2. Group Members
Presented By:-
Saad Saif (FA17-BSCS-359)
Ali Raza (FA17-BSCS-368)
Iqrar Ahmad (FA17-BSCS-345)
Rehan Ali (FA17-BSCS-292)
M. Imran (FA17-BSCS-267)
Presented To:-
Miss Aleen Chaudhary
3. The French Newton
Pierre-Simon Laplace
Developed mathematics in
astronomy, physics, and statistics
Began work in calculus which led
to the Laplace Transform
Focused later on celestial
mechanics
One of the first scientists to
suggest the existence of black
holes
4. History of the Transform
Euler began looking at integrals as solutions to differential equations
in the mid 1700’s:
Lagrange took this a step further while working on probability
density functions and looked at forms of the following equation:
Finally, in 1785, Laplace began using a transformation to solve
equations of finite differences which eventually lead to the current
transform
5. Definition
The Laplace transform is a linear operator
that switched a function f(t) to F(s).
Specifically:
where:
Go from time argument with real input to a
complex angular frequency input which is
complex.
6. Restrictions
There are two governing factors that
determine whether Laplace transforms
can be used:
f(t) must be at least piecewise continuous
for t ≥ 0
|f(t)| ≤ Meγtwhere M and γ are constants
7. Since the general form of the Laplace
transform is:
it makes sense that f(t) must be at least
piecewise continuous for t ≥ 0.
If f(t) were very nasty, the integral would
not be computable.
Continuity
8. Boundedness
This criterion also follows directly from the
general definition:
If f(t) is not bounded by Meγtthen the
integral will not converge.
10. Laplace Transforms
•Some Laplace Transforms
•Wide variety of function can be transformed
•Inverse Transform
•Often requires partial fractions or other
manipulation to find a form that is easy to
apply the inverse
11. Laplace Transform for ODEs
•Equation with initial conditions
•Laplace transform is linear
•Apply derivative formula
•Rearrange
•Take the inverse
12. Laplace Transform in PDEs
Laplace transform in two variables (always
taken with respect to time variable, t):
Inverse laplace of a 2 dimensional PDE:
Can be used for any dimension PDE:
The Transform reduces dimension by “1”:
ODEs reduce to algebraic equations
PDEs reduce to either an ODE (if original equation
dimension 2) or another PDE (if original equation dimension
>2)
15. Ex. Semiconductor Mobility
Motivation
semiconductors are commonly made
with superlattices having layers of differing
compositions
need to determine properties of carriers
in each layer
• concentration of electrons and holes
• mobility of electrons and holes
conductivity tensor can be related to
Laplace transform of electron and hole
densities
16. Notation
R = ratio of induced electric field to the product of
the current density and the applied magnetic field
ρ = electrical resistance
H = magnetic field
J = current density
E = applied electric field
n = concentration of electrons
u = mobility
22. Background
In 1807, Joseph Fourier showed that any periodic signal
could be represented by a series of sinusoidal functions
In picture: the composition of the first two functions gives the bottomone
23. What is FourierTransform?
Understandings
The Fourier Transform is a type of mathematical transform.
The Fourier Transform transforms a non-periodic function 𝑓(𝑡)in
time domain into a function 𝐹(𝑠)in its corresponding frequency
domain.
It is a tool to break a function into a sinusoidal forms characterized
by sine and cosine terms.
To understand the conduction of heat, wave propagation, digital
signal processing, image processing, filtering, etc.
24. Mathematical Definition:
defined on (−∞, ∞)
piecewise continuous in each finite
interval
absolutely integrable in (−∞, ∞)
The function 𝐹𝑤 ( ) is called the Fourier
Transformation of 𝑓(𝑡)
25. So We discussed Examples Of Fourier
Transform and Its Application in Cell Phones
Communication is all based on Mathematics, be it digital,
wired or wireless. Signal transmission is done through
modulation i.e. amplitude modulation (AM), frequency
modulation (FM) or phase modulation (PM). At the receiving
end the transmitted signal is demodulated to extract the
information. All these techniques are based on pure
mathematics.
26. Role of Fourier Transform (FT) in Cell
phone
One of the most prominent communication devices, the Cell Phone is
dramatically changing the way people interact and communicate with
each other. Cell-phones emit small amount of electromagnetic signals
via the radio waves through a low power transmitter. While talking over
the cell-phone, the transmitter takes the sound of voice & changes it into
a continuous sine wave. Sine wave is measured in terms of frequency.
Transmitter sends the sine wave to antenna. Antenna transmits the sine
wave in the form of electromagnetic signal to the BTS. Cell-phone works
by communication between service network through BTS or cell tower.
Cell towers divide the city into small areas or cells. As the user moves
from one cell to another, the signal along with the information is handed
over from tower to tower
27. How mathematics is involved in making
cell phones work and make calls
The cell phones are designed by using a lot of math in just about
every aspect of their design. Also cell phones operate by principles
of electromagnetic, which are described mathematically.
1.One has to dial a number that it is based in a protocol named
Internet Protocol (IP). Protocol is basically a set of rules.
2.The phone has to use coordinates to locate the Satellite to receive
and transmitted to the other end.
3.They have to convert from an electric system or wave system into a
voice system that it is based in alphabetical words, and then
translated between the 2 system based in a numerical system called
binaries.
