Regularized Reconstruction of the Order in Semilinear Subdiffusion with Memory

Abstract

For $\nu \in (0,1)$, we analyze the semilinear integro-differential equation on  the multidimensional space domain in the unknown $u=u(x,t)$:

$$\mathbf {D}_{t}^{\nu }u-\mathcal {L}_{1}u-\int _{0}^{t}\mathcal {K}(t-s)\mathcal {L}_{2}u(\cdot ,s)ds=f(x,t,u)+g(x,t)$$

where $\mathbf {D}_{t}^{\nu }$ is the Caputo fractional derivative and $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ are uniform elliptic operators of the second order with time-dependent smooth coefficients. We obtain the explicit formula reconstructing the order of the fractional derivative $\nu$ for small time state measurements. The formula gives rise to a regularization algorithm for calculating $\nu$ from possibly noisy measurements. We present several numerical tests illustrating the algorithm when it is equipped with quasi-optimality criteria for choosing the regularization parameters.

Appendix: Proof of Proposition 10.1

First, we note that the statements (i)–(iii) have been obtained in the proof of Proposition 1 [22]. Moreover, statement (v) follows immediately from statement (iv).

Hence, we are left to verify inequality (iv). To this end, we first rewrite the difference $(\mathbf {D}_{t}^{\nu _{2}}U-\mathbf {D}_{t}^{\nu _{1}}U)$ as follows:

$$ \mathbf {D}_{t}^{\nu _{2}}U(t)-\mathbf {D}_{t}^{\nu _{1}}U(t)=\mathbf {D}_{t}^{\nu _{2}}U(t)-\Theta _{\nu _{2}-\nu _{1}}\star \mathbf {D}_{t}^{\nu _{2}}U(t)$$

$$ = \Big [1-\frac{t^{\nu _{2}-\nu _{1}}}{\Gamma (\nu _{2}-\nu _{1}+1)}\Big ]\mathbf {D}_{t}^{\nu _{2}}U(t)$$

$$ + \int \limits _{0}^{t}\frac{(t-\tau )^{\nu _{2}-\nu _{1}-1}}{\Gamma (\nu _{2}-\nu _{1})}[\mathbf {D}_{t}^{\nu _{2}}U(t)- \mathbf {D}_{\tau }^{\nu _{2}}U(\tau )]d\tau $$

$$\begin{aligned} \equiv \sum \limits _{j=1}^{3}\mathcal {R}_{j}, \end{aligned}$$

(10.34)

where we put

$$ \mathcal {R}_{1}=\frac{\Gamma (\nu _{2}-\nu _{1}+1)-1}{\Gamma (\nu _{2}-\nu _{1}+1)}\mathbf {D}_{t}^{\nu _{2}}U(t),$$

$$ \mathcal {R}_{2}=\frac{1-t^{\nu _{2}-\nu _{1}}}{\Gamma (\nu _{2}-\nu _{1}+1)}\mathbf {D}_{t}^{\nu _{2}}U(t),$$

$$ \mathcal {R}_{3}=\int \limits _{0}^{t}\frac{(t-\tau )^{\nu _{2}-\nu _{1}-1}}{\Gamma (\nu _{2}-\nu _{1})}[\mathbf {D}_{t}^{\nu _{2}}U(t) -\mathbf {D}_{\tau }^{\nu _{2}}U(\tau )]d\tau . $$

Now it is enough to estimate each term $\mathcal {R}_{j}$ separately.

$\bullet$ By inequalities (i) in Proposition 10.1,

$$ \mathcal {R}_{1}\le C(\nu _{2}-\nu _{1})\Vert \mathbf {D}_{t}^{\nu _{2}}U\Vert _{\mathbf {C}[0,T]}, $$

where C is the positive constant.

$\bullet$ Concerning $\mathcal {R}_{2}$, the simple straightforward calculations give

$$ |1-t^{\nu _{2}-\nu _{1}}|\le C(\nu _{2}-\nu _{1}) t\quad \qquad \,\,\,\, \mathrm{if}\quad t>1,$$

$$ |1-t^{\nu _{2}-\nu _{1}}|\le C(\nu _{2}-\nu _{1}) |\ln t| \qquad \mathrm{if}\quad t\le 1, $$

and from this inequality, we easily draw the estimate

$$ \mathcal {R}_{2}\le C(\nu _{2}-\nu _{1})(t+|\ln t|)\Vert \mathbf {D}_{t}^{\nu _{2}}U\Vert _{\mathbf {C}[0,T]}. $$

$\bullet$ For $\mathcal {R}_{3}$, we have

$$ \mathcal {R}_{3} \le C\Big |\int \limits _{0}^{t}\frac{(t-\tau )^{\nu _{2}-\nu _{1}-1+\gamma }}{\Gamma (\nu _{2}-\nu _{1})}d\tau \Big | \langle \mathbf {D}_{t}^{\nu _{2}}U\rangle _{t,[0,T]}^{(\gamma )}$$

$$ \le C(\nu _{2}-\nu _{1})\frac{{t}^{\nu _{2}-\nu _{1}+\gamma }}{(\nu _{2}-\nu _{1}+\gamma )\Gamma (1+\nu _{2}-\nu _{1})} \langle \mathbf {D}_{t}^{\nu _{2}}U\rangle _{t,[0,T]}^{(\gamma )}. $$

Summarizing the above estimates, we obtain statement (iv).