The Laplace Transform of The Dirac Delta Function Bernd Schr oder - PowerPoint PPT Presentation

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check The Laplace Transform of The Dirac Delta Function Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Original DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original ✲ DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ Laplace transform of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform Solution ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. � ∞ 2. Sifting property: − ∞ f ( x ) δ ( x − a ) dx = f ( a ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. � ∞ 2. Sifting property: − ∞ f ( x ) δ ( x − a ) dx = f ( a ) ???? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. � ∞ 2. Sifting property: − ∞ f ( x ) δ ( x − a ) dx = f ( a ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. � ∞ 2. Sifting property: − ∞ f ( x ) δ ( x − a ) dx = f ( a ) 3. The delta function is used to model “instantaneous” energy transfers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check What is the Delta Function? 1. δ ( x ) = 0 for all x � = 0. � ∞ 2. Sifting property: − ∞ f ( x ) δ ( x − a ) dx = f ( a ) 3. The delta function is used to model “instantaneous” energy transfers. = e − as � � 4. L δ ( t − a ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check A Possible Application logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check A Possible Application (Dimensions are fictitious.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function

Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check A Possible Application (Dimensions are fictitious.) In an LRC circuit with L = 1 H , R = 8 Ω and C = 1 15 F , the capacitor initially carries a charge of 1 C and no currents are flowing. There is no external voltage source. At time t = 2 s , a power surge instantaneously applies an impulse of 4 δ ( t − 2 ) into the system. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta Function