The Laplace Transform of The Dirac Delta Function

1 Logo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckThe Laplace Transform of The DiracDelta FunctionBernd Schr oderBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)

2 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)

3 OriginalDE & IVPB ernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)OriginalDE & IVP-LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t)

4 OriginalDE & IVPA lgebraic equation forthe Laplace Transform -LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t) Transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace Transform -LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t) Transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace Transform -LAlgebraic solution,partial fractions?

5 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t) Transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solution-LAlgebraic solution,partial fractions?Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t) Transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solution- LL 1 Algebraic solution,partial fractions?

6 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the Domain (t) Transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solutionSolution- LL 1 Algebraic solution,partial fractions?Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?

7 1. (x)=0 for allx6= property: f(x) (x a)dx=f(a)????Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?1. (x)=0 for allx6= property: f(x) (x a)dx=f(a)????Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?1. (x)=0 for allx6= property: f(x) (x a)dx=f(a)????Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?

8 1. (x)=0 for allx6= property: f(x) (x a)dx=f(a)????Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?1. (x)=0 for allx6= property: f(x) (x a)dx=f(a) Delta Function is used to model instantaneous { (t a)}=e asBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?

9 1. (x)=0 for allx6= property: f(x) (x a)dx=f(a) Delta Function is used to model instantaneous { (t a)}=e asBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckWhat is the Delta Function ?1. (x)=0 for allx6= property: f(x) (x a)dx=f(a) Delta Function is used to model instantaneous { (t a)}=e asBernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckA Possible Application(Dimensions are fictitious.)

10 In an LRC circuit withL=1H,R=8 andC=115F, thecapacitor initially carries a charge of 1 Cand no currents areflowing. There is no external voltage source. At timet=2s, apower surge instantaneously applies an impulse of 4 (t 2)into the the charge of the capacitor over +Rq +qC=E(t)q +8q +15q=4 (t 2)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceThe Laplace Transform of The Dirac Delta Functionlogo1 Transforms and New FormulasA ModelThe Initial Value ProblemInterpretationDouble CheckA Possible Application(Dimensions are fictitious.)In an LRC circuit withL=1H,R=8 andC=115F, thecapacitor initially carries a charge of 1 Cand no currents areflowing.