In this paper, by arithmetic-geometric mean inequality and Riccati transformation, interval oscillation criteria are established for second-order forced impulsive differential equation with mixed nonlinearities of the formwhere t t 0 , k ∈ N; Φ * (u) = |u| * −1 u; {τ k } is the impulse moments sequence with 0 t 0 = τ 0 < τ 1 < τ 2 < · · · < τ k < · · · and lim k→∞ τ k = ∞; α = p/q, p, q are odds, and the exponents satisfy β 1 > · · · > β m > α > β m+1 > · · · > β n > 0.Some known results are generalized and improved. Examples are also given to illustrate the effectiveness and non-emptiness of our results.
We study the following second order mixed nonlinear impulsive differential equations with delay(r(t)Φα(x′(t)))′+p0(t)Φα(x(t))+∑i=1npi(t)Φβi(x(t-σ))=e(t),t≥t0,t≠τk,x(τk+)=akx(τk),x'(τk+)=bkx'(τk),k=1,2,…, whereΦ*(u)=|u|*-1u,σis a nonnegative constant,{τk}denotes the impulsive moments sequence, andτk+1-τk>σ. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.
A unified full-wave characterization of massive number of vias with or without circular pads is formulated analytically by means of equivalent magnetic frill array model and Galerkin's procedure. The proposed method takes advantage of the parallel-plate structure and employs image theory and Fourier transform to simplify the problem from 3-D configuration into 2-D frame. Based on the cylindrical symmetry with Bessel's functions and addition theorem, the final matrix equation is formulated analytically which can be used immediately for sensitivity analysis in both via dimensions and pad size. As a result, the new method is simple, efficient, and accurate. Numerical examples demonstrate good agreement between our analytical solution and the results obtained by commercial software (high frequency structure simulator) over a frequency range up to 20 GHz.
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