Acceleration by the plane shock wave diffusion approximation

Let us suppose, following to Berezhko et al. (M1988), that the plane shock wave propagates in the negative direction of the x-axis and the diffusion coefficient for particle scattering in the background plasma is k. In this case the transport equation in the system of coordinates of the shock front will be df^pA = A f K^kp*) - u - s(x)p ^P^ + Q(x, p ), (4.21.

where k is the diffusion coefficient, Q(x,p) is the source of particles, Au = u\ - . and i(x < 0) = ui, u(x > 0) = U2 .

Let us take into account that most probably the particle injection into the acceleration process is from the region of shock front the with of which l is supposed to be much smaller then the transport path À; in this case Q(x, p ) = Qo (p )£(x ). The other possibility which may be realized is the existence in the non-disturbed region 1 of fast particles with some spectrum f1^(p ); in this case we shall have the boundary condition at x ^ :

The boundary condition for f (x, p, t) at x = 0 will be

The solution of Eq. 4.21.12 with boundary conditions described by Eq. 4.21.14 and Eq. 4.21.15 was found in Krymsky (1977), Berezhko et al. (M1988) as follows:

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