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the first explanation ( or rather prediction ) of the power law of primary cosmic ray spectrum made by e. fermi was purely phenomenological : it was based on the exponential growth of energy with time and exponential distribution of ages of observed particles . both of them ignored the real space - time distribution of sources , highly inhomogeneous structure of the interstellar medium and boundedness of the galaxy . a few years later , the cr propagation was interpreted as some kind of brownian motion and began to be calculated in terms of the ordinary ( normal ) diffusion theory @xcite . it is easy to understand the origin of this model : fermi wrote about random walk of cr particles among magnetic clouds chaotically distributed in interstellar space , and brownian motion was the most expressive image of such a process . however , its mathematical representation in the form of the _ local diffusion _ ( lod ) equation ( or its other local modifications ) did not involve the particle velocity in an explicit form and for this reason failed in description of front propagation and some other pecularities of the cr transport . this is just the same task which inspired authors of the work @xcite to seek a new propagator in the frame of _ nonlocal diffusion _ ( nod ) model . we talk here on locality because the generic sign of the diffusion approximation is the local laplace operator @xmath0 . its application assumes uniformity and mutually independence of points of scattering like those we observe in ideal gases . no doubt , distribution of irregularities in the interstellar magnetic field do not possess this property , they are linked with each other by means of more or less long magnetic force lines . these lines are frozen into the material component of interstellar plasma and are involved into the common turbulent motion . it is rather strange , why the results in turbulent diffusion theory obtained in classical works followed the kolmogorov ( 1941 ) phenomenological basis of the turbulence @xcite were completely ignored in the early development of the cr - propagation theory . one of the generic features of the turbulent diffusion is the presence of nonlocal operators in its equation . involvement of non - local operator @xmath1 instead of its local counterpart @xmath2 has shown that in case of a single local pulse source the knee arises without any additional assumptions except the turbulent ( called also _ fractal _ ) character of the interstellar medium . during 2001 - 2010 years , a series of calculations were performed by lagutin et al . with the use of nod - model , but all results were obtained without account for the relativistic limitation of speed and the boundary conditions . the speed limitation was inserted into the nod - model in @xcite and this version was later named the _ nonlocal relativistic diffusion _ ( nord ) model @xcite . the physical interpretation of the nord model and first results obtained in its frame were described in reviews @xcite . here , we report some new results of numerical investigations in the frame of this model adding account of multiplicity of sources and boundedness of halo . modeling of cr sources was performed under following assumptions .    1 .   all sources are isotropic , point and instantaneous with equal powers and energy spectra @xmath3 . here @xmath4 is an exponent of injection spectrum , and @xmath5 characterizes maximum energies of @xmath6-type nuclei accelerated at supernova remnants . the sources are separated into two groups : relatively young supernovae ( less than @xmath7 years ) whose space - time coordinates were taken individually ( 15 objects : vela , monogem ring , geminga and others ) , and 30000 sources with spatial distribution generated according to approach proposed in @xcite and used in @xcite . the radial distribution of discrete sources was simulated according to the density of pulsars ( see @xcite ) , @xmath8 with @xmath9  kpc and @xmath10 , the @xmath11-distribution was modeled by exponential function with @xmath12  pc . to simulate the spiral structure , the algorithm proposed in  @xcite was used . the example realization is presented in fig . [ fig_sources ] . the area of cr diffusion is the cylinder of height @xmath13  kpc with radius @xmath14  kpc surrounded by a non - reflective environment . the galactic disk being in the middle of the cylinder parallel to bases has the thickness @xmath15 . as in ref . @xcite , we assume that the diffusion coefficient is independent of spatial and time variables , and determined by particle rigidity @xmath16 , @xmath17 , \quad r>3gv.\ ] ] results reported here are related to values @xmath18 and @xmath19  @xmath20 .      to account for the halo boundedness , solutions ( [ eq_lod ] ) should be modified : considering zero boundary conditions and neglecting by lateral surface of the cylinder , the image method can be used in local approach , @xmath23 @xmath24 here @xmath25 are @xmath26-coordinates of the image sources , @xmath27 are coordinates of an original source .    the nod - propagator in an infinite medium is expressed through fractional stable lvy - feldheim density @xcite . in case of @xmath28 it can be presented in the form @xcite @xmath29 @xmath30 here @xmath31 is a one - sided lvy stable density ( subordinator , @xmath32 ) and @xmath33 in ( [ eq_nod ] ) be taken with the diffusion coefficient linked with @xmath34 of the nod - model by relation @xmath35^{2/\alpha}.\ ] ] if @xmath36 , then @xmath37 , where @xmath38 . for example , for @xmath18 and @xmath39 , we have @xmath40 .    formula ( [ eq_nod ] ) is suitable for monte carlo representation of the nod - propagator @xmath41 where @xmath42 is a stable random variable generated according to ( see e.g. @xcite ) @xmath43 @xmath44^{2/\alpha-1}}{[\sin ( \pi u_1)]^{2/\alpha}[\ln u_2]^{2/\alpha-1}}.\ ] ] formulae ( [ eq_diff_coeff ] ) and ( [ eq_nod_mc ] ) indicate the correspondence principle between nod- and lod - models ( when @xmath45 ) and allow us to use the standard diffusion coefficient to estimate cr characteristics in frames of the nonlocal models .    in calculations of energy spectrum in frames of the nord - model ( for @xmath46 ) we have used approximate expression for propagators @xmath47 @xmath48 @xmath49 where @xmath50 is the heaviside step function . it should be noted that zero boundary conditions for nonlocal models are fulfilled approximately , if the mean scattering length of crs much smaller than thickness of the halo . to account for restriction of the diffusion domain ( halo ) , the generalized marshak boundary conditions can be used . the solutions have been modified by means of the image method . in this approximation , we use @xmath51 instead of @xmath33 in expressions  ( [ eq_nod ] ) and ( [ eq_nod_mc ] ) using procedure ( [ eq_nord ] ) . we estimated energy spectra accounting for spatial and temporal distribution of supernova remnants and diffusion in halo in frames of four models ( lod , lord , nod , nord ) . note , that such calculations for the local model were performed in many papers ( see e.g. @xcite ) and for nonlocal model in refs . all of them , however , utilize propagators ignoring the relativistic speed limit requirement . relativistic nature of crs sufficiently modify the propagators and spectra as well . nord - propagators are restricted to domain @xmath52 , have ballistic splash at this boundary whereas in the middle part they are spreading superdiffusively ( remind , that we restrict ourselves by values of superdiffusion parameter @xmath53 $ ] ) . spallation effects are taken into account according to blasi and amato @xcite using formulae provided in @xcite .              as is known , the standard assumptions about the origin of galactic crs : power law spectra of particles injected by snr , and diffusive propagation throughout the galaxy with a coefficient @xmath36 leads to the spectrum @xmath55 ( from the snr ensemble ) reflecting the balance between injection and escape of crs from the confinement volume . accounting for relativistic restriction ( in terms of relativistic propagators ) steepens the spectrum at high energies , and the position of the spectrum break , its sharpness and slopes of asymptotes depend on @xmath56 and  @xmath57 @xcite . the cause of this effect is universal and must be kept in mind in all calculations of propagation of high energy crs . explanation of the spectrum steepening at high energies in the nord model is schematically represented in fig . [ fig_steepening ] . the grey field corresponds to the space - time area containing cr sources which do not take part in the formation of the spectrum due to the relativistic restriction . their accounting in the ordinary diffusion model leads to a pure power law spectrum @xmath58 under the assumption about the pure power law spectrum of injected particles . taking the relativistic restriction into account in the nord model excludes automatically these sources and this exclusion affects predominantly on high energy part of crs due to @xmath36 . it is important to emphasize that , the nord model ( compared to the nod - version used by lagutin et al . ) operates with acceptable values of injection exponent @xmath4 ( @xmath59 ) ( in contrast with @xmath60 , e.g. @xmath61 , in recent calculations by lagutin et al . @xcite ) . [ proton ] , [ cno ] and [ total ] demonstrate computational results for spectrum of protons , cno - group and all nuclei obtained in frames of lod , lord , nod and nord models . relativistic speed limit requirement taken into account in lord and nord leads to steepening of spectra as mentioned above . the energy of the ` knee ' @xmath62 obtained in nord is very close to provided in @xcite by exponential cut - off of injection spectrum , but we observe power law nord - spectrum above @xmath62 in contrast to exponential decay obtained in  @xcite . sure , this requires greater maximum energies of particles accelerated in sources . the exponentially falling tails for all nuclei contradict to observed power law spectrum above @xmath62 , if we consider it as produced by gcrs ( without extragalactic component ) . from this point of view nord - model seems to be preferable . lagutin , a. a. , nikulin , y. a. , & uchaikin , v. v. ( 2001 ) . the ` knee ' in the primary cosmic ray spectrum as consequence of the anomalous diffusion of the particles in the fractal interstellar medium . _ nuclear physics b  proceedings supplements _ , 97(1 ) , 267 . erlykin , a. d. , sibatov , r. t. , uchaikin , v. v. , & wolfendale , a. w. ( 2015 ) a look at the cosmic ray anisotropy with the nonlocal relativistic transport approach . _ proceedings of science _ ( icrc2015 ) 463 . blasi , p. , & amato , e. ( 2012 ) . diffusive propagation of cosmic rays from supernova remnants in the galaxy . i : spectrum and chemical composition . _ journal of cosmology and astroparticle physics _ , 2012(01 ) , 010 . evoli , c. , gaggero , d. , grasso , d. , & maccione , l. ( 2008 ) . cosmic ray nuclei , antiprotons and gamma rays in the galaxy : a new diffusion model . _ journal of cosmology and astroparticle physics _ , 2008(10 ) , 018 . lagutin , a. a. , tyumentsev , a. g. , volkov , n. v. , & raikin , r. i. ( 2015 ) . spectra of cosmic - ray protons and nuclei from @xmath63 to @xmath64  ev within the galactic origin scenario of cosmic rays . _ bulletin of the russian academy of sciences : physics _ , 79(3 ) , 322 - 325 .
energy spectra of galactic cosmic rays calculated within the framework of the nord ( nonlocal relativistic diffusion ) model are presented . 
 the model accounts for the turbulent character of the interstellar medium and the relativistic speed limit requirement . 
 calculations account for spiral distribution of sources , boundedness of halo , spallation of nuclei , energy dependence of the diffusion coefficient , tempered power law injection spectrum . 
 we show that the knee in the background spectra from the ensemble of supernovae can arise due to relativistic speed limit requirement . 
 we compare our calculations with the results obtained in frames of the local models ( lod and lord ) and nonlocal nonrelativistic model ( nod ) .