• Previous Article
    Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space
  • DCDS-S Home
  • This Issue
  • Next Article
    Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion
August  2021, 14(8): 2893-2916. doi: 10.3934/dcdss.2020409

An extension of the landweber regularization for a backward time fractional wave problem

1. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

2. 

School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical, Modeling and High Performance Scientific Computing, Xiamen University, Xiamen 361005, China

3. 

Bordeaux INP, Laboratoire I2M UMR 5295, Pessac 33607, France

*Corresponding author: Chuanju Xu

Received  February 2020 Revised  May 2020 Published  August 2021 Early access  July 2020

Fund Project: This research is partially supported by NSFC grant 11971408 and NSFC/ANR joint program 51661135011/ANR-16-CE40-0026-01. The first author gratefully acknowledges the financial support from China Scholarship Council and the laboratory I2M UMR 5295 for hosting. The second author has received financial support from the French State in the frame of the "Investments for the future" Programme Idex Bordeaux, reference ANR-10-IDEX-03-02

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Citation: Bin Fan, Mejdi Azaïez, Chuanju Xu. An extension of the landweber regularization for a backward time fractional wave problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2893-2916. doi: 10.3934/dcdss.2020409
References:
[1]

O. P. Agrawal, Formulation of euler-lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[2]

O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.  doi: 10.1023/A:1016539022492.

[3]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.

[4]

D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005.

[5]

H. Cheng and C. L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Applied Mathematical Modelling, 36 (2012), 5642-5649.  doi: 10.1016/j.apm.2012.01.016.

[6]

E. CuestaM. Kirane and S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing, 92 (2012), 553-563.  doi: 10.1016/j.sigpro.2011.09.001.

[7]

Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14pp. doi: 10.1088/0266-5611/25/9/095004.

[8]

Y. Deng and Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Analysis: Real World Applications, 12 (2011), 156-166.  doi: 10.1016/j.nonrwa.2010.06.005.

[9]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-14574-2.

[10]

R. DuW. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34 (2010), 2998-3007.  doi: 10.1016/j.apm.2010.01.008.

[11]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996.

[12]

G. H. Gao and Z. Z. Sun, The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, Journal of Computational Physics, 236 (2013), 443-460.  doi: 10.1016/j.jcp.2012.11.011.

[13]

D. GerthE. KlannR. Ramlau and L. Reichel, On fractional tikhonov regularization, Journal of Inverse and Ill-posed Problems, 23 (2015), 611-625.  doi: 10.1515/jiip-2014-0050.

[14]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984.

[15]

Y. HanX. Xiong and X. Xue, A fractional landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, 78 (2019), 81-91.  doi: 10.1016/j.camwa.2019.02.017.

[16]

M. E. Hochstenbach and L. Reichel, Fractional tikhonov regularization for linear discrete ill-posed problems, BIT Numerical Mathematics, 51 (2011), 197-215.  doi: 10.1007/s10543-011-0313-9.

[17]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.

[18]

E. KlannP. Maass and R. Ramlau, Two-step regularization methods for linear inverse problems, Journal of Inverse and Ill-posed Problems, 14 (2006), 583-607.  doi: 10.1515/156939406778474523.

[19]

X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[20]

Y. M. Lin and C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788.  doi: 10.1080/00036810903479731.

[22]

R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006.

[23]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.

[24]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[25] I. Podlubny, Fractional Differential Equations, Acad. Press, New York, 1999. 
[26]

I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006.

[27]

M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016.

[28]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[29]

F. Y. Song and C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, Journal of Computational Physics, 299 (2015), 196-214.  doi: 10.1016/j.jcp.2015.07.011.

[30]

Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[31]

J. G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem, Numerical Methods for Partial Differential Equations, 30 (2014), 2029-2041.  doi: 10.1002/num.21887.

[32]

J. G. WangT. Wei and Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, 37 (2013), 8518-8532.  doi: 10.1016/j.apm.2013.03.071.

[33]

L. Wang and J. J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626.  doi: 10.1016/j.camwa.2012.10.001.

[34]

T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 603-621.  doi: 10.1051/m2an/2013107.

[35]

T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers & Mathematics with Applications, 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.

[36]

X. XiongX. Xue and Z. Qian, A modified iterative regularization method for ill-posed problems, Applied Numerical Mathematics, 122 (2017), 108-128.  doi: 10.1016/j.apnum.2017.08.004.

[37]

F. YangY. Zhang and X. X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numerical Algorithms, 83 (2020), 1509-1530.  doi: 10.1007/s11075-019-00734-6.

show all references

References:
[1]

O. P. Agrawal, Formulation of euler-lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.

[2]

O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.  doi: 10.1023/A:1016539022492.

[3]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.

[4]

D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005.

[5]

H. Cheng and C. L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Applied Mathematical Modelling, 36 (2012), 5642-5649.  doi: 10.1016/j.apm.2012.01.016.

[6]

E. CuestaM. Kirane and S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing, 92 (2012), 553-563.  doi: 10.1016/j.sigpro.2011.09.001.

[7]

Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14pp. doi: 10.1088/0266-5611/25/9/095004.

[8]

Y. Deng and Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Analysis: Real World Applications, 12 (2011), 156-166.  doi: 10.1016/j.nonrwa.2010.06.005.

[9]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-14574-2.

[10]

R. DuW. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34 (2010), 2998-3007.  doi: 10.1016/j.apm.2010.01.008.

[11]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996.

[12]

G. H. Gao and Z. Z. Sun, The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, Journal of Computational Physics, 236 (2013), 443-460.  doi: 10.1016/j.jcp.2012.11.011.

[13]

D. GerthE. KlannR. Ramlau and L. Reichel, On fractional tikhonov regularization, Journal of Inverse and Ill-posed Problems, 23 (2015), 611-625.  doi: 10.1515/jiip-2014-0050.

[14]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984.

[15]

Y. HanX. Xiong and X. Xue, A fractional landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, 78 (2019), 81-91.  doi: 10.1016/j.camwa.2019.02.017.

[16]

M. E. Hochstenbach and L. Reichel, Fractional tikhonov regularization for linear discrete ill-posed problems, BIT Numerical Mathematics, 51 (2011), 197-215.  doi: 10.1007/s10543-011-0313-9.

[17]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.

[18]

E. KlannP. Maass and R. Ramlau, Two-step regularization methods for linear inverse problems, Journal of Inverse and Ill-posed Problems, 14 (2006), 583-607.  doi: 10.1515/156939406778474523.

[19]

X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[20]

Y. M. Lin and C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788.  doi: 10.1080/00036810903479731.

[22]

R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006.

[23]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.

[24]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[25] I. Podlubny, Fractional Differential Equations, Acad. Press, New York, 1999. 
[26]

I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006.

[27]

M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016.

[28]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[29]

F. Y. Song and C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, Journal of Computational Physics, 299 (2015), 196-214.  doi: 10.1016/j.jcp.2015.07.011.

[30]

Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[31]

J. G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem, Numerical Methods for Partial Differential Equations, 30 (2014), 2029-2041.  doi: 10.1002/num.21887.

[32]

J. G. WangT. Wei and Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, 37 (2013), 8518-8532.  doi: 10.1016/j.apm.2013.03.071.

[33]

L. Wang and J. J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626.  doi: 10.1016/j.camwa.2012.10.001.

[34]

T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 603-621.  doi: 10.1051/m2an/2013107.

[35]

T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers & Mathematics with Applications, 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.

[36]

X. XiongX. Xue and Z. Qian, A modified iterative regularization method for ill-posed problems, Applied Numerical Mathematics, 122 (2017), 108-128.  doi: 10.1016/j.apnum.2017.08.004.

[37]

F. YangY. Zhang and X. X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numerical Algorithms, 83 (2020), 1509-1530.  doi: 10.1007/s11075-019-00734-6.

Figure 1.  The regularized solutions with $ \alpha = 1.1$ for Examples 5.1– 5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR
Figure 2.  Regularized solutions with $\alpha = 1.6 $ for Examples 5.1–5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR
Figure 3.  (ExFLR) Regularization parameter $ m $ and relative error $ e_r $ as functions of $ \nu $, vary $ \nu $ in the range $ \{0, 0.1, \cdots, 1\} $. (a) and (d) for Example 5.1 with $ \varepsilon = 10\% $; (b) and (e) for Example 5.2 with $ \varepsilon = 0.1\% $; (c) and (f) for Example 5.3 with $ \varepsilon = 0.1\% $
Figure 4.  The computed (regularized) solutions for Example 5.1-3 (from left to right) with $ \alpha = 1.6 $: smoothness comparison of the computed solutions for three different values of $ \nu $. (a)-(c) for ExFLR; (d)-(f) for ImFLR
Figure 5.  The computed initial conditions by ImFLR and absolute errors for Example 5.4. (a)—(c) for the regularized solutions $f_4^{m, \delta}$; (d)—(f) for the absolute errors $|f_4^{m, \delta}-f_4|$
Table 1.  Examples 5.1–5.3. Relative errors and regularization parameter versus relative noise levels
$\alpha = 1.1 $ $\alpha = 1.6 $
ExFLR (21) ImFLR (22) ExFLR (21) ImFLR (22)
$ f $ $ \varepsilon $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $
$ f_1 $ $ 1\% $ 7324 0.0103 7328 0.0103 147 0.0188 152 0.0187
$ 5\% $ 4775 0.0506 4778 0.0506 96 0.0634 99 0.0634
$ 10\% $ 3677 0.1003 3680 0.1002 73 0.1209 76 0.1188
$ f_2 $ $ 0.1\% $ 14913 0.0323 14915 0.0323 510 0.0647 511 0.0641
$ 0.5\% $ 3846 0.0674 3846 0.0674 8 0.1183 13 0.1187
$ 1\% $ 701 0.1068 706 0.1067 6 0.1211 11 0.1200
$ f_3 $ $ 0.1\% $ 232223 0.2947 232224 0.2947 69253 0.2728 69254 0.2728
$ 0.5\% $ 32815 0.3827 32817 0.3827 7456 0.3753 7458 0.3752
$ 1\% $ 20594 0.4186 20595 0.4186 4695 0.4032 4697 0.4032
$\alpha = 1.1 $ $\alpha = 1.6 $
ExFLR (21) ImFLR (22) ExFLR (21) ImFLR (22)
$ f $ $ \varepsilon $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $ $ m $ $ e_r(f^m_\delta, \varepsilon) $
$ f_1 $ $ 1\% $ 7324 0.0103 7328 0.0103 147 0.0188 152 0.0187
$ 5\% $ 4775 0.0506 4778 0.0506 96 0.0634 99 0.0634
$ 10\% $ 3677 0.1003 3680 0.1002 73 0.1209 76 0.1188
$ f_2 $ $ 0.1\% $ 14913 0.0323 14915 0.0323 510 0.0647 511 0.0641
$ 0.5\% $ 3846 0.0674 3846 0.0674 8 0.1183 13 0.1187
$ 1\% $ 701 0.1068 706 0.1067 6 0.1211 11 0.1200
$ f_3 $ $ 0.1\% $ 232223 0.2947 232224 0.2947 69253 0.2728 69254 0.2728
$ 0.5\% $ 32815 0.3827 32817 0.3827 7456 0.3753 7458 0.3752
$ 1\% $ 20594 0.4186 20595 0.4186 4695 0.4032 4697 0.4032
[1]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266

[2]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007

[3]

Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations and Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024

[4]

Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033

[5]

Mohamed Jleli, Bessem Samet. Nonexistence for time-fractional wave inequalities on Riemannian manifolds. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022115

[6]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013

[7]

Zhiyuan Li, Yikan Liu, Masahiro Yamamoto. Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022027

[8]

Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022017

[9]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[10]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269

[11]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

[12]

Xinchi Huang, Atsushi Kawamoto. Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates. Inverse Problems and Imaging, 2022, 16 (1) : 39-67. doi: 10.3934/ipi.2021040

[13]

Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations and Control Theory, 2020, 9 (3) : 773-793. doi: 10.3934/eect.2020033

[14]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[15]

Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223

[16]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[17]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282

[18]

Dong Li, Chaoyu Quan, Jiao Xu. Energy-dissipation for time-fractional phase-field equations. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3371-3387. doi: 10.3934/cpaa.2022104

[19]

Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems and Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027

[20]

Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems and Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (413)
  • HTML views (450)
  • Cited by (0)

Other articles
by authors

[Back to Top]