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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 and current significant study effects throughout a wide spectrum.

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When studying similarities or differences in the results, attention will be given to the accurate performance of the cubic spline interpolation. 3 ) in a 4D phase-space for the electron distribution function Fe ( x, p xe , p ye , p ze , t ) and the ion distribution function Fi ( x, p xi , p yi , p zi , t ) (one spatial dimension) as follows [52]: ∂Fe ,i ∂t with + me ,i p xe,i ∂Fe ,i γ e ,i ∂x ∓ (E + pxB γ e ,i )⋅ ∂Fe,i ∂p e ,i = 0. 35) (the upper sign is for electrons and the lower sign for ions, and subscripts e or i denote electrons or ions respectively).

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. (17). 478ω pe . 4(ω pe / c) for the scattered mode. 18ω pe and k h = 11(ω pe / c) . 3. 44 M. Shoucri Figure 31. Frequency spectrum of E+ Figure 32. Wavenumber spectrum of E+. 3, E = E 0 (0, cosψ , sin ψ ) , ψ = (k 0 x − ω 0 t ) . Faragay’s law is: ∂E y ∂E ∂B ). = (0, z ,− ∂x ∂t ∂x ∂a ⊥ and p ⊥ = a ⊥ , we get ∂t p ⊥ = p 0 (0,− sin ψ , cosψ ) .

Note again the clear picture of the vortices, in the low density region of the phase-space, with very little numerical noise appearing. Figs(36) and (37) show respectively the electric field and the charge (ni –ne ) across the box. (22) and (23). 46 M. Shoucri Figure 35. 8 at −1 t = 60ω pe . Figure 36. Longitudinal electric field. The Method of Characteristics for the Numerical Solution… 47 Figure 37. Charge (nI – ne ). So we have been able using the technique of cubic spline interpolation to get results from two different models for laser-plasma interaction, using two different numerical codes, which show similarities and differences in the physics associated with the scattering results.

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