Nonlinear inverse problem of control diffusivity parameter determination for a space-time fractional diffusion equation

doi:10.1016/j.amc.2020.125589

Applied Mathematics and Computation

Volume 390, 1 February 2021, 125589

https://doi.org/10.1016/j.amc.2020.125589 Get rights and content

Abstract

In paper the nonlinear inverse problem of finding a control parameter in a space-time fractional diffusion equation with varying degrees of heterogeneity is investigated. An iterative method for finding an unknown control parameter is proposed. Under the maximal regularity condition respecting input data on the Bessel potential scale, the existence and uniqueness of a solution of such inverse problem are shown. Results are illustrated by examples. An example of numerical simulation of an iterative algorithm is given.

Introduction

The aim of this paper is to find a computational method for finding an unknown control parameter Φ in the inverse hybrid problem for time (β)-fractional diffusion equation with spatial heterogeneity reflected by (α/2)-fractional Laplacian $D_{t}^{β} u (x, t) + a^{2} (- Δ)^{α / 2} u (x, t) - Φ (t) u (x, t) = F (x, t), (x, t) \in R^{n} \times (0, T]$ where the hybridity property lies in the inequality α > β with β ∈ (0, 1) (see, e.g. [3]) and $D_{t}^{β} u (x, t) = \frac{1}{Γ (1 - β)} (\frac{\partial}{\partial t} \int_{0}^{t} \frac{u (x, τ)}{(t - τ)^{β}} d τ - \frac{u (x, 0)}{t^{β}})$ is the regularized (or Caputo-Djrbashian-Nersessian) derivative of order β ∈ (0, 1), as well as, $(- Δ)^{α / 2}$ is defined by the Fourier transform $F [(- Δ)^{α / 2} ψ (x)] = | λ |^{α} F [ψ (x)] .$

The considered inverse coefficient problem arose from an attempt to apply a mathematical model of fractional diffusion with spatial heterogeneity. More precisely, the inverse coefficient problem is to find simultaneously the anomalous diffusion distribution u and an unknown coefficient Φ in the space-time fractional diffusion equation with spatial heterogeneity and a given source parameter F, additionally satisfied an over-determination conditions on the diffusion energy, denoted further by E.

Turned out that the investigated inverse problem is essentially nonlinear and in this formulation it not been studied before. The founded algorithm of solution is based on the principle of a fixed point and significantly uses the property of maximum regularity expressed in coercive inequalities (2.6) established in Lopushansky [13] for the case of the Cauchy problem $\begin{matrix} D_{t}^{β} u (x, t) + a^{2} (- Δ)^{α / 2} u (x, t) & = F (x, t), (x, t) \in R^{n} \times (0, T], \\ u (x, 0) & = u_{0} (x), x \in R^{n} \end{matrix}$

The Cauchy problem for different types of space-time fractional diffusion equations have been studied in Anh and Leonenko [2], Duan [5], Gorenflo et al. [7], Hanyga [8], Mainardi [16], Metzler and Nonnenmacher [19] and many others. Inverse problems for such equations arise in many branches of science and engineering from the requirement to interpret indirect measurements and were studied in Aleroev et al. [1], Cheng et al. [4], Rundell et al. [22], Zhang and Xu [23] and others.

The space-time fractional diffusion equations in various applications simulate anomalous diffusion and spatial heterogeneity (see [9], [10], [11], [12], [14], [15], [17], [18], [19], [20], [23]). Note that there are still many unresolved problems if spatial heterogeneity is described by the fractional Laplacian (see, e.g. [21]).

Section snippets

The main result

We investigate the nonlinear inverse problem for the space-time fractional diffusion equation $D_{t}^{β} u (x, t) + a^{2} (- Δ)^{α / 2} u (x, t) - Φ (t) u (x, t) = F (x, t), (x, t) \in Q_{T},$ $u (x, 0) = u_{0} (x), x \in R^{n},$ $〈 u (\cdot, t), φ_{0} (\cdot) 〉 = E (t), t \in [0, T]$ on $Q_{T} = R^{n} \times (0, T]$ with an unknown function Φ ∈ C[0, T] under the following assumptions:

(A)
${\begin{matrix} β \in (0, 1), α > β, 0 < θ < 1, s \in R, 1 < p < (1 - β θ)^{- 1} \\ F \in C ([0, T]; H^{s + α θ, p} (R^{n})), u_{0} \in H^{s + α, p} (R^{n}) \end{matrix}$
(B)
${\begin{matrix} E, D_{t}^{β} E \in C [0, T], φ_{0} \in H^{- s, p^{'}} (R^{n}) \\ | E (t) | \geq c_{0} = const > 0, \forall t \in [0, T] . \end{matrix}$

Wherein, the Bessel potential spaces with $s \in R$ and p > 1 are defined to be $H^{s, p} (R^{n}) = {v \in S^{'} (R^{n}) {: ∥ v ∥}_{H^{s, p}}$

Justification of the main result

The mentioned results from [13] and properties of the Green vector-function imply that the solution $u \in C_{α, β} ([0, T]; H^{s, p} (R^{n}))$ of (2.1)–(2.2) satisfies the Eq. (2.8).

Lemma 3.1

Lopushansky [13]

For ϱ ≤ α the functions $g_{j} (ξ, t, ϱ) = ω^{\frac{ϱ}{α}} (ξ) F [G_{j}] (ξ, t), j = 0, 1,$ are continuous and bounded in variable $ξ \in R^{n}$ for any t ∈ (0, T], and there exists a constant $c = c (p) > 0$ such that the following inequality holds, $∥ F^{- 1} [g_{j} (\cdot, t, ϱ) F [f]] ∥_{L_{p} (R^{n})} \leq c h_{j} (t, ϱ) {∥ f ∥}_{L_{p} (R^{n})}, j = 0, 1,$ for all p > 1, $f \in L_{p} (R^{n}),$ t ∈ (0, T], where is denoted $h_{0} (t, ϱ) = t^{β - 1} h_{1} (t, ϱ), h_{1} (t, ϱ) = \max {1, t^{-}$

Algorithm description

The proof of main Theorem 2.1 contains both the inverse problem solution algorithm based on the principle of a fixed point. Namely, if the inequality $T_{0}^{β} < \frac{c_{0}}{C_{1} M}$ holds, then for each initial iterative data $u_{0} \in M_{R} (Q_{T_{0}})$ the corresponding iteration algorithm $u_{n} = P^{n} u_{0} + w_{0}, n \in N$ is convergent in $C ([0, T_{0}]; H^{s + α, p} (R^{n}))$ to a unique solution $u \in C_{α, β} ([0, T_{0}]; H^{s, p} (R^{n}))$ of the Eq. (2.8) for any ${∥ Φ ∥}_{C [0, T_{0}]} \leq M$ such that $(Φ, v) \in M (Q_{T_{0}})$ .

Now, we have to substitute these iterations u_n in the formula (2.7) to calculate subsequent

Conclusions

The considered inverse coefficient problem can be used in an anomalous diffusion transfer process with varying degrees of spatial heterogeneity, where a source parameter is present (Fig. 1, Fig. 2, Fig. 3, Fig. 4).

Such problems arise when describing diffusion processes in nanomaterials with a highly inhomogeneous spatial structure, such as nanoceramics.

If we let u(x, t) represent the temperature distribution, then the above-mentioned problem can be regarded as a control problem with a given

Acknowledgements

The authors would like to thank referees for valuable comments which helped to improve the manuscript.

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Short CommunicationNonlinear inverse problem of control diffusivity parameter determination for a space-time fractional diffusion equation

Abstract

Introduction

Section snippets

The main result

Justification of the main result

Lopushansky [13]

Algorithm description

Conclusions

Acknowledgements

Appl. Anal.

Inverse Probl.

Determination of a source term for a time fractional diffusion equation with an integral type over-determination condition

Electron. J. Differ. Equ.

Spectral analysis of fractional kinetic equations with random datas

J. Stat. Phys.

Hybrid inverse problems and internal functionals

Inside Out II MSRI Publications

Uniqueness in an inverse problem for a one-dimentional fractional diffusion equation

Inverse Probl.

Time- and space-fractional partial differential equations

J. Math. Phys.

Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type

Time-fractional diffusion equation in the fractional Sobolev spaces

Fract. Calc. Appl. Anal.

Multi-dimensional solutions for space-time-fractional diffusion equations

Philos. Trans. R. Soc. A

Fractional Calculus: An Introduction for Physicists

Short Communication
Nonlinear inverse problem of control diffusivity parameter determination for a space-time fractional diffusion equation