Computational Physics - Department of Physics

8.7 Physics Project: the pendulum 269 out[1]=-in[1]/((double)Q)-sin(in[0])+A_roof*cos(omega_roof*t); //out[1] = (phi)'' else out ...

270 8 Differential equations t_h+=delta_t_roof; y_h[0]=yout[0]; y_h[1]=yout[1];} fout.close;} int main(){pendelum testcase; test ...

8.8 Exercises 271 Clearly, the relevant degrees of freedom will not be the samein the crust region of a neu- tron star, where th ...

272 8 Differential equations Dimensionless equations When we now attempt the numerical solution, we need however to rescale the ...

8.8 Exercises 273 and M 0 =^4 π ρs ( √ ρsG 4 π)^3 . However, since we would like to have the radius expressed in units of 10 km, ...

274 8 Differential equations Employ now the fourth order Runge-Kutta algorithm to obtain new values for the pressure and the ma ...

8.8 Exercises 275 Keep now the constants from the previous exercise fixed butset nowA= 1. 35 ,A= 1. 44 and A= 1. 465. Plotθ(in ...

276 8 Differential equations Check also for the case of a circular orbit that both the kinetic and the potential energies are co ...

8.8 Exercises 277 helion precession, when all classical effects (such as the perturbation of the orbit due to gravitational attr ...

278 8 Differential equations Å (solid argon),σ= 3. 405 Å,ε= 1. 0318 · 10 −^2 eV. Another common practice is puttingE 0 = 4 ε, af ...

8.8 Exercises 279 whereδ∈{− 1 , 0 , 1 }andLis the length of the simulation box in thexdirection. This limits the interaction ran ...

280 8 Differential equations f) In order to simulate the canonical ensemble, interactions with an external heat bath must be tak ...

8.8 Exercises 281 Estimateg(r)forr∈( 0 ,L 2 ]in your argon system. The easiest way is to divide the distance interval into bins, ...

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Chapter 9 Two point boundary value problems AbstractWhen differential equations are required to satisfy boundary conditions at m ...

284 9 Two point boundary value problems 9.2 Shooting methods In many physics applications we encounter differential equations li ...

9.2 Shooting methods 285 We start with the Taylor expansion of the desired solution y(x+h) =y(x)+hy(^1 )(x)+ h^2 2! y(^2 )(x)+ h ...

286 9 Two point boundary value problems y(a) =α,y(b) =β, the requirementy′(a) =δ, whereδ could be an arbitrary constant. In quan ...

9.2 Shooting methods 287 ∂^2 y ∂t^2 =λ ∂^2 y ∂x^2 . We discuss the solution of this equation in chapter 10. Ifλ> 0 the above ...

288 9 Two point boundary value problems Table 9.1Integrated and exact solution of the differential equationy′′+λy= 0 with bounda ...