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A modified kernel method for a time-fractional inverse diffusion problem

Songshu Liu, Lixin Feng
2015 Advances in Difference Equations  
We show that a time-fractional inverse diffusion problem is severely ill-posed and we further apply a modified kernel method to solve it based on the solution in the frequency domain.  ...  In this paper, we consider a time-fractional inverse diffusion problem, where the data is given at x = 1 and the solution is sought in the interval 0 ≤ x < 1.  ...  We now list two kernels of regularization methods for solving the time-fractional inverse diffusion problem.  ... 
doi:10.1186/s13662-015-0679-0 fatcat:snaxqffaznasrerzmmnd7em55y

Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion equation

Fan Yang, Xiao Liu, Xiao-Xiao Li, Cheng-Ye Ma
2017 Advances in Difference Equations  
This problem is ill-posed, and we use the Landweber iterative regularization method to solve this inverse source problem.  ...  In this paper, we investigate an inverse problem to determine an unknown source term that has a separable-variable form in the time-fractional diffusion equation, whereby the data is obtained at a certain  ...  Acknowledgements The authors would like to thank the editor and referees for their valuable comments and suggestions that improved the quality of our paper.  ... 
doi:10.1186/s13662-017-1423-8 fatcat:n5cwtow2yjdb3orkmy2lykj7za

A BACKWARD IDENTIFICATION PROBLEM FOR AN AXIS-SYMMETRIC FRACTIONAL DIFFUSION EQUATION

Xiangtuan Xiong, Xiaojun Ma
2017 Mathematical Modelling and Analysis  
We consider a backward ill-posed problem for an axis-symmetric fractional diffusion equation which is described in polar coordinates. A closed form solution of the inverse problem is obtained.  ...  For numerical stability, a general regularization principle is presented for constructing regularization methods.  ...  I.Podlubny for providing the program on Mittag-Leffler function on the web http://www.mathworks.com. The authors would like to thank the reviewers for their important comments.  ... 
doi:10.3846/13926292.2017.1309329 fatcat:ydowlwvefzdn5pwoaigt2kpefu

Obtaining sparse distributions in 2D inverse problems

A. Reci, A.J. Sederman, L.F. Gladden
2017 Journal of magnetic resonance (San Diego, Calif. 1997 : Print)  
We introduce a robust algorithm for solving the L 1 regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error  ...  In this work, we focus on the application of L 1 regularization to a class of inverse problems; relaxation-relaxation, T 1 -T 2 , and diffusion-relaxation, D-T 2 , correlation experiments in NMR, which  ...  The most common method of inversion is Tikhonov regularization [15] and successful algorithms have been developed to solve this regularized problem [16, 17] .  ... 
doi:10.1016/j.jmr.2017.05.010 pmid:28623744 fatcat:etdct2y6rnh7tmmti4odk3bbua

A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

Shengnan Wang, Zhendong Wang, Gongsheng Li, Yingmei Wang
2019 Mathematical Problems in Engineering  
This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the  ...  Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the  ...  There are quite a few of researches on numerical methods for solving the variable-order time/space fractional diffusion equation models. We refer to some work given by F.  ... 
doi:10.1155/2019/2562580 fatcat:favew7pdx5coxmpkquejzr7sqm

Water mobility spectral imaging of the spinal cord: Parametrization of model-free Laplace MRI

Dan Benjamini, Peter J. Basser
2019 Magnetic Resonance Imaging  
This approach allows for a compact representation of the spectrum, and it also resolves overlapping spectral peaks, which allows for a robust extraction of their signal fraction.  ...  The rich spectral information, combined with a three-dimensional image, presents a challenge because it tremendously increases the dimensionality of the data and requires a robust method for interpretation  ...  Liz Salak for editing the manuscript.  ... 
doi:10.1016/j.mri.2018.12.001 pmid:30584915 pmcid:PMC6800163 fatcat:7ibfuafz2bajrjvwis7royff4y

An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

Songshu Liu, Lixin Feng
2020 Mathematical Problems in Engineering  
In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data.  ...  A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization  ...  Zheng and Wei [30] applied a new regularization method to solve an inverse problem for a time-fractional diffusion equation in a onedimensional semi-infinite domain.  ... 
doi:10.1155/2020/5865971 fatcat:zzitrqaifre7ro7cmfosabpzum

An inverse problem for fractional diffusion equation in 2-dimensional case: Stability analysis and regularization

Xiangtuan Xiong, Qian Zhou, Y.C. Hon
2012 Journal of Mathematical Analysis and Applications  
In this paper we investigate an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting.  ...  Some new regularization methods are constructed for solving the inverse problem and the corresponding error estimates are proved.  ...  The work described in this paper was partially supported by a  ... 
doi:10.1016/j.jmaa.2012.03.013 fatcat:monnv2fn3vdahph7mnd7tvtysq

A Lagrange Regularized Kernel Method for Solving Multi-dimensional Time-Fractional Heat Equations

Edson Pindza, Jules Clement Mba, Eben Maré, Désirée Moubandjo
2017 International journal of nonlinear sciences and numerical simulation  
The proposed method is based on a fractional exponential integrator scheme in time and the Lagrange regularized kernel method in space.  ...  In this paper, we propose an accurate method for numerical solutions of multi-dimensional time-fractional heat equations.  ...  Pindza is thankful to Brad Welch for the financial support through RidgeCape Capital.  ... 
doi:10.1515/ijnsns-2016-0089 fatcat:eanmsrt5f5h5vi3f3szwd27ysa

Fundamental kernel-based method for backward space–time fractional diffusion problem

F.F. Dou, Y.C. Hon
2016 Computers and Mathematics with Applications  
Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space-time fractional diffusion problem (BSTFDP).  ...  For inverse problems of space-time fractional diffusion equation in one-dimension, some numerical solutions have been given by Aldoghaither et al. [18]; Wei et al. [19] and Zheng and Wei [20].  ...  Conclusion We develop in this paper a Kernel-Based Approximation method based to solve a backward problem of space-time fractional diffusion equation.  ... 
doi:10.1016/j.camwa.2015.11.023 fatcat:ecy7elcotndwlinjuvm7gpybpa

Image Deblurring Via Total Variation Based Structured Sparse Model Selection

Liyan Ma, Tieyong Zeng
2015 Journal of Scientific Computing  
Other related methods will also be addressed if time permitted. 10:45 C10-2(invited) Fast retinal image enhancement using robust inverse diffusion equation and blockwise filtering  ...  As a generalization, we give a modified model for deblurring under salt-and-pepper noise. The resulting algorithm also has a good performance.  ...  Fractional inverse diffusion.  ... 
doi:10.1007/s10915-015-0067-7 fatcat:cbpymhfkrrfz5cpqci4kjyqzbe

Spectral regularization method for a Cauchy problem of the time fractional advection–dispersion equation

G.H. Zheng, T. Wei
2010 Journal of Computational and Applied Mathematics  
We show that the Cauchy problem of TFADE is severely illposed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method.  ...  In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated.  ...  Acknowledgements The authors would like to thank the referees for their carefully reading, valuable comments and suggestions that improved the quality of their paper.  ... 
doi:10.1016/j.cam.2009.11.009 fatcat:eczq7ro4x5am7lawftpmpxbslu

Filter regularization method for a time-fractional inverse advection–dispersion problem

Songshu Liu, Lixin Feng
2019 Advances in Difference Equations  
A filter regularization method is developed to solve a time-fractional inverse advection-dispersion problem, which is based on the modified 'kernel' idea.  ...  Proofs of convergence are given under both priori and posteriori regularization parameter choice rules. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.  ...  Funding This work was partially supported by National Natural Science Foundation of China (11871198), the Fundamental Research Funds for the Universities of Heilongjiang Province Heilongjiang University  ... 
doi:10.1186/s13662-019-2155-8 fatcat:emvipg6om5exrkmeklytr5er7q

Direct and Inverse Source Problem for a Space Fractional Advection Dispersion Equation [article]

Abeer Aldoghaither, Taous-Meriem Laleg-Kirati, Da-Yan Liu
2014 arXiv   pre-print
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied.  ...  The inverse problem consists in determining the source term from a final observation.  ...  William Rundell for their valuable suggestions and comments.  ... 
arXiv:1401.3153v1 fatcat:ml5jxxhk3zaftdskdxhizdf4be

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Fan Yang, Ping Fan, Xiao-Xiao Li, Xin-Yi Ma
2019 Mathematics  
In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain.  ...  Therefore, we use the Fourier truncation regularization method to solve this problem.  ...  Acknowledgments: The authors would like to thanks the editor and the referees for their valuable comments and suggestions that improve the quality of our paper.  ... 
doi:10.3390/math7090865 fatcat:y6k74zxdpzgdtbmgdt7bmglfwa
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