COMPUTATION OF COSMIC-RAY ACCELERATION BY ITOS STOCHASTIC DIFFERENTIAL-EQUATIONS

We describe here a method to solve the general transport equation of c osmic rays numerically, including diffusive shock acceleration, second -order Fermi acceleration, adiabatic gains or losses, and synchrotron losses. We use the equivalence between Fokker-Planck equations and sto chastic (ordinary) differential equations (SDEs) to transform the tran sport equation to a set of SDEs which is easier to implement numerical ly than the original partial differential equation. We are able to com pute the cosmic-ray distribution function in the vicinity of a shock, determining its power-law slopes and cutoff energies for an arbitrary dependence on momentum, spatial position, and time of the diffusion co efficients. We use this method to analyse the influence of Kolmogorov Alfven wave turbulence on the characteristic properties of the solutio ns of the cosmic-ray transport problem near a strong shock. We have ob tained the following results: The momentum spectrum of the accelerated particles is divided into two different regions. At lower momenta the spectrum is dominated by the influence of the second-order Fermi acce leration by Alfven waves in the vicinity of the shock. There is a turn over to a second region at higher energy, which is governed by the fir st-order Fermi process of shock acceleration. This turnover results fr om the fact that the typical momentum diffusion-time scales increases faster with momentum than the mean residence time of the particles in a finite acceleration region near the shock. The ''first-order part'' at higher energies shows a negligible influence of the second-order pr ocess. We propose this part to be responsible for the synchrotron spec trum as observed in the radio lobes of extragalactic radio sources. Ne vertheless, Alfven speeds in these objects could be large, so that the turnover in the spectrum could move to observable frequencies if the size of the acceleration region is small.