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**SE 207: Modeling and Simulation Introduction to Laplace Transform**

Dr. Samir Al-Amer Term 072

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Why do we use them We use transforms to transform the problem into a one that is easier to solve then use the inverse transform to obtain the solution to the original problem

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**Laplace Transform L L-1 t is a real variable s is complex variable**

f(t) is a real function Time Domain s is complex variable F(s) is a complex valued function Frequency Domain L-1 Inverse Laplace Transform

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**Use of Laplace Transform in solving ODE**

Differential Equation Algebraic Equation Laplace Transform Solution of the Algebraic Equation Solution of the Differential Equation Inverse Laplace transform

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**Definition of Laplace Transform**

Sufficient conditions for existence of the Laplace transform

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**Examples of functions of exponential order**

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Example unit step

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Example Shifted Step

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Integration by parts

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Example Ramp

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**Example Exponential Function**

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Example sine Function

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**Example cosine Function**

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**Example Rectangle Pulse**

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**Properties of Laplace Transform Addition**

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**Properties of Laplace Transform Multiplication by a constant**

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**Properties of Laplace Transform Multiplication by exponential**

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**Properties of Laplace Transform Examples Multiplication by exponential**

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Useful Identities

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Example sin Function

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**Example cosine Function**

Laplace Transform Inverse Laplace Transform

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**Properties of Laplace Transform Multiplication by time**

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**Properties of Laplace Transform**

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**Properties of Laplace Transform Integration**

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**Properties of Laplace Transform Delay**

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**Properties of Laplace Transform**

Slope =A L

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**Properties of Laplace Transform4**

Slope =A _ _ Slope =A A L L L Slope =A = L

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Summary

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**SE 207: Modeling and Simulation Lesson 3: Inverse Laplace Transform**

Dr. Samir Al-Amer Term 072

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**Properties of Laplace Transform**

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**Solving Linear ODE using Laplace Transform**

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**Inverse Laplace Transform**

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Notation

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Notation

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Notation

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Examples

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**Partial Fraction Expansion**

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**Partial Fraction Expansion**

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**Partial Fraction Expansion**

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Example

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Example

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**Alternative Way of Obtaining Ai**

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Repeated poles

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Repeated poles

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Repeated poles

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Repeated poles

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Common Error

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Complex Poles

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Complex Poles

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**What do we do if F(s) is not strictly proper**

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**Solving for the Response**

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Final value theorem

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Final value theorem

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Step function A

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impulse function

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impulse function

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**Initial Value& Final Value Theorems**

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Initial Value Theorem

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Final Value Theorems

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SE 207: Modeling and Simulation Lesson 4: Additional properties of Laplace transform and solution of ODE Dr. Samir Al-Amer Term 072

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**Outlines What to do if we have proper function? Time delay**

Inversion of some irrational functions Examples

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Step function A

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impulse function

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impulse function You can consider the unit impulse as the limiting case for a rectangle pulse with unit area as the width of the pulse approaches zero Area=1

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impulse function

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**Sample property of impulse function**

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Time delay g(t) G(s) f(t) F(s)

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**What do we do if F(s) is not strictly proper**

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**What do we do if F(s) is not strictly proper**

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Example − − −

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Example

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**Solving for the Response**

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