Chapter 3
DIFFERENCE EQUATIONS
3.1 Introduction
Differential equations are applicable for continuous systems and cannot be used for
discrete variables. Difference equations are the discrete equivalent of differential
equations and arise whenever an independent variable can have only discrete values.
They are of growing importance in engineering in view of their applications in discrete
time- systems used in association with microprocessors.
Some Useful Results
‘ ’ is a Forward difference operator such that
Taking as one unit
Also
Generalizing
Properties of operator ‘ ’
, being a constant
=
Example 1 Evaluate the following:
i. ii. iii. iv.
Solution: i. = =
, if
ii. =
=
=
=
=
, =
iii. =
=
=
iv.
The shift operator ‘ ’ is defined as
Clearly effect of the shift operator is to shift the function value to the next
higher value or
Also
Moreover , where is the inverse operator.
Relation between and is given by
Proof: we know that
or
Factorial Notation of a Polynomial
A product of the form is called a factorial and is
denoted by
In case, the interval of differencing is , then
–
, The results of differencing are analogous to that differentiating
Also , and so on
Remark:
i. Every polynomial of degree can be expressed as a factorial polynomial of
the same degree and vice-versa.
ii. The coefficient of highest power of and also the constant term remains
unchanged while transforming a polynomial to factorial notation.
Example2 Express the polynomial in factorial notation.
Solution:
Example3 Express the polynomial in factorial notation.
Solution: Using remarks i. and ii.
Comparing the coefficients on both sides
,
,
or
We can also find factorial polynomial using synthetic division as shown below
Let
Now coefficients and can be found as remainders under and columns
1 3 0 –1 2
– 3 3
2 3 3
– 6
3 9=A
DIFFERENCE EQUATIONS
3.1 Introduction
Differential equations are applicable for continuous systems and cannot be used for
discrete variables. Difference equations are the discrete equivalent of differential
equations and arise whenever an independent variable can have only discrete values.
They are of growing importance in engineering in view of their applications in discrete
time- systems used in association with microprocessors.
Some Useful Results
‘ ’ is a Forward difference operator such that
Taking as one unit
Also
Generalizing
Properties of operator ‘ ’
, being a constant
=
Example 1 Evaluate the following:
i. ii. iii. iv.
Solution: i. = =
, if
ii. =
=
=
=
=
, =
iii. =
=
=
iv.
The shift operator ‘ ’ is defined as
Clearly effect of the shift operator is to shift the function value to the next
higher value or
Also
Moreover , where is the inverse operator.
Relation between and is given by
Proof: we know that
or
Factorial Notation of a Polynomial
A product of the form is called a factorial and is
denoted by
In case, the interval of differencing is , then
–
, The results of differencing are analogous to that differentiating
Also , and so on
Remark:
i. Every polynomial of degree can be expressed as a factorial polynomial of
the same degree and vice-versa.
ii. The coefficient of highest power of and also the constant term remains
unchanged while transforming a polynomial to factorial notation.
Example2 Express the polynomial in factorial notation.
Solution:
Example3 Express the polynomial in factorial notation.
Solution: Using remarks i. and ii.
Comparing the coefficients on both sides
,
,
or
We can also find factorial polynomial using synthetic division as shown below
Let
Now coefficients and can be found as remainders under and columns
1 3 0 –1 2
– 3 3
2 3 3
– 6
3 9=A
