# 2250-4 12:25pm Lecture Record Week 4 F2010

Last Modified: September 18, 2010, 13:38 MDT.    Today: September 23, 2018, 14:07 MDT.

## 13 Sep and 15 Sep: Michal K.

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Murphy's Lecture, time permitting: Maple Lab 2, problem 1,2,3 details.
Exam 1 date is 20 Sep or 22 Sep, 12:50pm in JWB 335. Also possible is 23 Sep at 7:25am in WEB 103.
Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.
Exams and exam keys for the last 5 years (17.7 K, html, 18 Dec 2010)

## 14 Sep: Numerical Solutions for y'=f(x,y)

```Numerical Solution of y'=F(x)
Example: y'=2x+1, y(0)=1
Symbolic solution y=x^2 + x + 1.
Dot table. Connect the dots graphic.
How to draw a graphic without knowing the solution equation for y.
Making the dot table by approximation of the integral of F(x).
Rectangular rule.
Dot table steps for h=0.1.
Answers: (x,y) = [0, 1], [.1, 1.1], [.2, 1.22], [.3, 1.36], [.4, 1.52],
[.5, 1.70], [.6, 1.90], [.7, 2.12], [.8, 2.36], [.9, 2.62], [1.0, 2.90]
The exact answers for y(x)=x^2+x+1 are
(x,y) = [0., 1.], [.1, 1.11], [.2, 1.24], [.3, 1.39], [.4, 1.56],
[.5, 1.75], [.6, 1.96], [.7, 2.19], [.8, 2.44], [.9, 2.71], [1.0, 3.00]
```
``` Maple support for making a connect-the-dots graphic.
Example: L:=[[0., 1.], [2,3], [3,-1], [4,4]]; plot(L);JPG Image: connect-the-dots graphic (11.2 K, jpg, 12 Sep 2010) ```
``` Maple code for the RECT rule, applied to quadrature problem y'=2x+1, y(0)=1.
# Group 1, initialize.
F:=x->2*x+1:
x0:=0:y0:=1:h:=0.1:Dots:=[x0,y0]:n:=10:

# Group 2, repeat n times. RECT rule.
for i from 1 to n do
Y:=y0+h*F(x0);
x0:=x0+h:y0:=Y:Dots:=Dots,[x0,y0];
od:

# Group 3, display dots and plot.
Dots;
plot([Dots]);
```
```Rect, Trap, Simp rules from calculus
Introduction to the Euler, Heun, RK4 rules from this course.
Example: y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.
Example: y'=2x+1, y(0)=3 with solution y=x^2+x+3.
Dot tables,  connect the dots graphic.
How to draw a graphic without knowing the solution equation for y.
Key example y'=sqrt(x)exp(x^2), y(0)=2.
Challenge: Can you integrate sqrt(x) exp(x^2)?
Making the dot table by approximation of the integral of F(x).
Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
```
```Example for your study:
Problem:  y'=x+1, y(0)=1
It has a dot table with x=0, 0.25, 0.5, 0.75, 1 and
y= 1, 1.25, 1.5625, 1.9375, 2.375.
The exact solution y = 0.5(1+(x+1)^2) has values
y=1, 1.28125, 1.625, 2.03125, 2.5000.
Determine how the dot table was constructed and identify
which rule, either Rect, Trap, or Simp, was applied.
```

### Second lecture on numerical methods

```Euler, Heun, RK4 algorithms
Computer implementation in maple
Geometric and algebraic ideas in the derivations.
Numerical Integration   Numerical Solutions of DE
RECT                    Euler
TRAP                    Heun [modified Euler]
SIMP                    Runge-Kutta 4 [RK4]

Numerical work maple L3.1, L3.2, L3.3, L4.1, L4.2, L4.3 will be
submitted after Semester Break. No numerical problems from ch 2
are assigned.
All discussion of maple programs will be based in the TA session
[Laura and Michal].
There will be one additional presentation of maple lab details
in the main lecture. The examples used in maple labs 3, 4 are
the same as those in exam review problems ER-1, ER-2. Each has
form dy/dx=f(x,y) and requires a non-quadrature algorithm, e.g.,
Euler, Heun, RK4.
```
```Numerical Solution of y'=f(x,y)
Two problems will be studied, in maple labs 3, 4.
First problem
y' = -2xy, y(0)=2
Symbolic solution y = 2 exp(-x^2)
Second problem
y' = (1/2)(y-1)^2, y(0)=2
Symbolic solution y = (x-4)/(x-2)
The work begins in exam review problems ER-1, ER-2, both due
before the first midterm exam. The maple numerical work is due
much later, after Semester Break.Exam Review: Problems ER-1, ER-2 (109.2 K, pdf, 17 Aug 2010)```
```Examples
Web references contain two kinds of examples.
The first three are quadrature problems dy/dx=F(x).
The fourth is of the form dy/dx=f(x,y), which requires a
non-quadrature algorithm like Euler, Heun, RK4.

y'=3x^2-1, y(0)=2, solution y=x^3-x+2
y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x).
y'=2x+1, y(0)=3 with solution y=x^2+x+3.
y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
```
Introduction: Maple Labs 3 and 4, due after Semester Break.
Maple lab 3 F2010. Numerical DE (152.5 K, pdf, 17 Aug 2010)
Maple lab 4 F2010. Numerical DE (138.2 K, pdf, 17 Aug 2010)
References for numerical methods:
Slides: Numerical methods (112.1 K, pdf, 03 Sep 2009)
Manuscript: Numerical methods (389.2 K, pdf, 13 Sep 2010)
Text: Maple L3 snips S2010 (maple text) (5.0 K, txt, 24 May 2007)
Maple Worksheet: Maple L3 snips F2010 (maple .mws) (6.6 K, mws, 25 May 2007)
Text: Maple code for maple labs 3 and 4 (5.1 K, txt, 17 Aug 2010)
Maple Worksheet: Sample maple code for Euler, Heun, RK4 (1.9 K, mws, 21 Aug 2010)
Maple Worksheet: Sample maple code for exact/error reporting (2.1 K, mws, 21 Aug 2010)
How to use maple at home (4.0 K, txt, 06 Jan 2010)
Maple lab 3 symbolic solution, ER-1 solution. (184.6 K, jpg, 08 Feb 2008)
Transparencies: Sample Report for 2.4-3. Includes symbolic solution report. (175.9 K, pdf, 02 Jan 2010)
F2010 notes on numerical DE report for Ch2 Ex 10 (97.9 K, pdf, 17 Aug 2010)
F2010 notes on numerical DE report for Ch2 Ex 12 (108.4 K, pdf, 17 Aug 2010)
F2010 notes on numerical DE report for Ch2 Ex 4 (97.8 K, pdf, 17 Aug 2010)
F2010 notes on numerical DE report for Ch2 Ex 6 (125.7 K, pdf, 17 Aug 2010)
Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)
Transparencies: ch2 Numerical Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (219.5 K, pdf, 29 Jan 2006)
```The work for book sections 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4.
The numerical work using Euler, Heun, RK4 appears in L3.1, L3.2, L3.3.
The actual symbolic solution derivation and answer check were
submitted as Exam Review ER-1. Confused? Follow the details in
the next link, which duplicates what was done in ER-1.Sample symbolic solution report for 2.4-3 (22.6 K, pdf, 19 Sep 2006)```
```Maple lab 3 report
Confused about what to put in your L3.1 report? Do the same
as what appears in the sample report for 2.4-3. Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)        ```
```        Include  the hand answer check. Include the maple code appendix. Then fill
in the table in maple Lab 3, by hand.
The example shows a  hand answer check and the maple code appendix.
Download all .mws maple work sheets to disk, then run the worksheet in xmaple.
In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...".
Some browsers require SHIFT and then mouse-click. Open the saved file in
xmaple or maple.

Extension .mws [or .mpl] allows interchange between
different versions of maple. Mouse copies of the worksheet pasted into email
allow easy transfer of code between versions of maple.
```

## 14 Sep: Linear Algebraic Equations. No matrices. Section 3.1.

```Maple lab 2 problem 1
Discussion: Option 1: Freezing pipes maple lab 2
Problem: u' + ku = kA(t)
Integration methods
Tables
Maple
```
Option 1: Maple Lab 2, Newton cooling freezing pipes (112.0 K, pdf, 16 Sep 2010)
Option 1: maple worksheet text freezing pipes (1.2 K, txt, 17 Aug 2010)
Option 2: Maple Lab 2, Newton cooling swamp cooler (240.6 K, pdf, 17 Aug 2010)
Option 2: maple worksheet text swamp cooler (1.3 K, txt, 17 Aug 2010)
For more on superposition y=y_p + y_h, see Theorem 2 in the link
Linear DE part I (152.7 K, pdf, 07 Aug 2009)
Manuscript: Linear equation applications, brine tanks, home heating (374.2 K, pdf, 28 Jul 2009)
Slides: Brink tanks (62.9 K, pdf, 30 Nov 2009)
Slides: Home heating (73.8 K, pdf, 30 Nov 2009)
```Linear Algebraic Equations sections 3.1, 3.2
Frame sequences
Toolkit: combo, swap, multiply
Plane and space geometry
The three possibilities
Unique solution
No solution
Infinitely many solutions
Method of elimination
Example for a unique solution
x + 2y =  1
x -  y = -2
Example for no solution
x + 2y = 1
x + 2y = 2
Example for infinitely many solutions
x + 2y = 1
0 = 0
Parameters in the general solution
Differential equations example, problem 3.1-26
y'' -121y = 0, y(0)=44, y'(0)=22
General solution given: y=A exp(11 x) + B exp(-11 x)
Substitute y into y(0)=44, y'(0)=22 to obtain a 2x2
system for unknowns A,B that has the unique solution
A=23, B=21.
```
Prepare 3.1 problems for next collection. See problem notes section 3.1:
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

## 16 Sep: Frame Sequences. Three Possibilities. No matrices. Sections 3.2, 3.3.

```Lecture: 3.1, 3.2, 3.3
Frame sequences
Toolkit: combo, swap, multiply
Plane and space geometry
The three possibilities
Unique solution
No solution
Infinitely many solutions
Free variable
Signal equation
Echelon form
The last frame test
The last frame algorithm
A detailed account of the three possibilities
Unique solution == zero free variables
No solution == signal equation
Infinitely many solutions == one+ free variables
```
```How to solve a linear system using the toolkit
Toolkit: swap, combo, mult
Toolkit operations neither create nor destroy solutions!
Frame sequence examples
Computer algebra systems and error-free frame sequences.
How to program maple to make a frame sequence without errors.
```
```Solved Problems
Example 4 in 3.2
Back-substitution should be presented as combo operations in a
frame sequence, not as isolated, incomplete algebraic jibberish.
Technically, back-substitution is identical to applying the
frame sequence method to variables in reverse order.
The textbook observes that an echelon matrix as frame one is
a special case, when only combo operations are required to
determine the last frame. Then, and only then, does the last
frame algorithm apply to write out the general solution.
Problem 3.2-24
The book's answer is wrong, it should involve k-4.
See references on 3 possibilities with symbol k.
```

Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
Slides: 3 possibilities with symbol k (72.8 K, pdf, 31 Jan 2010)
Manuscript: Example 10 in Linear algebraic equations no matrices (292.8 K, pdf, 01 Feb 2010)
In all your solved problems, to be submitted for grading, please use frame sequences to display the solution, as in today's lecture examples. Expected is a sequence of augmented matrices. Yes, you may use maple to make the frame sequence. The maple answer check for the last frame is rref(A).

## 16Sep: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

```Review
Last frame test. The RREF of a matrix.
Last frame algorithm.
Scalar form of the solution.
```
```Lecture: 3.3 and 3.4
Translation of equation models
Scalar equations to augmented matrix
Augmented matrix to scalar equations
Matrix toolkit: Combo, swap and multiply
```

## Last Frame Algorithm in Maple

How to use maple to compute a frame sequence. Example is Exercise 3.2-14 from Edwards-Penney.
Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 21 Aug 2010)
Maple Text: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)
Answer checks should also use the online FAQ.
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

## Frame Sequence with Symbols

Review: Answer checks with matlab, maple and mathematica. Pitfalls.
Review of the three possibilities and frame sequence analysis to find the general solution.