Systems of Differential Equations
Systems of differential equations have important applications as they arise quite frequently as models of many engineering
problems.
Solution by elimination
Successive elimination of unknown functions and its derivatives until it reduces a single higher order differential equation
containing one function and its derivatives
Solve the above single equation and then the other unknown functions are found in turn
. 1 1 1( ) (1) . 2 2 2( ) (2)
1 2 . , , , 1 2 1 2 tan .
1: Re (1)
General Method of Elimination to the Systems of the form
x a x b y f t y a x b y f t
where f and f are given functions a a b b are cons ts
SOLUTION
Step arrange eq
1 (. ) (3)
1 1
1
.. . . .
(4) 1 1 1
.. . .
( ) 1 (5)
1 1 2 2 2
2: (1)
3: (2) (4)
y x a x f b
x a x b y f
x a x b a x b y f f
for y
Step Differentiate eq
Step Substitute eq in eq
General Method of Elimination...continued
.. . . .
( ) 1 (6)
1 1 2 2 1 1 1 2
.. .
( ) ( ) (7)
1 2 1 2 2 1
4: (3) (5)
5: (7)
6: sin
x a x b a x b x a x f b f f x a b x a b a b x r
Step Substitute eq in eq
Step Solve eq for x Step Solve for y u g
(3) eq
Example 1:
. .
. (1) . (2)
1: Re (1)
. (3)
2: (1)
.. . (4)
3: (3
x y y x
x y y x
Step arrange eq for y y x
Step Differentiate eq x y
Step Substitute eq
Solution
Solve the following systems of differential equations
) (4) .. (5)
in eq x x
Example 1: Continued…
4: (5)
.. 0
( 2 1) 0 1
1 2
5: (2) .
1 2
1 2
Step Solve eq for x x x
D x
D t t
x c e c e
Step Solve eq for y
t t
y c e c e
x x
y c e c e
3 4
4 3
. 3 4 (1) . 4 3 (2)
1: Re (1) 4 . 3 (3)
2: (1)
.. 3 . 4 (4) .
Solve the following systems of differential equations
Dx x y
Dy x y
Solution
x x y
y x y
Step arrange eq for y y x x
Step Differentiate eq
x x y
Ste
3: (2) (4) .. 3 16 12 (5) .
p Substitute eq in eq x x x y
Example 2
4: (2) (4)
. .
.. 3 16 3 9 (5) 5: (5) .. 25 0
( 2 25) 0 5
5 5
1 2
6: (3) . 3
4 5 5
2 1 8 2 4
Step Substitute eq in eq
x x x x x
Step Solve eq for x
x x
D x
D
t t
x c e c e
Step Solve eq for x y x x
t t
c e c e y
NUMERICAL
METHODS
Analytical Methods:
Analytical solution provides excellent insight into the behavior of systems.
Analytical solution can be derived for only a limited class of problems
linear models and those that have simple geometry and dimensionality
Analytical solutions are of limited practical value because most real life problems are non-linear and complex shapes and processes
Today, computers and numerical methods provide an alternative for
such complicated calculations
Numerical Methods:
Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations.
They invariably involve large number of tedious arithmetic calculations.
With the development of fast, efficient digital computers, engineering problem solving by numerical methods has increased drastically in recent years.
For effective use of numerical methods understanding
the concept of error is very important.
Error Definitions
The numerical result is an approximate value of the exact result.
Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities.
The relationship between the true value and the actual value (approximation) can be defined as
( )
100 true value actual value approximation error
is also called the
error relative error
true value
error percentage relative error r true value
absolute error
Notation Represent (true value) Approximate (actu
al value) Error (true valu e – actual valu e)
1/7 0.142 857 142857 142857 … 0.142 857 0.000 000 142857 14
2857 …
ln 2 0.693 147 180 559 945 309 41... 0.693 147 0.000 000 180 559 94 5 309 41…
log10 2 0.301 029 995 663 981 195 21... 0.3010 0.000 029 995 663 981 195 21...
∛ 2 1.259 921 049 894 873 164 76... 1.25992 0.000 001 049 894 873 164 76...
√ 2 1.414 213 562 373 095 048 80... 1.41421 0.000 003 562 373 095 048 80…
e 2.718 281 828 459 045 235 36... 2.718 281 828 459 045 0.000 000 000 000 00 0 235 36...
Types of errors
1. Experimental errors
errors of data arising from measurements 2. Truncation errors
errors due to chopping off the remaining digits e.g.,
Consider the real numbers 5.6341432543653654 32.438191288
6.3444444444444
To truncate these numbers to 4 decimal digits, we only consider the 4 digits to the right of the decimal point.
The result would be:
5.6341 32.4381 6.3444
3. Round-off error (rounding error)
errors due to rounding off during computation Rule for rounding off a number to k decimals
1. Decide which is the last digit to keep.
2. Increase it by 1 if the next digit is 6 or more (this is called rounding up) 3. Leave it the same if the next digit is 4 or less (this is called rounding down) 4. Increase it by 1 if the next digit is a 5 followed by one or more non-zero
digits.
5. Round up or down to the nearest even digit if the next digit is a five followed (if followed at all) only by zeroes. That is, increase the rounded digit if it is currently odd; leave it if it is already even.
Example: 3.046 rounded to hundredths is 3.05 (because the next digit [6] is 5 or more)
Examples:
3.016 is rounded to hundredth is 3.02 (because the next digit (6) is 6 or more
3.013 rounded to hundredths is 3.01 (because the next digit (3) is 4 or less
3.015 rounded to hundredths is 3.02
(because the next digit is 5, and the hundredth digit (1) is odd)
3.045 rounded to hundredths is 3.04 (because the next digit (4) is even)
3.04501 rounded to hundredths is 3.05
(because the next digit is 5, but it is followed by non-zero digits)
• In decimal notation, every real number is represented by a finite or infin ite sequence of decimal digits
• For machine computation the number must be replaced by a number of f initely many digits
In digital computers the numbers are stored in two ways
1. Fixed point system: The numbers are represented with a fixed number o f decimal places
e.g., 16.78, 1.678, 0.1678, 1.000
2. Floating point system: The numbers are represented with a fixed number of significant digits
3 5 2
. ., 0.1213 10 , 0.1789 10 , -0.5000 10
e g
Rules For Significant Digits
Digits from 1-9 are always significant
Zeros between two other significant digits are always significant
One or more additional zeros to the right of both the decimal place and another significant digit are
significant
Zeros used solely for spacing the decimal point (placeholders) are not significant.
EXAMPLES
VALUE # OF SIG. DIG. COMMENT
453 3 All non-zero digits are always significant 5057 4 Zeros between 2 sig. dig. are significant 5.00 3 Additional zeros to the right of decimal and a sig. dig. are significant.
0.007 1 Placeholders are not significant
Multiplying and Dividing
RULE: When multiplying or dividing, your answer may only show as many significant digits as the multiplied or divided measurement showing the least number of significant digits.
Example:
25.47 x 4.50 x 75.75 = 8682.08625 25.47 4 significant digits.
4.50 3 significant digits.
75.75 4 significant digits.
answer can only show 3 significant digits because that is the least number of significant digits in the original problem.
round to the place in order to show only 3 significant digits.
Final answer becomes 8680
Adding and Subtracting
RULE: When adding or subtracting your answer can only show as many decimal places as the measurement having the fewest number of decimal places.
Example:
13.76 + 24.83 +2.1 = 40.69
2.1 shows the least number of decimal places.
round our answer, 20.69, to one decimal place.
Final answer is 40.7
Class Exercise
Identify the number of significant digits
1. 0.0005 2. 201.0 3. 6.02 X 105 4. 0.00530 5. 13450
Perform the following calculations and round according to the rule
1. 2.25 + 6
2. 18.640 + 670.445 3. 640 - 627.03 4. 12.09 - 6.7 5. 3.14 x 5.6 6. 0.059 x 6.95 7. 0.003/106 8. 8.5/0.356
What is a computer?
A computer is an electronic device that executes the instructions in a program.
A computer has four functions:
a. accepts data Input
b. processes data Processing c. produces output Output d. stores results Storage
Hardware: the physical parts of the computer.
Software: the programs (instructions) that tell the computer what to do and designed for end users
1. Display 2. Motherboard 3. CPU (Microprocessor) 4. Primary storage (RAM) 5. Expansion cards 6. Power supply 7. Optical disc drive 8. Secondary storage (HD) 8. Keyboard
10. Mouse
Input: for feeding and transfers information into the memory
Memory: For storing instructions and data
Control unit: Controls input information,
execution of arithmetic operations and output information
Arithmetic unit: Performs arithmetic operations (addition, subtraction, multiplication, division)
Output: For transferring information from memory onto the output sheet.
CONTROL UNIT
MEMORY
ARITHMERIC UNIT
INPUT OUTPUT
Components of a computer and main functions
Problem Solving Using Computer
1. Define the problem in full
2. Describe it using a program-oriented language such as Fortran (FORmula TRANslation)
C C++
and feed into the computer
3. Use a compiler which performs an automatic translation of the program into a machine language programs and execute it
Examples of Real life Problems:
Mathematical models in engineering and science mostly occurs in the form differential equations
1. Ordinary Differential Equations (ODE) Systems with one independent variable e.g.
Consider the reaction A B taking place in a batch reactor.
Then the rate of formation of B is given by
2. Partial Differential Equations (ODE) 3. e.g.
Systems with two or more independent variables e.g. Heat conduction in a slab
dC B kC dt A
2 T T
Finite Difference Methods
•Real life problems results in large sets of simultaneous differential equations, do not have analytical solutions but require the application of numerical techniques
•Finite difference method is the standard method used for the numerical solution of ODE and PDE.
•Finite difference enables to take a differential equation and integrate it
numerically by calculating the values of the function at discrete (finite) number of points
•If a set of finite values is available, such as experimental data, these may be differentiated, or integrated, using the calculus of finite differences.
Numerical differentiation is inherently less accurate
than numerical integration
Basic Concepts of Finite Difference
In differential calculus, the derivative of a function f(x) is given as
0 0
0 0
0 0
0 0
0 0
0
( ) ( )
( ) '( ) (1)
(1)
( ) ( )
'( )
( ), in t int
[ , ],
x x x
h
f x f x
df x f x lt
dx x x
let h x x
then eq may be approximated by f x h f x
f x lt
h
A function f x which is continuous and differentiable he erval x x can be re
2 3
0 0 0 0
0 0 0
( )
0 0
( ) ''( ) ( ) '''( )
( ) ( ) ( ) '( )
2! 3!
( ) ( )
... ( )
!
( ) sin
n n
n
n
presented by a Taylor series
x x f x x x f x
f x f x x x f x
x x f x
n R x
where R x is the remainder or truncation error and is represented u g the notation
1
2 3
0 0 0 0
0 0 0
( )
0 0 1
1
( )
( ) ''( ) ( ) '''( )
( ) ( ) ( ) '( )
2! 3!
( ) ( )
... ( )
! . .
n
n n
n
n
O h
x x f x x x f x
f x f x x x f x
x x f x
n O h
i e error of order h
The calculus of finite differences is used in conjunction with a series of discrete values, which can be either
experimental data, such as
3 2 1 1 2 3
or discrete values
y i y i y i y y i i y i y i
of a continuous function ( )
( -3 ) ( -2 ) ( - ) ( ) ( ) ( 2 ) ( 3 ) or by values of a function ( )
( -3 ) ( -2 ) ( - ) ( ) ( ) ( 2 ) ( 3 ) y x
y x h y x h y x h y x y x h y x h y x h f x
f x h f x h f x h f x f x h f x h f x h All these operato
, lg
rs satisfy distributive commutative and
associative laws of a ebra
0
2 3
' 0 '' 0 '''
0 0 0 0 0
( ), in t int [ , ]
exp :
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2! 3!
1.
A function f x which is continuous and differentiable he erval x x can be ressed by Taylor seris as
x x x x
f x f x x x f x f x f x
Taylor Series
Series Used
0
0
2 3
0
2 3
1 1
1
( )
... ( ) ( )
!
( ) .
1 ... ...
2! 3! ! !
ln(1 ) ... ( 1) ... ( 1)
2 3
2.
n n
n
n
n n
x
n n
n n
x x f x R x
n
where R x is called the remainder
x x x x
e x
n n
x x x x
x x
n n
Maclaurin Series
- - int
- - - -
-
D differential operator I egral operator
E shift operator
forward difference operator backward difference operator
central difference operator average operat
Linear Symbolic Operators
or
...
1. :
( ) ( ) '( )
2. :
( ) ( )
int
( )
Symbolic operators continued
Differential operator Dy x dy x y x
dx
Integral operator Iy x x h y x dx
x
the egral operator is equivalent to
Applying symbolic functions to the function y x
1
the inverse of the differential operator
I D
...
( ) ( )
2 ( ) ( 2 )
( ) ( )
1 ( ) ( )
( ) ( )
:
3. :
Symbolic operators continued
Ey x y x h E y x y x h E y x n y x nh
E y x y x h E n y x y x nh
The inverse shift operator
Shift operator
( )int
2 3
( ) ( ) '( ) ''( ) '''( ) ...
1! 2! 3!
sin
( )
( )
By Expanding the function y x h o a Taylor series about x
h h h
y x h y x y x y x y x
u g the differential
Expression of shift operator E
in terms of differential operator D
,
2 2 3 3
( ) ( ) ( ) ( ) ( ) ...
1! 2! 3!
Re
2 2 3 3
( ) 1 ... ( )
1! 2! 3!
operator D to indicate the derivatives of y
h h h
y x h y x Dy x D y x D y x arranging
h h h
y x h
D D D
y x
(Continued...)
( )
( )
. . ( ) ( )
( ) ( )
( ) ( )
1
Expression of shift operator E in terms of differential operator D
i e y x h e hD y x Ey x e hD y x
E e hD
y x h e hD y x E e hD
Similarly
3 2 1 1 2 3
( 3 ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( 3 )
4.
Consider the set of values
y i y i y i y i y i y i y i or the equivalent set
y x h y x h y x h y x y x h y x h y x h The first backward
Backward Finite Differences
1
( )
( ) ( ) ( )
i y yi i
or
difference of y at i or x is defined as y
y x y x y x h
2 1 1
1 1 2
2 1 2
2
...
sec ( )
i i i i i i
i i i i
i i i i
Continued
y y
y y y y
y y y
o
Backward Finite Differences
The ond backward difference of y at i or x is defined as
y y y y
y
2
0
( 1) !
( )! !
( ) ( ) 2 ( ) ( 2 ) general formula of the nth-order
exp
n m
n i i m
m
r
n y
n m m
y x y x y x h y x h
The backward
difference can be ressed as
y
( ) ( ) ( )
( )
in t ( )
Re
y x y x y x h
Expression of backward finite difference operator erms of differential operator D
lation between backward finite difference operator and differential operator
2
(1) ( ) ( ) (2)
(2) (1) ( ) (1 ) ( )
. . (1 )
2 3 3
...
but
y x h e hDy x Substituting in
y x e hD y x i e e hD
h D h D hD
Re
...
,
2 1 2 1 2 2
3 1 3 1 3 3 2 3
.. .
1
lation between backward finite difference operator and differential operator Continued
Similarly
hD hD hD
e e e
hD hD hD hD
e e e e
n e hD n
exp sec
2 2 2 3 3 7 4 4 ...
12
3 5
3 3 3 4 4 5 5 ...
2 4
Expansion of the onential terms
and rearrangement yields the following equations for the ond and third
backward difference operators
h D h D h D
h D h D h D
(1 ) (1) 1 (2) ln(1 ) (3) ln(1 )
( ) n t
( )
e hD then
e hD hD
Expression of differential operator D i erms of backward finite difference operator
2 3 4
2 3 4 5
... (4)
2 3 4
.(3) .(4)
... (5)
2 3 4 5
Combining eq with eq hD
( )in t ( )...
2 2 2 3 4 5
3 3
11 5 ... (6)
12 6
Expression of differential operator D erms of backward finite difference operator Continued
h D
h D
Similarly for higher order differential operator
n
3 4 5
2 3 4 5
3 7 ... (7)
2 4
.. .
... (8)
2 3 4 5
n n
h D
Backward difference operators Differential operators 2 2 3 3 ... 2 3 4 ...
2 6 2 3 4
7 11
2 2 2 3 3 4 4 ... 2 2 2 3 4 ...
12 12
3 3 3 3 4 4 2
h D h D
hD hD
h D h D h D h D
h D h D
5 5 5 ... 3 3 3 3 4 7 5 ...
4 2 4
2 3 4
(1 ) ...
2 3 4
h D h D
n e hD n h D n n
n
3 2 1 1 2 3
( 3 ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( 3 )
5.
Consider the set of values
y i y i y i y i y i y i y i or the equivalent set
y x h y x h y x h y x y x h y x h y x h The first forward d
Forward Finite Differences
1
( )
( ) ( ) ( )
i yi yi
or
ifference of y at i or x is defined as y
y x y x h y x
2 1 1
2 1 1
2 2 1
2
...
=
2
sec ( )
( ) ( 2 ) 2 ( ) ( )
i i i i i i
i i i i
i i i i
Forward Finite Differences Continued
y y
y y y y
y y y
or
The ond forward difference of y at i or x is defined as
y y y y
y
y x y x h y x h y x
m n y i m m
n n n
m i m n y
y i y i
y i y i
y i y i
y i y i
y i y i
y i
as ressed be
can difference
finite forward
order nth
the of formula general
The
derived
similarly Higher order forward difference s are
! )!
( !
0 1 ) (
4 1 6 2
4 3 4 4
3 1 3 2
3 3
exp
( ) ( ) ( ) (1
( )
in t ( )
Re
y x y x h y x
Expression of forward finite difference operator erms of differential operator D
lation between forward finite difference operator and differential operator
2
) ( ) ( ) (2)
(2) (1) ( ) ( 1) ( )
. . 1
2 3 3
...
2! 3!
but
y x h ehDy x Substituting in
y x ehD y x i e ehD
h D h D hD
Re
...
,
2 1 2 2 2 1
3 13 3 3 2 3 1
.. .
1
lation between forward finite difference operator and differential operator Continued
Similarly
hD hD hD
e e e
hD hD hD hD
e e e e
n ehD n
2 3 4 5
1 (1)
ln ln(1 ) (2)
ln(1 )
2 3 4 5
( ) n t
( )
ehD then
ehD hD but
Expression of differential operator D i erms of forward finite difference operator
2 3 4 5
... (3) .(2) .(3)
... (4)
2 3 4 5
Combining eq with eq hD
( )in t ( )...
2 2 2 3 4 5
3 3
11 5 ... (5)
12 6
Expression of differential operator D erms of forward finite difference operator Continued
h D
h D
Similarly for higher order differential operator
n
3 4 5
2 3 4 5
3 7 ... (6)
2 4
.. .
... (7)
2 3 4 5
n n
h D
Forward difference operators Differential operators 2 2 3 3 ... 2 3 4 ...
2 6 2 3 4
7 11
2 2 2 3 3 4 4 ... 2 2 2 3 4 ...
12 12
3 5
3 3 3 4 4
2
h D h D
hD hD
h D h D h D h D
h D h D
5 5 ... 3 3 3 3 4 7 5 ...
4 2 4
2 3 4
( 1) ...
2 3 4
h D h D
n e hD n h D n n
n
Form a finite difference table for the following function x = 0.6 0.7 0.8 0.9 1.0
f(x) = 5.9072 6.0092 6.35552 6.9992 8.0000
