The quasi-boundary value regularization method for identifying the initial value with discrete random noise

Abstract

In this paper, we study an inverse initial value problem for the fractional diffusion equation with discrete noise. This problem is ill-posed in the sense of Hadamard. We apply the trigonometric method in a nonparametric regression associated with the quasi-boundary value regularization method to deal with this ill-posed problem. The corresponding convergence estimate for this method is obtained. The numerical results show that this regularization method is flexible and stable. MSC: 35R25; 47A52; 35R30

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

“…The truncation method in this paper is similar to the method in [27,18]. The quasi-boundary value method in this section is more effective and useful than the one in [28]. The advantage of this method is that it allows us to estimate the norm on the Hilbert scales H σ (Ω).…”

“…where W k = W k1,k2,...,k d are mutually independent random variables, W k ∼ N (0, 1) and ε k = ε k1,k2,...,k d are positive constants bounded by a positive constant ε max . Some inverse problems when d = 1 were studied in [6,21,28]. Our main contributions in this paper is as follows:…”

Efficient Preconditioning for Time Fractional Diffusion Inverse Source Problems

“…The quasi-boundary value method (QBVM), also referred to as the non-local boundary value method, has been widely used in solving ill-posed inverse problems for parabolic equations, see e.g., [2,13,11,19] and the references therein. Recently, it has been successfully generalized for solving the fractional diffusion inverse problems [42,48,49,50]. Instead of working on (1) and (2), the method deals with a well-posed problem D α t v(x, t) = (Lv)(x, t) + f µ (x)q(t), x ∈ Ω, t ∈ (0, T ), v(x, t) = 0,…”

Regularization of a multidimensional diffusion equation with conformable time derivative and discrete data

“…For applications of partial differential equations and other areas with conformable derivatives, we refer the reader to previous works 1–5,8–28 and the references therein. However, we note 29 (fix and ) a function has a conformable derivative if and only if is differentiable at and .…”