2250 F2006 7:30 and 10:45 Lecture Record

Week 9, Oct 16,18,20: Sections 4.7,5.1,5.2,5.3.

Week 8, Oct 9,11,13: Sections 4.3,4.4,4.5,4.6.

Week 7, Oct 2,4: Sections 4.1,4.2.

Week 6, Sep 25,27,29: Sections 3.5,3.6.

Week 5, Sep 18,20,22: Sections 2.5,2.6,3.1.

> # This will compute a numerical solution to an ODE using Euler's method. 
> # Only Step 1-3 information needs to be changed for different problems.
> #
> # Step 1: Provide initial data
> y0 := 2:
> x0 := 0:
> 
> # Step 2: Define ODE 
> f := (x,y) -> 2*x*y:
> 
> # Step 3: Stepsize and number of nodes
> h := 0.5:
> n := 5:
> 
> # The remainder of this code need not be changed. 
> # Store data in 2 arrays
> Y := array(1..n):
> X := array(1..n):
> # Initialize first element in array
> Y[1] := y0:
> X[1] := x0:
> # Compute the solution using Euler's method      
> for i from 1 to n-1 do
> X[i+1] := x0 + h*i;
> Y[i+1] := Y[i] + h*f(X[i],Y[i]);
> end do:
> # Define a list to plot the information in.
> l := [[X[r],Y[r]] $r=1..n]:
> # Plot the solution
> plot(l,x=X[0]..X[n]);
<\pre>


20 Sep




Introduction to numerical
solutions of y'=f(x,y), y(x0)=y0. Euler, Heun and RK4 algorithms.
Predictor-corrector methods. Motivation for the algorithms. Reduction of
the algorithms to RECT, TRAP and SIMP in the case of a quadrature DE.
Explanation of computer algorithms in maple. Comparison of codes for
RECT, TRAP, EULER, HEUN.


Four examples again,
from last time, including symbolic solutions to 2.4-6 and 2.4-12:
y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2,
y=(x-4)/(x-2). Also included: y'=3x^2-1, y(0)=2, y=x^3-x+2 and
y'=exp(x^2), y(0)=2 with solution y=int(F,0..x)+0, F(x)=exp(x^2).
Dot table for x=0, 0.1, 0.2, 0.3 and y= 2, 2.1, 2.2 ,2.3.


Worked in class: y'=2xy, y(0)=2 by Euler and Heun methods. Showed how
to use dummy variable names x0, y0 to fill out the rows of the dot table
using a single formula (Euler, Heun).


Introduction to linear algebra. Systems in variables x,y. The three
possibilities. Cramer's rule and elimination. Signal equation "0=1".
Scope of our study of linear algebra: we stop at eigenanalysis.



References for numerical methods:
        
Slides on Rect,Trap,Simp,Euler,Heun,RK4 (39k pdf)
        
 Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
        
 Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)

  
 Introduction to Maple Lab 3, Numerical Solutions    
       
Maple Lab 3 Fall 2006 (pdf)
       
The actual symbolic solution derivation and answer check are
           submitted as dailies, separately. See the due dates page. A
           sample derivation and answer check appears in the sample
           symbolic solution report for 2.4-3 below.
        
 Report details on 2.4,2.5,2.6 prob 6 (pdf)
        
 Report details on 2.4,2.5,2.6 prob 12 (pdf)
        
 Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
        
 Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
        
 Sample Report for 2.4-3 (pdf 3 pages, 350k. Page 1 is symbolic sol.)
        
 Numerical DE coding hints, TEXT Document (1 pages, 2k)
       
 ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
       
Maple Worksheet files: Press SHIFT and then mouse-click to save the file to disk. Open the saved file in xmaple.
       In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...".
        
 Sample maple code for Euler, Heun, RK4 (maple worksheet)
        
 Sample maple code for exact/error reporting (maple worksheet)



22 Sep




Continued linear algebra and simultaneous linear equations. Reduced
echelon system. Cramer's rule and the three possibilities: (1) No sol,
(2) Infinitely many sols, (3) Unique solution. The result: determinant
not zero if and only if (3); determinant zero if and only (1) or (2).
The three rules for elimination. Snapshot sequence example 2x2. Frames.
First frame==original system, Last Frame==reduced echelon system. Logic:
in the unique solution case (3), the reduced echelon system is a list of
equations in variable list order, which assigns to each variable a
unique number. The list has this essential property: each nonzero
equation has a leading variable, i.e., a variable that appears just once
in the whole list, and it appears first, read left-to-right, with
coefficient 1. How to write the general solution in the infinitely many
solution case (2). Signal equation and no solution, which means no
equations for the variables and no answer check (there is no answer to
check!). Due Monday: 3.1 problems.

  
  Weeks 5,6 slides
       
 Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)
       
 Snapshot sequence and general solution, 3x3 system (1 page, pdf)
       
 Linear algebra,  no matrices, (typeset, 21 pages, pdf)
       
 PDF Document (11 pages, 113k)
       
 Matrix equations (typeset, 6 pages, 92k)
       
 Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)

 Week 4, Sep 11,13,15: Sections 2.5,2.6,3.1.

We are behind the lecture schedule, catching up.




On 1.5-34: there was a lecture in class. You were left to discover the
input concentration constant c_i, and solve the initial value problem
x'=r_i c_i - (r_0/v)x, x(0)=x_0. The initial value is x_0 =
(0.25/100)8000, using units of millions of cubic feet.

Covered in class for 2.1, 2.2: theory of autonomous DE y'=f(y),
stability, funnel, spout, phase diagram, asymptotic stability, unstable,
equil solution, verhulst models with harvesting and periodic free term.
You should be working on problem 2.1-8 and ask questions on 2.1 problems
on Tuesday.


References for 2.1, 2.2:
       
Verhulst logistic equation (typeset, 5 pages, pdf)
       
 Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)
       
 ch2 sections 1,2,3 Slides: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (8 pages, pdf)
       
 ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)
       
 ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)




12 Sep, Begin maple 2 in lab. 


Content [Geoffrey Hunter]:

We discussed question 1.5-34 in both sections in great detail. Started
with the dp/dt = stuff in - stuff out equation and derived the full DE,
partly intuitive. Discussed units and ways to check units, a sanity check.
Solved 2.1-15. Reviewed partial fractions, applied theory to a problem.
 
Both sections also wanted further clarification as to the purpose of the
integrating factor and when to use it. On Tues, I showed two different
ways that 1.5-20 could be solved (integrating factor and separation of
variables), while on Wed I just went through question 1.5-18.

Maple labs discussed: 1,2. Sources on www.


Content [Hwanyong Lee]:

I distributed maple lab2. We discussed about maple lab 1, like how to
control the size of plots and how to solve etc. Discussed how to find
the general solution of the linear first-order equation. Solved problem
1.5-18. In addition, we discussed about the existence of the unique
solution of the linear first-order equation by looking at Peano Thm and
Picard-Lindelof Thm. Sec.2.1, 2.2 next time, if at all.






Due 15 Sep, Page 86, 2.1:  8, 16 


7:30 class: discussed 2.2 problems, stability theory, phase diagrams,
calculus tools, DE tools, partial fraction methods.

10:45 class: Class ended after 5 minutes. As a result, 2.3 will be skipped.

  
  Heaviside coverup method manuscript 
       
 PDF Document (4 pages, 86k)




15 Sep Maple Lab 1 is due: Intro maple   L1.1, L1.2.
Due 18 Sep, Page 96, 2.2:   10, 14.

Never due: Page 106, 2.3:  10, 20. It will be extra credit, for Dec 7.


7:30 class: continued with partial fractions and introduction to
numerical solutions of quadrature problems y'=F(x), y(x0)=y0. Discussed
2.2 problems. Four examples, including symbolic solutions to 2.4-6 and
2.4-12: y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2,
y=(x-4)/(x-2). Also included: y'=3x^2-1, y(0)=2, y=x^3-x+2 and
y'=exp(x^2), y(0)=2 with solution y=int(F,0..x)+0, F(x)=exp(x^2).
Did the dot table for x=0, 0.1, 0.2, 0.3 and y= 2, 2.1, 2.2 ,2.3.

10:45 class: 
discussed 2.2 problems, stability theory, phase diagrams,
calculus tools, DE tools, partial fraction methods. Some work 
on RECT, TRAP rules.

References for numerical methods:
       
 ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
        
 Report details on 2.4,2.5,2.6 prob 6
        
 Report details on 2.4,2.5,2.6 prob 12
        
 Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
        
 Sample Report for 2.4-3 (pdf 3 pages, 350k)
        
 Numerical DE coding hints, TEXT Document (1 pages, 2k)
        
 Sample maple code for Euler, Heun, RK4 (maple worksheet)
        
 Sample maple code for exact/error reporting (maple worksheet)



 Week 3, Sep 4,6,8: Sections 2.1,2.2,2.3,2.4.


04 Sep: Holiday





05 Sep: Begin maple 1 in lab, Hunter and Lee. Discussed
exercises section 1.4. Trouble with room ST 104. Go to LS 101 for the
rest of the semester.


[Hwanyong Lee]: We discussed the maple lab1, due in about one week; see
the web due dates for the exact moment it is due. Discussed was the
separable D.E. is and #2 as its example. We talked about a D.E. which is
not separable, the definitions of implicit solution and explicit
solution and the difference between them. Then we discussed how to solve
y'=1+y and what meaning the solution y=-1 has comparing with a general
solution.


[Geoffrey Hunter]: Same as H. Lee, plus remarks on question 1.4-6. I was
asked about how to check if a solution was correct, a logic question. I
used that question to explain skipped solutions. Attendance: 25 at 7:30
Tuesday, 5 at 5:55 Wednesday. 





06 Sep: Collected in class, Page 41, 1.4:   6, 12.
Review of Picard-Lindelof and Peano theorems. More on problem 1.3-14.
Theory of variables separable equations. Solutions to 1.4-6,12,18,22,26.
Linear integrating factor method 1.5. Application to y'=1+y. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Integrating factor Lemma. Main theorem on linear DE
and explicit general solution.





08 Sep: 
Collected in class, Page 41, 1.4:   18, 22, 26.
Due Monday, 11 Sep, Page 54, 1.5:  8, 18, 20, 34
Three examples: y'=1+y, y'=x+y, y'=x+ (1/x)y. Variation of parameters
formula. General solution of the homogeneous equation. Superposition
principle. Slides for exercises 1.5-3, 1.5-5, 1.5-11, 1.5-33. Some
discussion of 1.5-34. Introduction to 2.1, 2.2 topics. Presentation of
Midterm 1 problems 1 to 4. Started topic of partial fractions, 2.1-2.2.
References for linear DE:
       
 Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing  (9 pages, pdf)
       
 Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
       
 Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
       
How to do a maple answer check for y'=y+2x (TEXT 1k)

 Week 2, Aug 28,30, Sep 1: Sections 1.3,1.4,1.5.


28 Aug: Collected in class Page 16, 1.2:  4, 6, 10. Lecture on 
Euler's direction field ideas. Example on y'=(1-y)y in class. See
Three Examples (pdf)
Projected slide from page 1 of the document
 
Direction fields (11 pages, pdf). Threading edge-to-edge solutions 
was based upon two rules: (1) Solution curves don't cross, and (2) 
Threaded solution curves must match tangents with nearby arrows of the 
direction field. See the direction field document above for 
explanations. Also stated in class was the Picard-Lindelof theorem, 
found in this slide set:
 
Peano and Picard Theory (3 pages, pdf).
For problem 1.3-8, xerox at 200 percent the textbook page and paste the 
figure. Draw threaded curves on this figure according to the rules in the
direction field document above. For problem 1.3-14, see 
       
Picard-Lindelof and Peano Existence theory (typeset, 9 pages, pdf)
and       
 Peano and Picard Theory (3 pages, pdf)
and       
Picard-Lindelof and Peano Existence Example, similar to 1.3-14 (1 page, pdf)





29 Aug, More examples of how to write a report, in lab. 
[from Hwanyong Lee] We discussed briefly the Format Suggestions for
submitted work. For Sec 1.1, we discussed the definition of the
differential equation and the difference between the ordinary
differential equation and the partial differential equation. For Sec
1.2, we discussed how to find the solution of y'=f(x) and why we can
find the solution explictly. we solved together problems #1, #6. For Sec
1.3, we discussed why we can't use the method used in the previous
section to find the solution of y'=f(x,y) and we discussed the Peano
theorem and the Picard Lindelof theorem and the difference between the
two theorems. And we discussed together the problems #13, #14 and the
difference between two problems.

[Geoffrey Hunter]
We discussed the format requirements for assignments. Referenced the
sample work on the web site and referred specific questions about format
to MWF classes. I clarified some points of confusion regarding the
time/location of the Maple labs and reminded the Wed class of the Maple
lab tutorial notice sent by e-mail. I answered questions on 1.3-14 (Tues
and Wed) and did 1.3-11 to reinforce the ideas (Wed). I also did a
verbal comparison of 1.3-14 with 1.3-13 (Tues and Wed). Briefly
discussed (with 1.3-14) how you can have existence but not necessarily
uniqueness of solutions.






30 Aug: Collected Page 26, 1.3:  8 in class. Discussion of 1.3-8
graphic. More on threading edge-to-edge solutions and Rules 1,2. 
Rule 1: Solution curves don't cross, and Rule 2:
Threaded solution curves must match tangents with nearby arrows of the
direction field. See the direction field document above for
explanations. More on Picard-Lindelof and Peano theorems. 
See 
Peano and Picard Theory (3 pages, pdf) for the statement of results.
For a discussion of background material on functions and continuity, 
Click here.
For the discussion of
1.3-14, which is due Friday 1 Sep, see part (a) of the link

Picard-Lindelof and Peano Existence Example (1 page, pdf). 
Started in the 10:45 class only is section 1.4, on variables separable DE.
Variables separable method references:
       
 Separable Equations. Separable DE test is here. (typeset, 9 pages, PDF)
       
 1.4 Page 40 Exercise slides (4 pages, 500k)
       
How to do a maple answer check for y'=y+2x (TEXT 1k)





01 Sep: Collected in class, Page 26, 1.3: 14. The dailies on Page
41, 1.4: 6, 12 will be due Wednesday 6 Sep. Theory of separable
equations started, section 1.4. Separation test: F(x)=f(x,y0)/f(x0,y0),
G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable. Basic
theory discussed. Venn diagram of first order equations of type
quadrature, separable and linear. Tests for quadrature (f_y=0) and linear
(f_y indep of y) types. Skipped solutions y=constant and how to find
them using G(c)=0. Non-equilibrium solutions from y'/G(y)=F(x) and a
quadrature step. Implicit and explicit solutions.
References: see 30 Aug notes.

 Week 1, Aug 23,25: Sections 1.1,1.2.


Aug 23: Sections 1.1, 1.2. Examples for problems 1.2-1, 1.2-2. Details about exams and dailies.
Info about maple tutorials next week. Preview of "The Three Examples" for next time. 
Info about maple lab 1 due soon, get your print from the web site.
  

       
Week 1 references (documents, slides)
       
 Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulae (6 pages, pdf)
       
Three Examples (pdf)
       
 Three Examples, solved 1.2-1,2,5,8,10 by Tyson Black, Jennifer Lahti, GBG (11 pages, pdf)
       

       Recipes for 1st, 2nd order and 2x2 Systems of DE PDF Document (2 pages, 50kb)
       
 Log+exponential Background+Direction fields PDF Document (4 pages, 450k). Decay law derivation. Problem 1.2-2. Direction field examples. 
       
 For more on direction fields, print  Direction fields document (typeset, 11 pages, pdf)





Aug 25: Problems 1.2-2, 1.2-6, 1.6-10 discussed in class. Integration
details and how to document them using handwritten calculations like u-subst,
parts, tabular. Maple and Matlab methods. Integral table methods. The
Three Examples
were introduced (#1, #2 only). Proof that "0=1" and logic errors in
presentations. Panels 1 and 2 in the answer check for an initial value
problem like 1.2-2: y'=(x-2)^2, y(2)=1.