Problems 8
  1. Theory.
    1. How can you find out the declaration of the struct gsl_odeiv2_system?
      Hints:
      • #include<gsl/gsl_odeiv2.h> and gcc -E;
      • gsl-config --cflags and view.
  2. Practice.
    1. Solve, using GSL, the linear system (from Numerisk Fysik 2011)
      x1 -x2 +2x3 =2
      2x2 -x3 =-1
      x1 +x2 +3x3 =3
      Hint: probably the best way to solve such system is via LU-decomposition using the GSL functions 
      gsl_linalg_LU_decomp
gsl_linalg_LU_solve
      or via QR-decomposition using the GSL functions 
      gsl_linalg_QR_decomp
gsl_linalg_QR_solve
      See
      GSL manual → Linear Algebra → LU Decomposition;
      GSL manual → Linear Algebra → QR Decomposition;
      GSL manual → Linear Algebra → Linear Algebra Examples.
    2. Consider the Lotka-Volterra equations (predator-prey equations) (from Numerisk Fysik 2011) (look it up in Wikipedia),
      dr/dt = r(α-βf)
      df/dt = -f(γ-δr)
      where r is the number of prey (for example, rabbits); f is the number of some predator (for example, foxes); dr/dt and df/dt represent the growth rates of the two populations over time; t represents time; and α, β, γ and δ are parameters describing the interaction of the two species.
      • Build a function for the Lotka-Volterra equations to be used as gsl_odeiv2_system.function .
        Hint: 
        struct my_params {double alpha,beta,gamma,delta;};
int lotka(double t, const double y[], double dydt[], void* params){
	struct my_params p = *(struct my_params *)params;
	dydt[0]= y[0]*(p.alpha-p.beta*y[1]);
	dydt[1]=-y[1]*(p.gamma-p.delta*y[0]);
return GSL_SUCCESS;
}
      • Choose for example α=1, β=0.1, γ=2, δ=0.05, and solve the system numerically using GSL with the initial condition, for example, x=100, y=10. Plot the solutions x(t) and y(t) as functions of t on the same plot. Make also a parametric plot y versus x for different values of the parameters (let them approach the steady-state solution).