This program computes the solution of the Cauchy problem dy(t)/dt = f(t,y) with initial condition y(t0)=u0 until final time T. In particular it considers f(t,y) = -t*exp(-y).
The program makes use of the theta-method to approximate the time derivative. Moreover it exploits the Newton method for the solution of the arising non-linear equation at each time step (the derivative of the non-linear function in the Newton method is approximated by centered finite differences).
The following parameters are taken in input from command line thanks to GetPot: 1) theta = parameter of the theta-method; 2) N = number of time steps; 3) t0 = initial time; 3) T = final time; 5) u0 = initial condition.
Example of execution: ./main N=150 theta=0.5 t0=0.0 T=1.0 u0=0.0.
By default: N = 100, theta = 1 (Backward Euler scheme), t0 = 0.0, T = 1.0, u0 = 0.0.
In this directory, make produces the executable which is just called main.
The program is structured so that it will plot automatically the numeric solution, alongside the exact one, using GnuPlot and it can be run by calling the command ./main.