la place trransform

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Week 4 TopicLaplace Transform

• Control System Engineering
• PE-3032
• Prof. CHARLTON S. INAO
• Defence Engineering College,
• Debre Zeit , Ethiopia

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The French NewtonPierre-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

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History of the Transform

• Euler began looking at integrals as solutions to
  differential equations in the mid 1700s
• 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

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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.

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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) Me?t where M and ? are constants

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Continuity

• 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.

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Boundedness

• This criterion also follows directly from the
  general definition
• If f(t) is not bounded by Me?t then the integral
  will not converge.

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• Transfer function - the Laplace transform of the
  differential equation

The Laplace transform of a function f(t), defined
for all real numbers t 0, is the function F
(s), defined by The parameter s is a complex
number With real
numbers s and ?. S sigma plus I omega
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• In mathematics and with many applications in
  physics and engineering and throughout the
  sciences, the Laplace transform is a widely used
  integral transform.
• Denoted , it is a linear operator of a function
  f(t) with a real argument t (t 0) that
  transforms it to a function F(s) with a complex
  argument s. The respective pairs of f(t) and F(s)
  are matched in tables.

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• The Laplace transform has the useful property
  that many relationships and operations over the
  originals f(t) correspond to simpler
  relationships and operations over the images
  F(s).
• It is named for Pierre-Simon Laplace, who
  introduced the transform in his work on
  probability theory.

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• The Laplace transform is used for solving
  differential and integral equations. In physics
  and engineering it is used for analysis of linear
  time-invariant systems such as electrical
  circuits, harmonic oscillators, optical devices,
  and mechanical systems.
• In this analysis, the Laplace transform is often
  interpreted as a transformation from the
  time-domain, in which inputs and outputs are
  functions of time, to the frequency-domain, where
  the same inputs and outputs are functions of
  complex angular frequency, in radians per unit
  time.

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Time domain versus Frequency domain
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Laplace Transform Table
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Laplace transform
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Inverse Laplace transform
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Properties of Laplace Transform
Linearity
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Multiplication
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Derivatives
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Integrals
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First shift theorem
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Second shift theorem , time shifting
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Initial Value theorem
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The final value theorem
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The inverse transform
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Example Inverse Laplace transform using Partial
Fraction
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Laplace Transform for ODEs

• Equation with initial conditions
• Laplace transform is linear
• Apply derivative formula
• Rearrange
• Take the inverse

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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 gt2)

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Consider the case where uxutt with u(x,0)0
and u(0,t)t2 and Taking the Laplace of the
initial equation leaves Ux U1/s2 (note that the
partials with respect to x do not disappear)
with boundary condition U(0,s)2/s3 Solving this
as an ODE of variable x, U(x,s)c(s)e-x
1/s2 Plugging in B.C., 2/s3c(s) 1/s2 so
c(s)2/s3 - 1/s2 U(x,s)(2/s3 - 1/s2) e-x
1/s2 Now, we can use the inverse Laplace
Transform with respect to s to find u(x,t)t2e-x
- te-x t
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Sample Exercises
factorial
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Example..
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Sample Problem
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Seatwork/Boardwork
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Answer
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Sample Solved problems
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Further Example of Inverse Laplace Transforms
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Solved problems
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