The determination of an unknown boundary condition in a fractional diffusion equation

References

• Adams , EE and Gelhar , LW . 1992 . Field study of dispersion in heterogeneous aquifer 2. Spatial moments analysis . Water Resources Res. , 28 : 3293 – 3307 .
   Web of Science ®Google Scholar
• Zhou , L and Selim , HM . 2003 . Application of the fractional advection-dispersion equations in porous media . Soli. Sci. Soc. Am. J. , 67 : 1079 – 1084 .
   Web of Science ®Google Scholar
• Metzler , R and Klafter , J . 2000 . The random walk's guide to anomalous diffusion: A fractional dynamics approach . Phys. Rep. , 339 : 1 – 77 .
   Web of Science ®Google Scholar
• Chaves , AS . 1998 . A fractional diffusion equation to describe Levy flights . Phys. Lett. A , 239 : 13 – 16 .
   Web of Science ®Google Scholar
• Mainardi , F , Luchko , Y and Pagnini , G . 2001 . The fundamental solution of the space-time fractional diffusion equation . Fract. Calculus Appl. Anal. , 4 ( 2 ) : 153 – 192 .
   Google Scholar
• Mainardi , F and Pagnini , G . 2007 . The role of the Fox – Wright functions in fractional sub-diffusion of distributed order . J. Comput. Appl. Math. , 207 : 245 – 257 .
   Web of Science ®Google Scholar
• Nigmatullin , RR . 1986 . The realization of the generalized transfer equation in a medium with fractal geometry . Phys. Stat. Sol. B , 133 : 425 – 430 .
   Google Scholar
• Luchko , Y . 2009 . Maximun principle for the generalized time-fractional diffusion equation . J. Math. Anal. Appl. , 351 : 218 – 223 .
   Web of Science ®Google Scholar
• Luchko , Y . 2010 . Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation . Comput. Math. Appl. , 59 : 1766 – 1772 .
   Web of Science ®Google Scholar
• Sakamoto , K and Yamamoto , M . 2011 . Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems . J. Math. Anal. App. , 382 : 426 – 447 .
   Web of Science ®Google Scholar
• Sun , ZZ and Wu , XN . 2006 . A full discrete difference scheme for a diffusion-wave system . Appl. Numer. Math. , 56 : 193 – 209 .
   Web of Science ®Google Scholar
• Chen , C , Liu , F , Turner , I and Anhb , V . 2007 . A Fourier method for the fractional diffusion equation describing sub-diffusion . J. Comput. Phys. , 227 : 886 – 897 .
   Web of Science ®Google Scholar
• Lin , Y and Xu , C . 2007 . Finite difference/spectral approximation for the time-fractional diffusion equation . J. Comp. Phys. , 225 : 1533 – 1552 .
   Web of Science ®Google Scholar
• Brunner , H , Ling , L and Yamamoto , M . 2010 . Numerical simulations of 2D fractional subdiffusion problems . J. Comp. Phys. , 229 : 6613 – 6622 .
   Web of Science ®Google Scholar
• Cheng , J , Nakagawa , J , Yamamoto , M and Yamazaki , T . 2009 . Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation . Inverse Probl. , 25 : 115002 – 115018 .
   Web of Science ®Google Scholar
• Liu , JJ and Yamamoto , M . 2010 . A backward problem for the time-fractional diffusion equation . Appl. Anal. , 89 : 1769 – 1790 .
   Web of Science ®Google Scholar
• Xu , X , Cheng , J and Yamamoto , M . 2011 . Carleman estimate for a fractional diffusion equation with half order and application . Appl. Anal. , 90 ( 9 ) : 1355 – 1371 .
   Web of Science ®Google Scholar
• Zhang , Y and Xu , X . 2011 . An inverse source problem in a fractional diffusion equation . Inverse Probl. , 27 : 035010 – 035022 . 035010
   Web of Science ®Google Scholar
• Pilant , M and Rundell , W . 1989 . An iteration method for the determination of an unknown boundary condition in a parabolic initial boundary value problem . Proc. Edin. Math. Soc. , 32 : 59 – 71 .
   Web of Science ®Google Scholar
• Pilant , M and Rundell , W . 1986 . An inverse problem for a nonlinear parabolic equation . Commun. Partial Differ. Eqns , 11 ( 4 ) : 445 – 457 .
   Web of Science ®Google Scholar
• Pilant , M and Rundell , W . 1986 . Undetermined coefficient problems for nonlinear elliptic and parabolic equations . Int. Ser. Numer. Math. , 77 : 139 – 154 .
   Google Scholar
• Pilant , M and Rundell , W . 1987 . An inverse problem for a nonlinear elliptic equation . SIAM J. Math. Anal. , 18 ( 6 ) : 1801 – 1809 .
   Web of Science ®Google Scholar
• Pilant , M and Rundell , W . 1987 . Iteration schemes for unknown coefficient problems arising in parabolic equations . Numer. Methods PDE , 3 : 313 – 325 .
   Google Scholar
• Podlubny , I . 1999 . Fractional Differential Equations , San Diego : Academic .
   Google Scholar
• Cannon , J . 1984 . The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Application , Menlo Park, CA : Addison-Wesley .
   Google Scholar
• Lipnevich , V and Luchko , Y . 2010 . The Wright function: Its properties, applications, and numerical evaluation . Appl. Math. Tech. Natural Sci. , 1301 : 614 – 622 .
   Google Scholar