Euler, Heun, RK4 algorithmsComputer 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 Spring Break. No numerical problems from ch 2 are assigned. All discussion of maple programs will be based in the TA session [Cox and Murphy]. 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 Spring Break.: Problems ER-1, ER-2 (46.9 K, pdf, 01 Jan 2010)Exam Review

ExamplesWeb 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).

Rect, Trap, Simp rules from calculusIntroduction 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.

- Introduction: Maple Labs 3 and 4, due after Spring Break.

Maple lab 3 S2010. Numerical DE (82.5 K, pdf, 01 Jan 2010)

Maple lab 4 S2010. Numerical DE (78.9 K, pdf, 01 Jan 2010)

- References for numerical methods:

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)

S2010 notes on numerical DE report for Ch2 Ex 10 (49.5 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 12 (64.4 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 4 (49.5 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 6 (65.9 K, pdf, 01 Jan 2010)

Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)

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)

Confused about what to put in your L3.1 report? Do the same as what appears in the sample report for 2.4-3 (link below). 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.

Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)

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.

Maple lab 2 problem 1Discussion: 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:

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.

Linear Algebraic Equations sections 3.1, 3.2Frame 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.

Murphy's Lecture, time permitting: Maple Lab 2, problem 1,2,3 details.

Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.

Lecture: 3.1, 3.2, 3.3Frame 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 toolkitToolkit: 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 ProblemsExample 4 in 3.2Back-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 3.2-24The book's answer is wrong, it should involve k-4. See references on 3 possibilities with symbol k.

ReviewLast frame test. The RREF of a matrix. Last frame algorithm. Scalar form of the solution. Vector form of the solution.

Lecture: 3.3 and 3.4Translation of equation models Equality of vectors Scalar equations to augmented matrix Augmented matrix to scalar equations Matrix toolkit: Combo, swap and multiply Frame sequences for matrix models. Special matrices Zero matrix identity matrix diagonal matrix upper and lower triangular matrices square matrix THEOREM. Homogeneous system with a unique solution. THEOREM. Homogeneous system with more variables than equations. Equation ideas can be used on a matrix A. View matrix A as the set of coefficients of a homogeneous linear system Ax=0. The augmented matrix B for this homogeneous system would be the given matrix with a column of zeros appended: B=aug(A,0).

Answer checks should also use the online FAQ.

Review of the three possibilities and frame sequence analysis to find the general solution.

- Frame sequences with symbol k.

MatricesVector. Matrix multiply The college algebra definition Examples. Matrix rules Vector space rules. Matrix multiply rules.

- Examples: how to multiply matrices on paper.