Laplace transform

1. STUDY OF
LAPLACE TRANSFORM
AND
IT’S APPLICATIONS
Presented By
Amit Kundu 161-15-7149
Mehedul Ahsan 161-15-7181
Ankan lahiri 161-15-7074

2. Index
■ Introduction
■ Laplace transform
■ Properties of Laplace transform
■ Transform of derivatives
■ Application of LaplaceTransform

3. Abstract
The concept of Laplace Transforma plays a vital role in wide fields of science and
technology such as electric & communication engineering, quantum physics, solution of
partial differential operation, etc. This paper provides the reader with a solid foundation in
the fundamentals of Laplace Transform and gain an understanding of some of the very
important and basic applications of this transformation.

4. Introduction
■ Transformation in mathematics deals with the conversion of one function to
another function that may not
be in the same domain.
■ Laplace transform is a powerful transformation
tool, which literally transforms the original
differential equation into an elementary
algebraic expression.This latter can then
simply be transformed once again, into the
solution of the original problem.
■ This transform is named after the mathematician and renowned astronomer
Pierre Simon Laplace who lived in France.

5. Definition of LaplaceTransform
Suppose that, f is a real or complex valued function of the variable t
> 0 and s is a real or complex parameter.We define the Laplace
transform of f as
F(s) = L{f(t)} = 𝟎
∞
𝒆−𝒔𝒕
𝒇(𝒕)𝒅𝒕
{ ( )}f tL =F(s)
t domain s domain

6. Properties of Laplace
Transforms
■ Linearity
■ Scaling in time
■ Time shift
■ “frequency” or s-plane shift
■ Multiplication by tn
■ Integration
■ Differentiation

7. 7
Properties of Laplace transform
1) Linearity property :
If a, b be any constants and f, g any functions of t, then
2) First shifting property :
If then
         af t bg t a f t b g t    L L L
    f t F sL
    at
e f t F s a L

8. 8
3) Multiplication by
If
4) Change of scale property :
If then,
n
t
      1
n
nn
n
d
t f t F s
ds
 L
    f t F sL
    f t F sL
   1 s
f at F
a a
 
  
 
L

9. 9
5) Division of f(t) by t :
If then,
provided the integral exists.
    f t F sL
   
1
s
f t F s ds
t

 
 
 
L

10. Transform of derivatives
10
■ Laplace transform of the derivative of any order :
Let f, f’, ..., be continuous for all and be piecewise
continuous on every finite interval on the semi-axis
then transform of given by,
n
f
 1n
f 
0t 
n
f
0t 
n
f
         1 2 1
0 ' 0 ... 0n n n n n
f s f s f s f f  
    L L

11. LaplaceTransformOf Some Basic Function

12. Application of LaplaceTransform

13. Solving Ordinary Differential Equation
Problem:
Y" + aY' + bY = G(t) subject to the initial conditionsY(0) = A,Y' (0) = B
where a, b, A, B are constants.
Solution:
■ Laplace transform ofY(t) be y(s), or, more concisely, y.
■ Then solve for y in terms of s.
■ Take the inverse transform, we obtain the desired solutionY.

14. Solving Partial Differential Equation
Problem: Solve
with the boundary conditions U(x, 0) = 3 sin 2πx, U(0, t) = 0 and U(1, t) = 0
where 0 < x < 1, t > 0.
Solution:
■ Taking Laplace transform of both sides with respect to t,
■ Substituting in the value of U(x, 0) and rearranging, we get
where u = u(x, s) = L[U(x, t].
■ The general solution of (1) is
■ Determine the values of c1 and c2.Taking the Laplace transform of those boundary conditions that involve t, we
obtain c1 =0, c2 = 0. Thus (2) becomes
■ Inversion gives

15. Solving Electrical Circuits Problem
Problem: From the theory of electrical circuits we know,
where C is the capacitance, i = i(t) is the electric current , and v = v(t) is the voltage.
We have to find the correct expression for the complex impedance of a capacitor.
Solution:
■ Taking the Laplace transform of this equation, we obtain,
Where, and
■ Solving forV(s) we have
■ We know,
So we find:
which is the correct expression for the complex impedance of a capacitor.

16. Other Application of LaplaceTransform
■ To determine structure of astronomical object from spectrum
■ To find moment generating function in statistics
■ To determine the present value of consol or bond in economics
■ To solve the problem related to communication and network analysis.
■ To make a equation in simple form from hard equation like vibration of spring.
■ To solve Mixing Problem InvolvingTwoTanks

17. Limitation of LaplaceTransform
Only be used to solve differential equations with known constants. An
equation without the known constants, then this method is useless.

18. Conclusion
Laplace Transformation is powerful tool using in different areas of
mathematics, physics and engineering.
With the ease of application of Laplace transforms in many applications,
many research software have made it possible to simulate the Laplace
transformable equations directly which has made a good advancement in
the research field.

19. References
Books:
■ The LaplaceTransform:Theory and Applications by Joel L. Schiff
■ An Introduction to LaplaceTransforms and Fourier Series by Dyke, Phil
■ “Continuum spectra and brightness contours“ - Monthly Notices of the Royal Astronomical Society by Salem & Seaton
Websites:
■ http://en.wikipedia.org/wiki/Laplace_transform
■ http://electrical4u.com/laplace-transformation/
■ http://www.sosmath.com/diffeq/laplace/basic/basic.html
■ http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf
■ http://www.solitaryroad.com/c913.html
■ http://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx