Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

Abstract

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.

Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

“…Here we consider the inverse problem of solving the unknowns g 1 , g 2 2 and denote the numerical approximations asĝ 1 ,ĝ 2 2 respectively. After measuring the noisy data h σ (t, ω) disturbed by white noise σ from h(t, ω) = u(x 0 , t, ω), we obtain the mean and variance moments of the fractional integral H σ := I 1−α t h σ (t, ω), denoting as E σ (t) and V σ (t), respectively.…”

“…However, from the properties of W, which are displayed in section 2, the sign of g 2 can not affect the stochastic process g 2 (t)Ẇ(t). Sequentially, g 2 2 is concerned instead of g 2 .…”

“…The precise mathematical description of this inverse problem is given as follows: using the statistical moments of the realizations of h(t, ω) := u(x 0 , t, ω), x 0 ∈ D to recover g 1 and g 2 2 simultaneously.…”

“…In the field of inverse problems, for an extensive review, [24] is referred. For time-fractional inverse problems, see [33,25,2,47,57,55,58,48,22,61,36,60]; for fractional Calderón problems, [18] is one of the first works, and see [49,11,29,17,13,12,20] for more details. Furthermore, if we extend the assumption (∆x) 2 ∝ t α to a more general case (∆x) 2 ∝ F (t), the multi-term fractional diffusion equations and even the distributed-order differential equations will be generated, see [50,34,14] for details.…”

“…The choice of η will depend on the value of d. Numerical reconstruction. In this section, we illustrate the numerical reconstruction of the unknowns g 1 , g2 2 from equation(6). Firstly we setD × [0, T ] = [0, 1] 2 , α = 0.8, x 0 = 1/2, A = −∆, f (x) = sin (πx),then we consider the following experiments:(e1) : g 1 (t) = t + sin (2πt) + sin (3πt), g 2 (t) = sin (2πt) − 0.3, t ∈ [0, 1/2), sin (2πt) + 0.3, t ∈ [1/2, 1],(e2): g 1 (t) = t + sin (2πt) + sin (3πt), g 2 (t) = sin (πt), (e3) : g 1 (t) = sin (2πt) − 0.3, t ∈ [0, 1/2), sin (2πt) + 0.3, t ∈ [1/2, 1], g 2 (t) = sin (πt), (e4) : g 1 (t) = t + sin (2πt) + sin (3πt), g 2 (t) =   t ∈ [0, 0.3), 2, t ∈ [0.3, 0.6), 1, t ∈ [0.6, 1], (e5) : g 1 (t) = sin (2πt) − 0.3, t ∈ [0, 1/2), sin (2πt) + 0.3, t ∈ [1/2, 1], g 2 (t) =   t ∈ [0, 0.3), 2, t ∈ [0.3, 0.6), 1, t ∈ [0.6, 1].…”