Lecture 1 Sujin Khomrutai – 1 / 17
Method of Applied Math
Lecture 7: Laplace Transform
Sujin Khomrutai, Ph.D.
Heaviside Functions
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 2 / 17
Definition. The Heaviside function (or unit step function ) is
H ( t ) =
( 0, t < 0 1, t > 0 It follows that
kH (t − a) =
( 0 t < a
k t > a
The Laplace Transform of kH ( t − a )
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 3 / 17
Theorem. Let a ≥ 0 and k be a real number. Then
L[kH (t − a)] = k e −as s
Proof. By definition, L[kH (t − a)] =
Z ∞
0
e −st kH (t − a) dt
= k
Z ∞
a
e −st dt
= k e −as
s .
Example 1
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 4 / 17
EX. Find the Laplace transform of the function
f (t) =
( 0 0 < t < 3
5 t > 3
Example 2
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 5 / 17
EX. Show that the function g(t) =
( 2 0 < t < 4 0 t > 4
can be expressed as g(t) = 2(H (t) − H (t − 4)). Then find the
Laplace transform L[g(t)].
Example 11
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 6 / 17
EX. Find the inverse Laplace transform of the function F (s) = e − 3 s
s .
Shift in t
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 7 / 17
Definition. Let f ( t ) be a function and a ≥ 0 be a real number.
The function
f (t − a)H (t − a)
is called the shifting of f ( t ) by a .
Shift in t
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 8 / 17
Theorem Let F ( s ) = L[ f ]. Then
L[f (t − a)H (t − a)] = e −as F (s).
Thus
L − 1 [e −as F (s)] = f (t − a)H (t − a)
Example 3
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 9 / 17
EX. Find the Laplace transform L h
sin
t − π 4
H
t − π 4
i
.
Shift in t
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 10 / 17
Theorem. We have
L[f (t)H (t − a)] = e −as L[f (t + a)].
Example 4
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 11 / 17
EX. Find the Laplace transform of the function
f (t) = tH (t − 3).
Example 5
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 12 / 17
EX. Find the inverse Laplace transform L − 1
e − 3 s s 2 + 4
.
Example 6
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 13 / 17
EX. Solve the IVP:
y ′′ + y = (t − 4)H (t − 4), y(0) = 0, y ′ (0) = 0.
Differentiation of Laplace Transform
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 14 / 17
Theorem. Let F ( s ) = L[ f ( t )]. Then
L [t n f (t)] = (−1) n F ( n ) (s), where n ∈ {0, 1, 2, . . .}.
Proof. Apply the integration by parts.
Example 7
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 15 / 17
EX. Find
L [t cos at] , L[t sin at]
Integrals of Laplace Transform
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 16 / 17
Theorem. Let F ( s ) = L[ f ( t )]. Then L
f (t) t
=
Z ∞
s
F ( x ) dx.
Thus
L − 1
Z ∞
s
F (x) dx
= f (t)
t .
Example 8
Heaviside EX 1.
EX 2.
EX 11.
Prop 5: t-shifting EX 3.
EX 4.
EX 5.
EX 6.
Prop 6: s-diff EX 7.
Prop 7: s-integrating EX 8.
Lecture 1 Sujin Khomrutai – 17 / 17
EX. Find L
1 − cos at t
.
