By Albert R. Baswell
Advances in arithmetic learn offers unique study effects at the cutting edge of arithmetic study. every one article has been conscientiously chosen in an try to current immense learn effects throughout a large spectrum.
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Extra resources for Advances in Mathematics Research
Shoucri B = B0 e z + ∇ψ x e z , where B0 is constant and ψ ( x, y, t ) is the magnetic flux function. 48) and the Lagrangian invariants G ± are conserved fields advected along the characteristic curves, x ± (t ) : dx ± (t ) / dt = υ ± ( x ± , t ) , υ ± ( x ± , t ) = e z x ∇φ ± where φ ± = ϕ ± ( ρ s / d e )ψ . 49) is the electron collisionless skin depth and ρ s = (M e / M i )1 / 2 v the / ω ci is the ion sound Larmor radius, where vthe is the electron ω ci is the ion cyclotron frequency. 48) have the form of 1D Vlasov equations, with x and y playing the role of the coordinate and the conjugate momentum for the equivalent ‘ distribution functions’ G ± of two ‘particle ‘ species with opposite charges in the Poissontype equation for ϕ , and equal charges in the Yukawa-type equation for ψ [62,63,64].
47) X e ,i = (x, p xe,i ) , vector and ⎛ p xe,i me ,i ∂ ( a ⊥( n +1 / 2) ) 2 ⎞ ⎟ . 10). 12). 3 and in . The pump wave is penetrating the plasma at the left boundary at x=0 where we set E ( x = 0) = 2 E0 sin(ω 0 t ) and + F − ( x = 0) = 2 E0 cos(ω 0 t ) . 5(ω pe / c ) . 4(ω pe / c) . 86(ω pe / c) . 5 . 265 , and N x = 5000 grid points in space, N v = 256 grid points in velocity space for electrons and 128 for ions. 0105 . (31-37) show the results obtained when using the present fully relativistic model.
Wavenumber spectrum for the plasma wave. 4(ω pe / c) . e. the harmonic of the pump wave ω 0 . e. the harmonic of the pump wave k 0 . 43). 43) applied to the problem of inductive coupling. Indeed, if we assume a linearly polarized wave: E = (0, E y ,0) , we can write in a linear analysis E y = E 0 cos(ψ ), ψ = (k 0 x − ω 0 t ) . Faraday’s law is: ∂E y ∂B = (0,0,− ). 61) Then B = (0,0, B z ) with B z = B0 cos(ψ ), and B0 = E 0 k 0 / ω 0 . 44) v = (0, v y ,0) with v y = −v 0 sin(ψ ), and v0 = E 0 / ω 0 .
Advances in Mathematics Research by Albert R. Baswell