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2280 Lectures Week 8 S2018

Last Modified: March 07, 2018, 12:17 MST.    Today: December 14, 2018, 00:27 MST.
  Sections 7.1 to 7.6
  The textbook topics, definitions, examples and theorems
Edwards-Penney Ch 7, 7.1 to 7.5 (21.0 K, txt, 22 Feb 2015)

Week 7, Sections 7.1, 7.2, 7.3

Monday-Tuesday: Ch 7

Laplace Theory
Forward table
Backward table

 Basic Theorems of Laplace Theory
   Functions of exponential order
      Vector space theory
      Piecewise continuous and piecewise smooth functions
      Existence theorem for Laplace integrals
      Euler solution atoms have a Laplace integral
   Forward table
   Backward table
   Lerch's theorem
   Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
   Parts theorem L(y')=sL(y)-y(0)
     Parts formula derivation.
   Unit step, pulse and ramp
   General piecewise continuous function as a sum of modulated pulses
   Convolution on (-inf,inf) and Laplace theory convolution definition
   Convolution theorem [no proof ever ...]
   Maple Lab 2 session LCB115, last 20 min. Lab due next Monday.
   The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).

 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. (188.3 K, pdf, 14 Mar 2016)
Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)
Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)
Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)
Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016)
Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)
Manuscript: DE systems, examples, theory (730.9 K, pdf, 09 Apr 2014)
Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)
Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 20 Aug 2008)
Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)
Manuscript: Laplace theory 2008-2017 (546.8 K, pdf, 24 Feb 2017)
Transparencies: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)

Wednesday and Monday: Gamma, Piecewise Functions, Convolution, Resolvent

DEF: Piecewise Continuous Function
   Existence of the Laplace integral.
 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
 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 (Monday)
      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))