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An undetermined time-dependent coefficient in a fractional diffusion equation
Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA |
$^{C}D_{t}^{\alpha } u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$ |
$a(t)$ |
$a(t),$ |
$Ω$ |
$F(x,t)$ |
$a(t),$ |
$K$ |
$a(t)$ |
$a(t)$ |
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. |
[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. |
[3] |
B. Jin, R. 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. |
[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. |
[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. |
[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. |
[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. |
[8] |
K. S. Miller and S. G. Samko,
Completely monotonic functions, Integral Transform. Spec. Funct., 12 (2001), 389-402.
doi: 10.1080/10652460108819360. |
[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. |
[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. |
[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. |
[12] |
W. R. Schneider,
Completely monotone generalized Mittag-Leffler functions, Exposition. Math., 14 (1996), 3-16.
|
[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. |
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. |
[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. |
[3] |
B. Jin, R. 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. |
[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. |
[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. |
[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. |
[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. |
[8] |
K. S. Miller and S. G. Samko,
Completely monotonic functions, Integral Transform. Spec. Funct., 12 (2001), 389-402.
doi: 10.1080/10652460108819360. |
[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. |
[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. |
[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. |
[12] |
W. R. Schneider,
Completely monotone generalized Mittag-Leffler functions, Exposition. Math., 14 (1996), 3-16.
|
[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. |









Iteration algorithm to recover the coefficient |
1: Set up the right-hand side function 2: Set the initial guess as 3: for k = 1, ..., N do 4: Using the L1 time-stepping [13] to compute 5: Update the coefficient 6: Check stopping criterion 7: end for 8: output the approximate coefficient function |
Iteration algorithm to recover the coefficient |
1: Set up the right-hand side function 2: Set the initial guess as 3: for k = 1, ..., N do 4: Using the L1 time-stepping [13] to compute 5: Update the coefficient 6: Check stopping criterion 7: end for 8: output the approximate coefficient function |
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