October  2017, 11(5): 875-900. doi: 10.3934/ipi.2017041

An undetermined time-dependent coefficient in a fractional diffusion equation

Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA

Received  July 2016 Revised  October 2016 Published  July 2017

Fund Project: The author is supported by NSF Grant DMS-1620138

In this work, we consider a FDE (fractional diffusion equation)
$^{C}D_{t}^{\alpha } u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$
with a time-dependent diffusion coefficient
$a(t)$
. This is an extension of [13], which deals with this FDE in one-dimensional space. For the direct problem, given an
$a(t),$
we establish the existence, uniqueness and some regularity properties with a more general domain
$Ω$
and right-hand side
$F(x,t)$
. For the inverse problem–recovering
$a(t),$
we introduce an operator
$K$
one of whose fixed points is
$a(t)$
and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for
$a(t)$
is created and some numerical results are provided to illustrate the theories.
Citation: Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
References:
[1]

L. C. Evans, Partial Differential Equations vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[2]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and fractional calculus in continuum mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, Springer, Vienna, 1997,223–276. Google Scholar

[3]

B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221. doi: 10.1093/imanum/dru063. Google Scholar

[4]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem Inverse Problems 28 (2012), 075010, 19pp. doi: 10.1088/0266-5611/28/7/075010. Google Scholar

[5]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, 2006. Google Scholar

[6]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001. Google Scholar

[7]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. Google Scholar

[8]

K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transform. Spec. Funct., 12 (2001), 389-402. doi: 10.1080/10652460108819360. Google Scholar

[9]

H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_a(-x)$, Bull. Amer. Math. Soc., 54 (1948), 1115-1116. doi: 10.1090/S0002-9904-1948-09132-7. Google Scholar

[10]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar

[11]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Gordon and Breach Science Publishers, Yverdon, 1993, Theory and applications, Edited and with a foreword by S. M. Nikol'skiĭ, Translated from the 1987 Russian original, Revised by the authors. Google Scholar

[12]

W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Exposition. Math., 14 (1996), 3-16. Google Scholar

[13]

Z. Zhang, An undetermined coefficient problem for a fractional diffusion equation Inverse Problems 32 (2016), 015011, 21pp. doi: 10.1088/0266-5611/32/1/015011. Google Scholar

show all references

References:
[1]

L. C. Evans, Partial Differential Equations vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[2]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and fractional calculus in continuum mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, Springer, Vienna, 1997,223–276. Google Scholar

[3]

B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221. doi: 10.1093/imanum/dru063. Google Scholar

[4]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem Inverse Problems 28 (2012), 075010, 19pp. doi: 10.1088/0266-5611/28/7/075010. Google Scholar

[5]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, 2006. Google Scholar

[6]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001. Google Scholar

[7]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. Google Scholar

[8]

K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transform. Spec. Funct., 12 (2001), 389-402. doi: 10.1080/10652460108819360. Google Scholar

[9]

H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_a(-x)$, Bull. Amer. Math. Soc., 54 (1948), 1115-1116. doi: 10.1090/S0002-9904-1948-09132-7. Google Scholar

[10]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar

[11]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Gordon and Breach Science Publishers, Yverdon, 1993, Theory and applications, Edited and with a foreword by S. M. Nikol'skiĭ, Translated from the 1987 Russian original, Revised by the authors. Google Scholar

[12]

W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Exposition. Math., 14 (1996), 3-16. Google Scholar

[13]

Z. Zhang, An undetermined coefficient problem for a fractional diffusion equation Inverse Problems 32 (2016), 015011, 21pp. doi: 10.1088/0266-5611/32/1/015011. Google Scholar

Figure 1.  Experiment (a1): the initial guess and first three iterations
Figure 2.  Experiment (a1): the exact and approximate coefficients for $\alpha=0.9$ and $\epsilon_0=10^{-6}$
Figure 3.  the amounts of iterations $N$ for different $\alpha$
Figure 4.  $\|a-\overline{a}_N\|_{L^2[0, T]}$ for different $\epsilon_0$ under $\alpha=0.9$
Figure 5.  Experiment (a2): the initial guess and first three iterations
Figure 6.  Experiment (a2): the exact and approximate coefficients for $\alpha=0.9$ and $\epsilon_0=10^{-6}$
Figure 7.  Experiment (a1): the exact and approximate coefficients with $\alpha=0.9, $ $\epsilon_0=10^{-6}$ and $\delta=3\%$
Figure 8.  Experiment (a2): the exact and approximate coefficients with $\alpha=0.9, $ $\epsilon_0=10^{-6}$ and $\delta=3\%$
Figure 9.  $\|a-\overline{a}_{\delta, N}\|_{L^2[0, T]}/\|a\|_{L^2[0, T]}$ for different $\delta$ under $\alpha=0.9$ and $\epsilon_0=10^{-6}$
Figure 10.  Experiment (a1) in two dimensional case
Figure 11.  Experiment (a2) in two dimensional case
Table 1.  Numerical Algorithm
Iteration algorithm to recover the coefficient $a(t)$
1: Set up the right-hand side function $F(x,t)$ and the initial condition $u_0(x)$, then measure the output flux data $g(t).$ $F$, $u_0$ and $g$ should satisfy Assumption 4.1;
2: Set the initial guess as $\overline{a}_0(t)=g(t)\Big[\frac{\partial u_0}{\partial \mathbf{\vec{n}}}(x_0) + I_{t}^{\alpha }[\frac{\partial F}{\partial \mathbf{\vec{n}}}(x_0,t)]\Big]^{-1};$
3: for k = 1, ..., N do
4: Using the L1 time-stepping [13] to compute $u(x,t;\overline{a}_{k-1})$, which is the weak solution of FDE (1) with coefficient function $\overline{a}_{k-1}$;
5: Update the coefficient $\overline{a}_{k-1}$ by $ \overline{a}_{k}=K\overline{a}_{k-1};$
6: Check stopping criterion $\|\overline{a}_{k}- \overline{a}_{k-1}\|_{L^2[0,T]}\le \epsilon_0$ for some $\epsilon_0>0$;
7: end for
8: output the approximate coefficient function $\overline{a}_{N}$.
Iteration algorithm to recover the coefficient $a(t)$
1: Set up the right-hand side function $F(x,t)$ and the initial condition $u_0(x)$, then measure the output flux data $g(t).$ $F$, $u_0$ and $g$ should satisfy Assumption 4.1;
2: Set the initial guess as $\overline{a}_0(t)=g(t)\Big[\frac{\partial u_0}{\partial \mathbf{\vec{n}}}(x_0) + I_{t}^{\alpha }[\frac{\partial F}{\partial \mathbf{\vec{n}}}(x_0,t)]\Big]^{-1};$
3: for k = 1, ..., N do
4: Using the L1 time-stepping [13] to compute $u(x,t;\overline{a}_{k-1})$, which is the weak solution of FDE (1) with coefficient function $\overline{a}_{k-1}$;
5: Update the coefficient $\overline{a}_{k-1}$ by $ \overline{a}_{k}=K\overline{a}_{k-1};$
6: Check stopping criterion $\|\overline{a}_{k}- \overline{a}_{k-1}\|_{L^2[0,T]}\le \epsilon_0$ for some $\epsilon_0>0$;
7: end for
8: output the approximate coefficient function $\overline{a}_{N}$.
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