordinary differential equations student notes

y

x

xi xi+1


Improved polygon method

Improved Polygon Method

  • Another modification of Euler’s Method

  • Uses Euler’s to predict a value of y at the midpoint of the interval

  • This predicted value is used to estimate the slope at the midpoint


Improved polygon method1

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 xi+1 using Euler’s algorithm


Ordinary differential equations student notes

y

f(xi)

x

xi


Ordinary differential equations student notes

y

h/2

h

x

xixi+1/2 xi+1


Ordinary differential equations student notes

y

h/2

x

xixi+1/2


Ordinary differential equations student notes

y

f(xi+1/2)

x

xixi+1/2


Ordinary differential equations student notes

y

f’(xi+1/2)

x

xixi+1/2


Ordinary differential equations student notes

y

Extend your slope

now to get f(x i+1)

h

x

xixi+1/2 xi+1


Ordinary differential equations student notes

y

f(xi+1)

x

xixi+1/2xi+1


Runge kutta methods

Runge-Kutta Methods

Both Heun’s and the Improved Polygon Method have been introduced graphically. However, the algorithms used are not as straight forward as they can be.

Let’s 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.


Runge kutta methods1

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


Incremental function can be interpreted as a representative slope over the interval

, Incremental Functioncan be interpreted as a representative slope over the interval


Ordinary differential equations student notes

NOTE:

k’s 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


Second order rk methods

Second Order RK Methods


Second order rk methods1

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 yi+1 and f(xi,yi)


Ordinary differential equations student notes

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


Ordinary differential equations student notes

By setting these two equations equal to each other and

recalling:

we derive three equations to evaluate the four unknown

constants


Ordinary differential equations student notes

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?


Ordinary differential equations student notes

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.


Ordinary differential equations student notes

Consider the following:

Case 1: a2 = 1/2

Case 2: a2 = 1

These two methods

have been previously

studied.

What are they?


Ordinary differential equations student notes

Case 1: a2 = 1/2

This is Heun’s Method

Note that k1 is the slope at

the beginning of the interval and k2 is the slope at the

end of the interval.


Ordinary differential equations student notes

Case 2: a2 = 1

This is the Improved Polygon Method (also called Mid-Point Technique).


Ordinary differential equations student notes

Ralston’s 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


Example1

Example

Evaluate the following

ODE using Heun’s

Methods (e.g. a2 = ½)

STRATEGY


Strategy1

Strategy

  • Calculate k1=f(x,y) = f = 4x2y

    • Use initial values x=1, y=1

  • Calculate the x and y values for k2

    • x= xi + h

    • y= yi + 4(xi)2(yi)h

  • Calculate k2 = 4(xi+1 )2(yi+1)

  • Calculate

    • yi+1 = yi +0.5(k1 + k2)h

  • Start process again using this value of y and x+h as the new initial values


Third order runge kutta methods

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 Simpson’s 1/3 Rule


Fourth order runge kutta

Fourth Order Runge Kutta

  • The most popular

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


Ordinary differential equations student notes

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


Example2

Example

Use 4th Order RK to solve the following differential equation:

using an interval of h = 0.1

STRATEGY


Strategy2

Strategy

  • Calculate k1 = f(x,y)

  • Determine the x and y value for k2, then calculate k2

  • Determine the x and y value for k3, then calculate k3

  • Determine the x and y value for k4, then calculate k4

  • Estimate

    • yi+1 = yi +1/6(k1 + 2k2+2k3+k4)h


Higher order rk methods

Higher Order RK Methods

  • When more accurate results are required, Bucher’s (1964) fifth order RK method is recommended

  • There is a similarity to Boole’s Rule

  • The gain in accuracy is offset by added computational effort and complexity


Systems of equations

Systems of Equations

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


Ordinary differential equations student notes

  • 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


Boundary value problems

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


Boundary value problems1

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


General methods for boundary value problems

Ta

T1

T2

Ta

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:


Ordinary differential equations student notes

Ta

T1

T2

Ta

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


Ordinary differential equations student notes

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.


Shooting method

Shooting Method

Given:

We need an initial value

of z.

For the shooting method, guess

an initial value.

Guessing z(0) = 10


Ordinary differential equations student notes

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.


Ordinary differential equations student notes

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?


Ordinary differential equations student notes

We can now use this to solve the first order ODE


Ordinary differential equations student notes

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..


Finite difference methods

Finite Difference Methods

The finite divided difference approximation for

the 2nd derivative can be substituted into the

governing equation.


Ordinary differential equations student notes

D x = 2 m

T1

T2

L = 10 m

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


Ordinary differential equations student notes

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?


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

Notice the labeling for numbering Dx and i


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

40

200

Note also that the dependent values are known at the boundaries (hence the term boundary value problem)


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

40

200

Apply the governing equation at node 1


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

40

200

Apply the equation at node 2


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

40

200

We get a similar equation at node 3


Ordinary differential equations student notes

x=0 2 4 6 8 10

T(0)

T(10)

i=0 1 2 3 4 5

At node 4, we consider the

boundary at the right.

40

200


Ordinary differential equations student notes

For the four interior nodes, we get the following

4 x 4 matrix


Example3

Example

Consider the previous example, but with Dx=1. What is the matrix?


Specific study objectives2

Specific Study Objectives

  • Understand the visual representation of Euler’s, Heun’s 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.


Specific study objectives3

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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