Design and construct a computer program in one of the approved languages ( Java, ) that...

 

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Design and construct a computer program in one of the approved languages ( Java, ) 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-Ku_a 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 Design and construct a computer program in one of the approved languages ( Java, ) 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-Ku_a 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