Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition

Abstract

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.

Inverse problem of determining the order of the fractional derivative in the Rayleigh-Stokes equation

“…Based on physical considerations, usually the authors consider this problem in the domain Ω ⊂ R N , N = 1, 2, 3, and for N > 1 it is assumed that the boundary ∂Ω of the domain Ω is sufficient smooth (see e.g. [1], [3], [4]). This equation for α = 1 is called the Haller equation and is a mathematical model of water movement in capillary-porous media, which include soils (see, for example, in [5], formulas (1.4) and (1.84) on p. 137 and 158, [6], formula (9.6.4) on p. 255, and [7], formula (2.6.1) on p. 59).…”

“…A review of some works in this direction is contained in the above-mentioned paper [1]. See also recent papers [13], [14] and references therein; 4) Many works are devoted to the study of the inverse problem of determining the right-hand side of the Rayleigh-Stokes equation (see, for example, [3], [15], [16], and the bibliography cited there). Since this inverse problem is Hadamard ill-posed, various regularization methods and numerical methods for finding the right-hand side of the equation are proposed in these works; 5) If the initial condition u(x, 0) = ϕ(x) in the problem (1.1) is replaced by u(x, T ) = ϕ(x), then the resulting problem is called backward problem.…”

A non-local problem for the fractional order Rayleigh-Stokes equation

“…The study of the inverse problem of determining the right-hand side of the Rayleigh-Stokes equation is the subject of many works (see, for example, [15], [16], [17] and the bibliography cited there). Since this inverse problem is ill-posed in the sense of Hadamard, various regularization methods are considered in the above works, as well as numerical methods for finding the right-hand side of the equation are proposed.…”

Backward and Non-Local Problems for the Rayleigh-Stokes Equation

“…Quite a lot of works are devoted to the study of the inverse problem of determining the right-hand side of the Raleigh-Stokes equation (see, e.g., [13][14][15] and the bibliography therein). However, the case when the right-hand side has the general form f (x, t) has not yet been considered by anyone.…”

“…(1) We will give a mathematically justified solution to the forward problem. A formal formula for the solution in the form of eigenfunction expansions was given in the paper [10], cited above (see also [15,19,20]), but the convergence of the series itself and the differentiated series has not been investigated. (2) We will pay special attention to the backward problem, since in previous papers (see, e.g., [19,20]) the authors considered only the case N ≤ 3.…”