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Advances in Mathematics Research by Albert R. Baswell

By Albert R. Baswell

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

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Periodic boundary conditions are used in the y direction, and Dirichlet boundary conditions are used in the x direction with φ ( x = 0, L x , y ) = 0 , ρ ( x = 0, L x , y ) = 0 . 1) can be integrated along the characteristic curves given by the solution of the equations: dr = VD (r, t ) . 11)): ρ (r, t n+1 ) = ρ (r (t n−1 ), t n −1 ) . 4) We assume the value of r (t = t n +1 ) = ri , where the ri are grid points. 12)): ri − r (t n−1 ) ⎛ r + r (t n−1 ) ⎞ = VDn ⎜ i ⎟. 5) ∂φ (r, t n ) , and E(r, t n ) is computed by solving Poisson equation for ∂r ρ (r, t n ) at t = t n .

16) Details of these calculations have been presented in [58,59], and lead to the following result for the real and imaginary parts of ω : 34 M. Shoucri Re ω 3 = (k ys − k y ) . 17) Im ω 3 = (k ys − k y ) . 18) Figure 24. Growth and saturation of the potential at the position x=3Lx /4. Figure 25. Contour plot of the potential at t=80. In this case k y < k ys , and the system is unstable [58,59]. 2058. 2 . (24) shows the growth and oscillation of the potential , monitored at the position x = 3Lx / 4; y = 0 .

12). 3 and in [53]. 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. (17). 478ω pe . 4(ω pe / c) for the scattered mode. 18ω pe and k h = 11(ω pe / c) .

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