Lecture 2: Introduction to Ordinary Differential Equation (ODE) and Euler Method KKEK214:2 Numerical Methods for Engineers 2

Department of Chemical Engineering

Differential Equation

Definition: a mathematical equation that consists of an unknown function and its derivatives.

It expresses the relation between a function and its derivatives.

Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.

Examples: reaction rate, mass flow, heat transfer rate, acceleration, velocity, etc.

vm

cg

dt

dv

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2

E.g.: Newtons Second Law (1)

Where: = velocity g = gravitational constant t = time m = mass c = drag coefficient

- dependent variable

t- independent variable

Differential Equation

When a function involves one independent variable, the equation is called an ordinary differential equation (or ODE).

A partial differential equation (or PDE) involves two or more independent variables.

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Example: ODE in chemical reaction problem

A chemical reaction of the type of

(2)

takes place in a reactor, the material balance can be applied as:

Input = Output +

+ Generation Accumulation

(3)

For a batch reactor,

Input = Output +

+ Generation Accumulation

CBAk

k

1

2

Hence,

ODE

CBAC

CBAB

CBAA

CkCCkdt

dC

CkCCkdt

dC

CkCCkdt

dC

21

21

21

Classification of ODEs

ODEs are classified according to their order and their linearity.

The order of an ODE is the order of the highest derivative present in that equation.

1st order

2nd order

3rd order kx

dx

dyb

dx

yda

dx

yd

kxdx

dyy

dx

yd

kxydx

dy

2

2

2

3

3

2

2

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Classification of ODEs

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ODEs also can be classified according to its linearity, i.e. linear and nonlinear ODE.

An ODE is nonlinear if it contains product of the dependent variable or its derivatives or both.

Linear

Nonlinear

Nonlinear

2

2

23 2

3 2

dyy kx

dx

d y dyy kx

dx dx

d y d y dya b kx

dx dx dx

Degree of ODEs

The degree of an ODE is the largest power to which the highest order derivative is raised.

Has degree of 5

Has degree of 2

kxdx

dyb

dx

yda

dx

yd

kxdx

dyb

dx

yda

dx

yd

105

2

22

3

3

2

2

25

3

3

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Runge-Kutta Methods

However not all ODEs can be solved using analytical methods of calculus.

Need to use numerical methods.

Methods used to solve ODEs using iterative methods.

Developed by German mathematicians C. Runge and M.W. Kutta.

),( yxf

dx

dy

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Runge-Kutta Methods

Eulers method

Heuns method

Midpoint method

Higher Order Runge-Kutta (RK) method

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Runge-Kutta Methods

Solve ordinary differential equations of the form

Numerical method in the general form of

= +

or in mathematical terms

+1 = +

Applied step by step to compute out the value in the future (trajectory of the solution).

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),( yxfdx

dy

Runge-Kutta Methods

The type of RK methods can be differentiated by the manner in which the slope is estimated.

The easiest one-step method: Eulers method.

However, Eulers method has the least accurate predictions compared to other RK methods.

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Eulers Method

Named after Leonhard Euler (1707-1783).

Swiss mathematician and physicist.

He introduced several mathematical notations such as f(x), e (also known as Eulers number),

for summation and i.

He also had introduce the Eulers method to solve ordinary differential equation.

It is the simplest method to solve an ODE with a given initial value, yo.

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Eulers Method

The first derivative provides a direct estimate of the slope at xi

where f(xi,yi) is the differential equation evaluated at xi and yi. This estimate can be substituted into the equation:

A new value of y is predicted using the slope to extrapolate linearly over the step size h.

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Figure 25.2 Eulers method

),( ii yxf

hyxfyy iiii ),(1

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Example (pg. 708 in Text Book):

Use Eulers method to numerically integrate

From x=0 to x=4 with a step size of 0.5. The initial

condition at x=0 is y=1.

5.820122 23 xxxdx

dy

Exercise (homework)

Calculate the values from previous example for the subsequent steps, i.e. y(1.0), y(1.5), y(2.0), y(2.5), y(3.0), y(3.5) and y(4.0).

Compare the value that you obtained using Eulers method with those obtained using exact solution of

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= 0.54 + 43 102 + 8.5 + 1

Error Analysis of the RK Method

Numerical solutions of ODEs involves two types of error:

i) Truncation/discretisation errors

Caused by the nature of the techniques employed to approximate the value of y.

ii) Round off errors

Caused by the limited numbers of significant digits

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Error Analysis of the RK Method

Truncation errors:

i) Local truncation error that results from an application of the method in question over single step.

ii) Propagated truncation error that results from the approximation produced during the previous steps.

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Error Analysis of the Eulers Method (Cont.)

Total error = Local truncation error + Propagated truncation error

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Global truncation error

Error Analysis of the RK Method (Cont.)

Conclusions:

i) The global error can be reduced by decreasing h.

ii) The method will provide error-free predictions if the underlying function is linear.

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MATLAB

Short for MATrix LABoratory

Optimized to perform engineering and scientific calculations.

Initially designed to perform matrix mathematics

Provides a very extensive library of predefined functions to make technical programming tasks easier and more efficient.

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Advantages:

1. Ease of use

2. Platform independence

3. Pre-defined functions

4. Device-independent plotting

5. Graphical User Interface

6. MATLAB Compiler

Disadvantages:

1. Interpreted language execute more slowly

2. Cost

MATLAB Environment: MATLAB Desktop

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The Command Window: User can enter interactive commands at the command prompt (>>) and it will be executed on the spot. It is easier to use script files / M-files

The Command History Window

Displays the commands that user has entered

The MATLAB Workspace

Stores the variable name and value

Getting Help

Help Browser

Type help on the command window

Using lookfor command

Other Important Functions

Type demo to run MATLABs built-in demonstrations.

clc : clear the Command Windows content.

clear : clear the variables in the Workspace

abort command by typing control-c

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Important Programming Pitfalls

Never use a variable with the same name as a MATLAB function or command. If you do so, that function or command will become inaccessible.

If there is more than one function or command with the same name, the first one found on the search path will be executed.

Never create an M-file with the same name as a MATLAB function or command.

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