Assignment:

Design and construct a computer program in C++ that will illustrate the use of a third-order explicit Runge-Kutta method of your own design. In other words, you will first have to solve the Runge-Kutta equations of condition for the coefficients of a third-order Runge-Kutta method. Then, you will use these coefficients in a computer program to solve the ordinary differential equation below. Be sure to follow the documentation and programming style policies of the Computer Science Department.

The initial value problem to be solved is the following:

x'(t) =1 + sin(12 t) x(t)

subject to the initial condition: x(0) = 1.0

Obtain a numerical solution to this problem over the range from t=0.0 to t=2.0

for seven different values of the stepsize,

h=0.1, 0.05 , 0.025 , 0.0125 , 0.00625 , 0.003125 , and 0.0015625 .

In other words, make seven runs with 20, 40, 80, 160, 320, 640, and 1280 steps, respectively. For each run, print out the value of h and then a table of t and x. The true solution of this differential equation resembles the following plot of x(t) as a function of t.

The answer at the end of the integration is about 2.9769173907278 Hint: It is often helpful to test your program on simple differential equations (such as x' = 1 or x'=t or x'=x) as a part of the debugging process. Once you have worked these simple cases, then try working the nonlinear differential equation given above for the assignment (with a small stepsize). Also, check your coefficients to make sure that they satisfy the equations of condition and that you have assigned these correct values to the variables or constants in your program properly. For example, a common error is to write something like: a2 = 1/2; when you meant to write a2 = 1.0/2.0; so please be careful.

Write down (in your output file or in a text file) any conclusions that you can make from these experiments (e.g., what happens as h is decreased?).