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2280 12:55pm Lectures Week 7 S2015

Last Modified: February 25, 2015, 11:40 MST.    Today: December 16, 2018, 06:28 MST.
  Sections 7.1 to 7.6
  The textbook topics, definitions, examples and theorems
Edwards-Penney Ch 3, 7.1 to 7.5 (20.1 K, txt, 17 Apr 2015)

Week 7, Sections 7.1, 7.2, 7.3

Monday: Ch 7

Forward table
Backward table

 Basic Theorems of Laplace Theory
   Functions of exponential order
      Existence theorem for Laplace integrals
      Euler solution atoms have a Laplace integral
   Forward table
   Backward table
   Lerch's theorem
   The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
   Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
   Parts theorem L(y')=sL(y)-y(0)
     Parts formula derivation.

 Solving differential equations by Laplace's method.
   Slide: Solving y' = -1, y(0)=2 with Laplace's method
 Laplace's method and quadrature for higher order equations and systems
 Solving x'' + 4x = t exp(-t), x(0)=1, x'(0)=0 by the Laplace method
 Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012)
Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012)
Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012)
Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012)
Slides: Laplace examples (149.1 K, pdf, 03 Mar 2012)
Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013)
Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)
Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012)
Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014)
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)
Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)
Manuscript: Laplace theory 2008 (500.9 K, pdf, 16 Mar 2014)
Transparencies: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)

Tuesday: Gamma, Piecewise Functions, Convolution, Resolvent

DEF: Piecewise Continuous Function
   Existence of the Laplace integral.
   One-sided and two-sided Laplace integral
      Freeway example, suspension collides with a ramp.
 DEF. Gamma function
    Gamma(t) = integral x=0 to x=infinity  x^{t-1} e^{-x}
    Gamma(n)=(n-1)!, generalizes the factorial function
 DEF. Mellin transform
   {Mf}(s)= phi(s)=integral x=0 to x=infinity  x^{s-1} f(x)
 DEF. Two-sided Laplace transform
   {Bf}(s) = {Mf(-ln(x))}(s) = integral x=0 to x=infinity x^{s-1}f(-ln x)
 DEF. Unit step u(t-a)=1 for t>=a, else zero
 DEF. Ramp t->(t-a)u(t-a)
 Backward table problems: examples
 Forward table problems: examples
 Computing Laplace integrals L(f(t)) with rules
 Solving an equation L(y(t))=expression in s for y(t)
    Complex roots and quadratic factors
    Partial fraction methods
 Trig identities and their use in Laplace calculations
 Hyperbolic functions and Laplace calculations
   Why the forward and backward tables don't have cosh, sinh entries

Wednesday: Gamma, Piecewise Functions, Convolution, Resolvent

 Piecewise Functions
   Unit Step: u(t)=1 for t>=0, u(t)=0 for t<0.
   Pulse: pulse(t,a,b)=u(t-a)-u(t-b)
   Ramp: ramp(t-a)=(t-a)u(t-a)
   L(u(t-a)) = (1/s) exp(-as) [for a >= 0 only]
Integral Theorem
   L(int(g(x),x=0..t)) = s L(g(t))
   Applications to computing ramp(t-a)
    L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]
 Piecewise defined periodic waves
   Square wave: f(t)=1 on [0,1), f(t)=-1 on [1,2), 2-periodic
   Triangular wave: f(t)=|t| on [-1,1], 2-periodic
   Sawtooth wave: f(t)=t on [0,1], 1-periodic
   Rectified sine: f(t)=|sin(kt)|
   Half-wave rectified sine: f(t)=sin(kt) when positive, else zero.
   Parabolic wave
 Periodic function theorem
      Proof details
      Laplace of the square wave. Problem 7.5-25.
      Answer: (1/s)tanh(as/2)

Applications of Laplace's method from 7.3, 7.4, 7.5
Convolution theorem 
    DEF. Convolution of f and g = f*g(t) = integral of f(x)g(t-x) from x=0 to x=t
    THEOREM. L(f(t))L(g(t))=L(convolution of f and g)
    Application:   L(cos t)L(sin t) = L(0.5 t sin(t))
 Second shifting Theorems
   e^{-as}L(f(t))=L(f(t-a)u(t-a)) Backward table
   L(g(t)u(t-a))=e^{-as}L(g(t+a)) Forward table
   Forward table
   L(sin(t)u(t-Pi)) = e^{-Pi s} L(sin(t)|t->t+Pi)
                    = e^{-Pi s} L(sin(t+Pi))
                    = e^{-Pi s} L(sin(t)cos(Pi)+sin(Pi)cos(t))
                    = e^{-Pi s} L(-sin(t))
                    = e^{-Pi s} ( -1/(s^2+1))
   Backward table
   L(f(t)) = e^{-2s}/s^2
           = e^{-2s} L(t)
           = L(t u(t)|t->t-2)
           = L((t-2)u(t-2))
   Therefore f(t) = (t-2)u(t-2) = ramp at t=2.

Laplace Resolvent Method.
  ==> This method is a shortcut for solving systems by Laplace's method.
  ==> It is also a convenient way to solve systems with maple.
Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012)

Friday: Exam 1

Start at 12:50pm and end at 1:50pm.
Exam Prep Materials:
Sample Exam
Sample Exam Solutons