3. Laplace Transform

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3. Laplace Transform
3.1 Introduction

• Laplace transform(L.T.)
• a mathematical tool that can significantly
  reduce the effort required to solve linear
  differential equation model.
• Major benefit this transformation converts
  differential equations to algebraic equations,
  which can simplify the mathematical manipulations
  required to obtain a solution.

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3.2 Laplace Transform of Representative Functions

• Definition
• Where the function f(t) must satisfy conditions
  which include being piecewise continuous for
  .
• Inverse transform
• Property Linear operator it satisfies the
  general superposition principle.

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• Examples
• 1. Constant function
• , ( is a constant)
• 2. Unit step function

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• Delta function (the first derivative of the step
  function)

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• 3. Derivatives
• n-th order derivative
• where is the i-th derivative
  evaluated at t0.

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• 4. Exponential function
• if blt0, the real part of s must be restricted to
  be larger than -b for the integrated to be
  finite.
• 5. Trigonometric functions
• Euler Identity

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• 6. Rectangular pulse function

Figure 3.1. The rectangular pulse function.
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3.3 Solution of Differential Equations by
Laplace Transform Techniques

• Example Solve the differential equation,
• using Laplace transform.
• Solution) Take Laplace transform both side of
  (3.19).

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• General solution procedure
• 1. Take the Laplace transform of both sides of
  the differential equation.
• 2. Solve for .
• If the expression for dose not appear in
  table.
• If the expression for not appear in
  table, go to Step 5.
• 3. Factor the characteristic equation polynomial.
• 4. Perform the Partial Fraction Expansion.
• 5. Use the inverse Laplace transform relations to
  find y(t).

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3.4 Partial Fraction Expansion

• Example) Consider
• How to find and .
• Method 1.
• Multiply both sides of (3.24) by
  .
• Equating coefficients of each power of s, we have
• Solving for and simultaneously. ?
  , .

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• Method 2. Since (3.24) should hold for all
  values of s, we can specify two values of s and
  solve for the two constant.
• Solving, .
• Method 3. Heaviside expansion
• The fastest and most popular method.
• We multiply both sides of the equation by one of
  the denominator terms
• and then set , which causes all
  terms except one to be multiplied by zero.

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• Heaviside Expansion
• - General form
• Case 1) If all are distinct.
• Case 2) Repeated Factors If a denominator term
  occur times in
  the denominator, terms must be included in
  the
• expansion that incorporate the factor
• in addition to the other factors.

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• How to Find for repeated factor case.
• Example
• For
• set up the partial fractions and evaluate their
  coefficient.
• Approach 1
• Substituting the value in (3.34)
  yields

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• Approach 2 Differentiation of the transform.
• Multiply both sides of (3.34).
• Then (3.39) is differentiated twice with respect
  to ,
• , so that
  .
• Differentiation approach
• If the denominator polynomial contains
  the repeated factor
• , first form the quantity.
• The general expression for is given as follows

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• Case 3) Complex Factors
• Apply Inverse Laplace transform (3.43).
• With the use of the identities,
• (3.44) becomes

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3.5 Other Laplace Transform Properties

• Final value theorem
• If exits,
• Proof) Use the relation for the Laplace transform
  of a derivative.
• Taking the limit as and assuming that
  is continuous and has a limit for
  all , we find
• Integrating the left side and simplifying yields

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• Initial value theorem
• In the analogy to the final value theorem,
  initial value theorem can be stated as
• Proof) The proof of this theorem is similar to
  the case of the final value theorem.
• Transform of an integral

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• Time Delay(Translation in Time)
• Time delays are commonly encountered in process
  control problems because of the transport time
  required for a fluid to flow through piping.

Figure 3.2. A time function with and without time
delay. (a) Original function(no delay) (b)
Function with delay.