By Albert R. Baswell

Advances in arithmetic learn offers unique study effects at the cutting edge of arithmetic examine. each one article has been rigorously chosen in an try to current large study effects throughout a wide spectrum.

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**Extra info for Advances in Mathematics Research **

**Example text**

Charge (ni–ne). 8 at t=60. (23) shows the charge (ni –ne ). We note the system is solved for ions and electrons, but the response of the ions at t=60 is still negligible. In the conclusion of this section 2 , we stress the importance of the cubic spline and the good performance of the interpolation technique, having low numerical diffusion and dispersion and high accuracy. We also note the numerical stability of the numerical code. 3. Problems Involving the Interpolation along the Characteristic Curves in Two Dimensions The problems studied in section 2 for the Vlasov equation dealt essentially with the fractional step methods where the interpolation along the characteristic curves was carried out in 1D using a cubic spline.

8 at t=60. (21)) . 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.

We see also from Fig. 1 < vθ > , while along the gradient the electric field is essentially balanced by ∇Pi / ni . Figs. 2 (taking into consideration the mirroring due to the difference in the positive direction). 2 obtained in Cartesian geometry by a fractional step method associated with 1D cubic spline interpolation are the same as those obtained in this section using a cylindrical geometry , and associated with 2D interpolation in velocity space with a tensor product of cubic B-spline. By using two different numerical techniques based on the cubic spline interpolation with two different coordinate systems, we get identic results for the same problem.