In this paper, we study the following nonlinear eigenvalue problem: { Δ 2 p u = λ m ( x ) u i n Ω , u = Δ u = … Δ 2 p − 1 u = 0 o n ∂ Ω . \left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\Omega ,} \cr {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 0\,\,\,\,on\,\,\partial \Omega .} \cr } } \right. Where Ω is a bounded domain in ℝ N with smooth boundary ∂Ω, N ≥1, p ∈ ℕ*, m ∈ L ∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2 pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.
In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. We give some experiments to illustrate the success of such an approach. Finally, we apply the higher-order spline interpolants to solve the first kind of Fredholm integral equation and Pocklington’s integro-differential equation to treat the signal reconstruction inside the wire antennas.
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