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2250 7:30am Lectures Week 3 S2013

Last Modified: February 12, 2013, 05:36 MST.    Today: September 24, 2018, 01:33 MDT.
```Topics
Sections 2.4, 2.5, 2.6
The textbook topics, definitions, examples and theoremsEdwards-Penney 2.4, 2.5, 2.6 (11.1 K, txt, 26 Dec 2012)
```

Tuesday: Sections 2.4, 2.5, 2.6. Algorithms for y'=F(x)

```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, but before 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 (109.6 K, pdf, 04 Dec 2012)
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.
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)
Example: Find y(2) when y'=x exp(x^3), y(0)=1.
No symbolic solution!
How to draw a graphic with no solution formula?
Make the dot table by approximation of the integral of F(x).
REFERENCE:Manuscript: Numerical methods manuscript (149.9 K, pdf, 03 Mar 2012)
RECTANGULAR RULE
int(F(x),x=a..b) = F(a)(b-a) approximately
for small intervals [a,b]
Geometry and the Rectangular Rule
Example: y'=2x+1, y(0)=1
Rectangular rule applied to y(1)=1+int(F(x),x=0..1)
for y'=F(x), in the case F(x)=2x+1.
Dot table steps for h=0.1, using the rule 10 times.
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 correct answer y(2)=3.00 was approximated as 2.90.
Where did [.2,1.22] come from?
y(.2) = y(.1)+int(F(x),x=0.1 .. 0.2) [exactly]
= y(.1)+(0.2-0.1)F(0.1) [approximately, RECT RULE]
= y(.1)+0.1(2x+1) where x=0.1
= 1.1 + 0.1(1.2)  [approx, from data [.1,1.1]]
= 1.22
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'=x exp(x^3), y(0)=2.
Challenge: Can you integrate F(x) = x exp(x^3)?
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 the 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 1, 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.
Example 2, for your study:
Problem:  y'=x exp(x^3), y(0)=2
Find the value of y(2)=2+int(x*exp(x^3),x=0..1) to 4 digits.
Elementary integration won't find the integral, it has to be done
numerically. Choose a method and obtain 2.781197xxxx.
MAPLE ANSWER CHECK
F:=x->x*exp(x^3);
int(F(x),x=0..1);  # Re-prints the problem. No answer.
evalf(%);          # ANS=0.7811970311 by numerical integration.
```

Wednesday: Sections 2.5, 2.5, 2.6. Algorithms for y'=f(x,y)

``` Second lecture on numerical methods
Study problems like y'=-2xy, which have the form y'=f(x,y).
New algorithms are needed. Rect, Trap and Simp won't work,
because of the variable y on the right.
Euler, Heun, RK4 algorithms
Computer implementation in maple
Geometric and algebraic ideas in the derivations.
Numerical Integration   Numerical Solutions of y'=f(x,y)
RECT                    Euler
TRAP                    Heun [modified Euler]
SIMP                    Runge-Kutta 4 [RK4]
Reference for the ideas is thisManuscript: Numerical methods manuscript (149.9 K, pdf, 03 Mar 2012)
Numerical work maple lab 3 and lab 4 will be submitted after the
Semester Break. No Ch 2 numerical exercises will be submitted.
All discussion of maple programs will be based in the TA session,
or in office hours. The Math Center free tutors can help you.
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)
MAPLE TUTOR for NUMERICAL METHODS
# y'=-2xy, y(0)=2, by Euler, Heun, RK4
with(Student[NumericalAnalysis]):
InitialValueProblemTutor(diff(y(x),x)=-2*x*y(x),y(0)=2,x=0.5);
# The tutor compares exact and numerical solutions.

The work begins in exam review problems ER-1, ER-2, both due before
the first midterm exam.Exam Review: Problems ER-1, ER-2 (109.6 K, pdf, 04 Dec 2012)
The maple numerical work is due much later, to allow computer access,
but it is due before Semester Break.
Examples
Web references contain two kinds of examples.
The first three are quadrature problems dy/dx=F(x).
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.
The fourth is of the form dy/dx=f(x,y), which requires a
non-quadrature algorithm like Euler, Heun, RK4.
y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
WORKED EXAMPLE
y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
We will make a dot table by hand and also by machine.
The handwritten work for the Euler Method and the Improved Euler
Method [Heun's Method] are available:Jpeg: Handwritten example y'=1-x-y, y(0)=3 (427.8 K, jpg, 16 Dec 2012)
The references for the maple code areMaple Text: Maple lab 3 codes S2013 (maple text) (5.3 K, txt, 10 Dec 2012)Manuscript: Numerical methods manuscript (149.9 K, pdf, 03 Mar 2012)
EULER METHOD
Let f(x,y)=1-x-y, the right side of the differential equation.
Use step size h=0.2 from x=0 to x=0.4. The dot table has 3 rows.
Table row 1: x0=0, y0=3
Taken from initial condition y(0)=3
Table row 2: x1=0.2, y1=2.6
Compute from x1=x0+h, y1=y0+hf(x0,y0)=3+0.2(1-0-3)
Table row 3: x2=0.4, y2=2.24
Compute from x2=x1+h, y2=y1+hf(x1,y1)=2.6+0.2(1-0.2-2.6)
MAPLE EULER: [0, 3], [.2, 2.6], [.4, 2.24]

HEUN METHOD
Let f(x,y)=1-x-y, the right side of the differential equation.
Use step size h=0.2 from x=0 to x=0.4. The dot table has 3 rows.
Table row 1: x0=0, y0=3
Taken from initial condition y(0)=3
Table row 2: x1=0.2, y1=2.62
Compute from x1=x0+h, tmp=y0+hf(x0,y0)=2.6,
y1=y0+h(f(x0,y0)+f(x1,tmp))/2=2.62
Table row 3: x2=0.4, y2=2.2724
Compute from x2=x1+h, tmp=y1+hf(x1,y1)=2.62+0.2*(1-0.2-2.62)
y2=y1+h(f(x1,y1)+f(x2,tmp))/2=2.2724
MAPLE HEUN: [0, 3], [.2, 2.62], [.4, 2.2724]

RK4 METHOD
Let f(x,y)=1-x-y, the right side of the differential equation.
Use step size h=0.2 from x=0 to x=0.4. The dot table has 3 rows.
The only Honorable way to solve RK4 problems is with a calculator or
computer. A handwritten solution is not available (and won't be).
Table row 1: x0=0, y0=3
Taken from initial condition y(0)=3
Table row 2: x1=0.2, y1=2.618733333
Compute from x1=x0+h, 5 lines of RK4 code
Table row 3: x2=0.4, y2=2.270324271
Compute from x2=x1+h, 5 lines of RK4 code
MAPLE RK4: [0, 3], [.2, 2.618733333], [.4, 2.270324271]

EXACT SOLUTION
We solve the linear differential equation by the integrating factor
method to obtain y=2-x+exp(-x).
# MAPLE evaluation of y=2-x+exp(-x)
F:=x->2-x+exp(-x);[[j*0.2,F(j*0.2)] \$j=0..2];
# Answer [[0., 3.], [0.2, 2.618730753], [0.4, 2.270320046]]

COMPARISON GRAPHIC
The three results for Euler, Heun, RK4 are compared to the exact
solution y=2-x+exp(-x) in theGRAPHIC: y'=1-x-y Compare Euler-Heun-RK4 (17.8 K, jpg, 10 Dec 2012)
The comparison graphic was created with thisMAPLE TEXT: y'=1-x-y by Euler-Heun-RK4 (1.1 K, txt, 10 Dec 2012)    ```

Thursday: Leif

```Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Sample Exam: S2012 Exam 1 key. See also F2010 Exam 1.
Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18.
Maple Lab 2, problem 1 details [maple L2.1].
```

Friday: Sections 2.4, 2.5, 2.6. Maple lab examples.

```Third lecture on numerical methods.
Theory for RK4
Historical events: Heun, Runge and Kutta
How Simpson's Rule provides RK4, using Predictors and Correctors.
Why we don't read the proof of RK4.Manuscript: Numerical methods, RK4 predictors and correctors (149.9 K, pdf, 03 Mar 2012)
Maple Labs 3 and 4, due before Semester Break.Maple lab 3 S2013. Numerical DE (152.4 K, pdf, 04 Dec 2012)Maple lab 4 S2013. Numerical DE (138.2 K, pdf, 04 Dec 2012)
How to get started with the maple numerical labs.
Maple details for maple labs 3 and 4
Maple projection presentation, laptop

References for numerical methods:Manuscript: Numerical methods (149.9 K, pdf, 03 Mar 2012)Maple Text: Euler-Heun-RK4 y'=1-x-y [Maple Labs 3, 4] (5.3 K, txt, 10 Dec 2012)Maple Text: Sample for exact/error reporting (1.5 K, txt, 16 Dec 2012)How to use maple at home (7.3 K, txt, 06 Dec 2012)Maple lab 3 symbolic solution (ER-1) (184.6 K, jpg, 08 Feb 2008)Transparencies: Sample Report for 2.4-3 with symbolic solution (175.9 K, pdf, 15 Mar 2012)Jpeg: Handwritten example y'=1-x-y, y(0)=3 (427.8 K, jpg, 16 Dec 2012)Transparencies: Exercises 2.4-5,2.5-5,2.6-5. Rect, Trap, Simp rules (219.5 K, pdf, 29 Jan 2006)```