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2250-1 7:30am Lecture Record Week 4 S2012

Last Modified: January 30, 2012, 06:01 MST.    Today: December 15, 2017, 04:59 MST.

Jan 30: Jules Verne Problem. Numerical Solutions for y'=F(x)

Jules Verne problem. A rocket from the earth to the moon.
Slides: Jules Verne Problem (127.3 K, pdf, 03 Mar 2012)
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. Here's the statements for the
   exam review problems, which review chapter 1 methods to find the
   symbolic solutions:
Exam Review: Problems ER-1, ER-2 (109.3 K, pdf, 10 Dec 2011)
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)
Rect, Trap, Simp rules from calculus
   RECT
     Replace int(F(x),x=a..b) by rectangle area (b-a)F(a)
   TRAP
     Replace int(F(x),x=a..b) by trapezoid area (b-a)(F(a)+F(b))/2
   SIMP
     Replace int(F(x),x=a..b) by quadratic area 
             (b-a)(F(a)+4F(a/2+b/2)+F(b))/6

   The Euler, Heun, RK4 rules from this course: how they relate to
     calculus rules RECT, TRAP, SIMP
     Numerical Integration   Numerical Solutions of DE
     RECT                    Euler
     TRAP                    Heun [modified Euler]
     SIMP                    Runge-Kutta 4 [RK4]

   Example: y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.
   Example: y'=2x+1, y(0)=1 with solution y=x^2+x+1.
   Dot tables,  connect the dots graphic.
   How to draw a graphic without knowing the solution equation for y.

   What to do when int(F(x),x) has no formula?
     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).
   Accuracy: Rect, Trap, Simp rules have 1,2,4 digits resp.
 Maple code for the RECT rule, applied to quadrature problem y'=2x+1, y(0)=1.
    # Quadrature Problem y'=F(x), y(x0)=y0.
    # 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]);
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.

Jan 31: Numerical Solutions for y'=f(x,y)

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
     [Patrick].
   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 (109.3 K, pdf, 10 Dec 2011)
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 S2012. Numerical DE (60.0 K, pdf, 09 Dec 2011)
    Maple lab 4 S2012. Numerical DE (56.4 K, pdf, 09 Dec 2011)
    References for numerical methods:
    Manuscript: Numerical methods manuscript (149.9 K, pdf, 03 Mar 2012)
    Text: Maple L3 snips S2012 (maple text) (5.0 K, txt, 24 May 2007)
    Maple Worksheet: Maple L3 snips S2012 (maple .mws) (6.6 K, mws, 25 May 2007)
    Text: Maple code for maple labs 3 and 4 (5.1 K, txt, 10 Dec 2011)
    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 (6.8 K, txt, 28 Dec 2011)
    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)
    S2012 notes on numerical DE report for Ch2 Ex 10 (35.5 K, pdf, 09 Dec 2011)
    S2012 notes on numerical DE report for Ch2 Ex 12 (51.8 K, pdf, 09 Dec 2011)
    S2012 notes on numerical DE report for Ch2 Ex 4 (35.0 K, pdf, 09 Dec 2011)
    S2012 notes on numerical DE report for Ch2 Ex 6 (47.9 K, pdf, 09 Dec 2011)
    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)

    1 Feb: Linear Algebraic Equations. No matrices. Section 3.1.

    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 S2012 (5.2 K, html, 08 Apr 2012)
      References
      Exam Review: Problems ER-1, ER-2 (109.3 K, pdf, 10 Dec 2011)
      Transparency: Sample solution ER-1. (184.6 K, jpg, 08 Feb 2008)
      Slides: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)
      Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)
      Slides: No solution case (79.7 K, pdf, 03 Mar 2012)
      Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)
      Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)
      Transparencies: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)
      Transparency: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)

    2 Feb: Patrick

    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 23 Sep at 7:25am in WEB 103. Also possible are 20 Sep and 22 Sep, 12:50pm in JWB 335.
    Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.
    Exams and exam keys for the last 5 years (18.9 K, html, 02 May 2012)

    3 Feb: Frame Sequences. Three Possibilities. No matrices. Sections 3.2, 3.3.

    Maple lab 2 problem 1
       Discussion: Option 1: Freezing pipes maple lab 2
       Problem: u' + ku = kA(t)
       Integration methods
          Tables
          Maple
       Answer check by computer
    
      Links for maple lab 2:
      Option 1: Maple Lab 2, Newton cooling freezing pipes (135.7 K, pdf, 09 Dec 2011)
      Option 1: maple worksheet text freezing pipes (1.3 K, txt, 10 Dec 2011)
      Option 2: Maple Lab 2, Newton cooling swamp cooler (135.7 K, pdf, 09 Dec 2011)
      Option 2: maple worksheet text swamp cooler (1.3 K, txt, 10 Dec 2011)
      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)
      For more about home heating models, read the following links.
      Manuscript: Linear equation applications, brine tanks, home heating (374.2 K, pdf, 28 Jul 2009)
      Slides: Brink tanks (95.3 K, pdf, 04 Mar 2012)
      Slides: Home heating (109.8 K, pdf, 04 Mar 2012)
    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
       Lead variable
       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 (98.8 K, pdf, 03 Mar 2012)
    Manuscript: Example 10 in Linear algebraic equations no matrices (460.2 K, pdf, 10 Feb 2012)
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).
    References for this lecture.
    Slides: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)
    Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)
    Slides: No solution case (79.7 K, pdf, 03 Mar 2012)
    Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)
    Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)

6 Feb: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

Maple lab 2 problem 1
   Discussion: Option 1: Freezing pipes maple lab 2
   Problem: u' + ku = kA(t)
   Integration methods
      Tables
      Maple
   Answer check by computer
    Links for maple lab 2:
    Option 1: Maple Lab 2, Newton cooling freezing pipes (135.7 K, pdf, 09 Dec 2011)
    Option 1: maple worksheet text freezing pipes (1.3 K, txt, 10 Dec 2011)
    Option 2: Maple Lab 2, Newton cooling swamp cooler (135.7 K, pdf, 09 Dec 2011)
    Option 2: maple worksheet text swamp cooler (1.3 K, txt, 10 Dec 2011)
    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)
    For more about home heating models, read the following links.
    Manuscript: Linear equation applications, brine tanks, home heating (374.2 K, pdf, 28 Jul 2009)
    Slides: Brink tanks (95.3 K, pdf, 04 Mar 2012)
    Slides: Home heating (109.8 K, pdf, 04 Mar 2012)
Review
  Last frame test. The RREF of a matrix.
  Last frame algorithm.
   Scalar form of the solution.
  Translation of equation models
   Scalar equations to augmented matrix
   Augmented matrix to scalar equations
  Matrix toolkit: Combo, swap and multiply

References for chapters 3 and 4, Linear Algebra


    Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)
    Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)
    Manuscript: Linear equations, reduced echelon, three rules (45.8 K, pdf, 22 Sep 2006)
    Manuscript: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
    Manuscript: Vectors and Matrices (426.4 K, pdf, 07 Feb 2012)
    Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)
    Transparencies: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)
    Transparency: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)
    Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
    Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)
    Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)
    Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
    Slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)
    Slides: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)
    Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)
    Slides: No solution case (79.7 K, pdf, 03 Mar 2012)
    Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)
    Maple: Lab 5, Linear algebra (71.4 K, pdf, 09 Dec 2011)
    Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
    Transparencies: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)
    html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
    Slides: Determinants 2008 (227.1 K, pdf, 03 Mar 2012)
    Manuscript: Determinants, Cramers rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
    Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)
    Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)
    Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)
    Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)
    Slides: Rank, nullity and elimination (154.1 K, pdf, 03 Mar 2012)
    Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)
    Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)
    Slides: Partial fraction theory (160.7 K, pdf, 03 Mar 2012)
    Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.4 K, txt, 30 Dec 2009)