Evolutionary Systems Representations Based on Later Time Samples and Applications to PDEs

Abstract

Let $\{ƒ_n\}^∞_{n=1}$ be a basis for $L_2([0, 1])$ and $\{g_n\}^∞_{n=1}$ be a system of functions of controlled decay on [0,∞). Considering a function u(x, t) that can be the represented in the form

where a_n ∈ R, x ∈ [0, 1] and t ∈ [0,∞), we investigate whether the function ƒ(x) = u(x, 0) can be approximated, in a reasonable sense, by using data u(x₀, t₁), u(x₀, t₂),..., u(x₀, t_N). A mathematical framework and efficient computational schemes are developed to determine approximate solutions for various classes of partial differential equations via sampled data by first establishing a near-best approximation of ƒ .