## 2280 Lectures Week 8 S2018

Last Modified: March 07, 2018, 12:17 MST.    Today: September 23, 2018, 13:49 MDT.
```Topics
Sections 7.1 to 7.6
The textbook topics, definitions, examples and theoremsEdwards-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
Linearity.

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)
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.