# 2250 8:05am Lectures Week 11 S2015

Last Modified: March 27, 2015, 05:38 MDT.    Today: September 23, 2018, 14:48 MDT.
``` Edwards-Penney, sections 5.6, 10.1, 10.2, 10.3, 10.4
The textbook topics, definitions and theoremsEdwards-Penney 5.5, 5.6 (15.5 K, txt, 18 Dec 2013)Edwards-Penney 10.1, 10.2, 10.3, 10.4, 10.5 (20.5 K, txt, 18 Dec 2013)```

### Week 11: Sections 10.1, 10.2, 10.3, 10.4

#### Monday (after the Break): Sections 10.1, 10.2

``` Lecture: Basic Laplace theory.
Read ch6, ch7, ch8, ch9 later.
Direct Laplace transform == Laplace integral.
Def: Direct Laplace
transform == Laplace integral
== int(f(t)exp(-st),t=0..infinity)
== L(f(t)).
Introduction and History of Laplace's method
Photos of Newton and Laplace: portraits of the Two Greats.Slides: Laplace and Newton calculus. Photos of Newton and Laplace. (200.2 K, pdf, 03 Mar 2012)
The method of quadrature for higher order equations and systems.
Calculus for chapter one quadrature versus the Laplace calculus.
The Laplace integrator dx=exp(-st)dt.
The abbreviation L(f(t)) for the Laplace integral of f(t).
Lerch's cancelation law and the fundamental theorem of calculus.
Intro to Laplace TheorySlides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015)
A Brief Laplace Table
1, t, t^2, t^n, exp(at), cos(bt), sin(bt)
Some Laplace rules: Linearity, Lerch
Laplace's L-notation and the forward table
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012)
Problems 10.1: 18, 22, 28
```
```Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (133.2 K, pdf, 27 Mar 2015) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Optional Maple Lab 7. Laplace applications (151.5 K, pdf, 29 Nov 2014) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) 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: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes S2013 (17.7 K, txt, 30 Apr 2015) Text: Final exam study guide (8.3 K, txt, 06 Jan 2015) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012)```

#### Mon-Wed-Fri: Laplace Theory. Tables, Rules of Laplace's Calculus. Sections 10.1,10.2,10.3.

```   Problems 10.1: 18, 22, 28

Text: Laplace theory problem notes (17.7 K, txt, 30 Apr 2015)

History of the Laplace Transform
REF: Deakin (1981), Development of the Laplace transform 1737 to 1937
EULER
LAPLACE 1784
End of WWII 1945
Fourier Transform
Mellin Transform and Gamma function
Laplace transform: one-sided and 2-sided transform
Applications: DE, PDE, difference equations, functional equations
Diffusion equation for spatial diffusion problems

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)

A brief Laplace table.
Forward table.
Backward table.
Extensions of the Table.
Laplace rules.
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem.
Parts theorem.
Finding Laplace integrals using Laplace calculus.
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012)  Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)
Solving differential equations by Laplace's method.
Basic Theorems of Laplace Theory
Functions of exponential order
Existence theorem for Laplace integrals
Euler solution atoms have a Laplace integral
Lerch's theorem
Linearity.
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)

Slide: Solving y' = -1, y(0)=2 with Laplace's method
Examples:

```

#### Wed-Fri: Piecewise functions. More Laplace theory. Sections 10.4, 10.5

```Forward table

Parts formula derivation.
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

DEF: Piecewise Continuous Function
Functions of exponential order.
Existence of the Laplace integral.
One-sided and two-sided Laplace integral
Freeway example, suspension collides with a ramp.
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))
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.

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (133.2 K, pdf, 27 Mar 2015) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (151.5 K, pdf, 29 Nov 2014) 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: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes F2008 (17.7 K, txt, 30 Apr 2015) Text: Final exam study guide (8.3 K, txt, 06 Jan 2015)```

#### Thursday: Ziwen Zhu

```Lab 11.
The Week 11 Lab will have Laplace problems. Details Wed-Thu of Week 11.
There is no Lab 10, because that was the week of Spring Break.
Lab 9 problems are due Thursday.
See week 9 in the CALENDAR for the PDF link.
```

#### Friday: Convolution, Resolvent

```Applications of Laplace's method from 10.3, 10.4, 10.5
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
EXAMPLES.
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)
```
``` 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)
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
Laplace of the square wave. Problem 10.5-25.

Applications of Laplace's method from 10.3, 10.4, 10.5
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
EXAMPLES.
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.
Application to computing ramp(t-a)
L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]

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)
Intro to the Laplace resolvent shortcut for 2x2 systems
Problem: Write a 2x2 dynamical system as a vector-matrix equation u'=Au.
Problem: Solve a 2x2 dynamical system in vector-matrix form u'=Au.
The general vector-matrix DE Model u'=Au
Laplace of u(t) = Resolvent times u(0)
Resolvent = inverse(sI - A)
Chapter 1 methods for solving 2x2 systems
Solve the systems by ch1 methods for x(t), y(t):
x' = 2x, x(0)=100,
y' = 3y, y(0)=50.
Answer: x = 100 exp(2t), y = 50 exp(3t)
x' = 2x+y, x(0)=1,
y' = 3y, y(0)=2.
Answer: y(t) = 2 exp(3t) and x(t) is the solution of the linear
integrating factor problem x'(t)=2x(t)+2 exp(3t).
```