TopicsSections 2.4, 2.5, 2.6 The textbook topics, definitions, examples and theorems

Edwards-Penney 2.4, 2.5, 2.6 (11.1 K, txt, 26 Dec 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, but before Semester Break. Here's the statements for the exam review problems, which review chapter 1 methods to find the symbolic solutions:: Problems ER-1, ER-2 (109.6 K, pdf, 04 Dec 2012)Exam ReviewNumerical 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);: connect-the-dots graphic (11.2 K, jpg, 12 Sep 2010)JPG ImageExample: 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:: 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 RuleManuscriptExample: 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]] = 1.22Rect, Trap, Simp rules from calculusRECT 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 ruleApplied 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.

Second lecture on numerical methodsStudy 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 algorithmsComputer 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]Referencefor the ideas is this: 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.ManuscriptNumerical 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.: 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.Exam ReviewExamplesWeb 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 EXAMPLEy'=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:: Handwritten example y'=1-x-y, y(0)=3 (427.8 K, jpg, 16 Dec 2012) The references for the maple code areJpeg: Maple lab 3 codes S2013 (maple text) (5.3 K, txt, 10 Dec 2012)Maple Text: Numerical methods manuscript (149.9 K, pdf, 03 Mar 2012)ManuscriptEULER METHODLet 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 METHODLet 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 METHODLet 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 SOLUTIONWe 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 GRAPHICThe three results for Euler, Heun, RK4 are compared to the exact solution y=2-x+exp(-x) in the: y'=1-x-y Compare Euler-Heun-RK4 (17.8 K, jpg, 10 Dec 2012) The comparison graphic was created with thisGRAPHIC: y'=1-x-y by Euler-Heun-RK4 (1.1 K, txt, 10 Dec 2012)MAPLE TEXT

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].

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.: Numerical methods, RK4 predictors and correctors (149.9 K, pdf, 03 Mar 2012)ManuscriptMaple 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:: Numerical methods (149.9 K, pdf, 03 Mar 2012)Manuscript: 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)Maple Text

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): Sample Report for 2.4-3 with symbolic solution (175.9 K, pdf, 15 Mar 2012)Transparencies: Handwritten example y'=1-x-y, y(0)=3 (427.8 K, jpg, 16 Dec 2012)Jpeg: Exercises 2.4-5,2.5-5,2.6-5. Rect, Trap, Simp rules (219.5 K, pdf, 29 Jan 2006)Transparencies