By V. A. Tkachenko

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B reaction scheme, however now with immobile A and mobile B) corresponds to the target problem, or scavenging. In the ﬁrst case the survival probability ˚(N) for the A-particle after N steps goes for p small as ˚(t) / hexp( pS N )i, while for the second case it behaves as ˚(t) / exp( phS N i). The simple A C B ! B reaction problems can be also interpreted as the ones of the distribution of the times necessary to ﬁnd one of multiple hidden “traps” by one searching agent, or as the one of ﬁnding one special marked site by a swarm of independent searching agents.

Anomalous Diffusion The Einstein’s postulates leading to normal diﬀusion are to no extent the laws of Nature, and do not have to hold for a whatever random walk process. Before discussing general properties of such anomalous diﬀusion, we start from a few simple examples [10]. consider the random walk to evolve in discrete time; the time t and the number of steps n are simply proportional to each other. Diﬀusion on a Comb Structure The simplest example of how the subdiﬀusive motion can be created in a more or less complex (however yet non-fractal) geometrical structure is delivered by the diﬀusion on a comb.

U3 (x; t) D L t 1 (u2 (x; t)) x x t C L t 1 ( 3 A2 C 2B2 C C2 ) 1 4 x 3 D c e t ; 3! u4 (x; t) D L t 1 (u3 (x; t)) x x t C L t 1 ( 3 A3 C 2B3 C C3 ) 1 D c 4 ex t 4 ; 4! (55) Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory KdV equation, named K(m,n), of the form u t C (u m )x C (u n )x x x D 0 ; Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 3 Figure 3 shows the peakon graph u(x; t) D e j(x ct)j ; c D 1; 2 Ä x; t Ä 2 and so on.