Recovery of an initial temperature of a one-dimensional body from finite time-observations

Abstract

Under the Dirichlet boundary setting, Aryal and Karki (2022) studied an inverse problem of recovering an initial temperature profile from known temperature measurements at a fixed location of a one-dimensional body and at linearly growing finitely many later times within a bounded interval. This paper studies the problem under the Neumann boundary conditions. That is, under this boundary setting, we suitably select a fixed location x0 on the body of length π and construct finitely many times tk, k = 1, 2, 3, . . . , n that grow linearly with k and are in [0, T] such that from the temperature measurements taken at x0 and at these n times, we recover the initial temperature profile f(x) with a desired accuracy, provided f is in a suitable subset of L2[0, π].