Abstract:We study an inverse initial data problem for a convection-diffusion equation with memory, where the goal is to recover the unknown initial condition from final-time data. The model includes convection, an instantaneous Laplacian term, and a nonlocal-in-time memory term involving the Laplacian of the past states, which leads to a severely ill-posed backward problem. We prove uniqueness in a spatially independent coefficient setting by applying the Fourier transform and using an analyticity argument for a scalar Volterra equation. For the variable-coefficient case, we develop a computational method based on Legendre spatial dimensional reduction and Tikhonov regularization. The solution is approximated by a finite tensor-product Legendre expansion, thereby reducing the inverse problem to a finite-dimensional terminal-value system for the time-dependent coefficients. We solve the reduced problem by a Tikhonov-regularized least-squares method with an $H^2$ penalty. For a fixed truncation order, we prove that the regularized minimizers converge to the finite-dimensional minimum-norm solution as the noise level and the regularization parameter vanish, under a suitable choice of the regularization parameter. Some two-dimensional numerical examples are presented to illustrate the performance of the proposed method.
From: Loc Hoang Nguyen [view email]
[v1]
Thu, 18 Jun 2026 19:06:16 UTC (2,887 KB)
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