MCS 471 Computer Problem 4: Numerical Solution of ODEs using Maple

5.3 Euler Methods
5.4 Runge-Kutta Methods
5.5 Multistep Methods
5.7 Adams-Moulton Multistep Method
5.9-5.10 Convergence and Errors
6.1-6.2 Shooting Method and Solution of Sets of Equations

Refer to the Class Maple Pages, or in particular to the

RK4, Euler's, and Modified Euler's Methods Maple DEtools Package Worksheet for 2D System
(Else Click Here for Plain Text).

See also the Maple Language Help for DEtools, DEplot and their options.  

• x'(t) = 1 - (1 + b(t) - a*x(t)*y(t))*x(t)
• y'(t) = (b(t) - a*x(t)*y(t))*x(t)

• a=(2*pi)**2;
• pi=4*atan(1.0e0);
• b(t)=(2 + a)*(1 + 0.5*sin(pi*t))

• x(0) = 1. & y(0) = 0.5

Remark 1: Since ODE methods work as well for systems of vector ODEs, you should compare the results for the following methods with starting value t = 0.00 and stepsize h = 0.02 until t = 2.00. Use the following three methods of the Maple DEtools package with DEplot:

1. YEM:. Euler's method (EM; called 'method=euler' option in DEplot).
    
   
2. YMODEM:. Modified Euler's method (ModEM; called Improved or 'method=impeuler' option in DEplot).
    
   
3. YRK4:. Runge-Kutta 4th order method (RK4, default method in DEplot).
    
   
4. (revised) Extra Credit ONLY: YPECE. Adams "PECE" method using y(n,i) for i = 1 to 2 components, n = 0 from the initial conditions, and n = 1 to 3 from an RK4 subroutine of your own as needed to start it up. (PECE = predict+evaluate+correct+(re-)evaluate). You will have to use a Maple Procedure such as in the Maple Example ODE worksheet:
   
   Euler's, Modified Euler's and Exact Methods Maple Worksheet
   Caution: See the remarks on the Class Maple Tools Page.
   (Else Click Here for Plain Text).