Regularization for a fractional sideways heat equation

We consider a sideways problem for a fractional heat equation which is highly ill-posed. This study gives a new dynamic method for choosing a regularization parameter. By using the spectral methods, some convergence rates on the temperature and heat flow are given. For illustration, several numerical examples are constructed to show the feasibility and efficiency of the proposed methods. Comparing with the traditional stationary methods for choosing regularization parameter, the proposal methods are more accurate and effective.

[1]  T. Szabo,et al.  A model for longitudinal and shear wave propagation in viscoelastic media , 2000, The Journal of the Acoustical Society of America.

[2]  T. Wei,et al.  Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation , 2010, J. Comput. Appl. Math..

[3]  D. S. Ivaschenko,et al.  Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient , 2009 .

[4]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[5]  Fredrik Berntsson,et al.  Wavelet and Fourier Methods for Solving the Sideways Heat Equation , 1999, SIAM J. Sci. Comput..

[6]  Ulrich Tautenhahn,et al.  Optimality for ill-posed problems under general source conditions , 1998 .

[7]  Fredrik Berntsson,et al.  A spectral method for solving the sideways heat equation , 1999 .

[8]  Hongguang Sun,et al.  Fractional diffusion equations by the Kansa method , 2010, Comput. Math. Appl..

[9]  Fotios C. Harmantzis,et al.  Fractional Lévy motion and its application to network traffic modeling , 2002, Comput. Networks.

[10]  Brian Clark,et al.  Physics in Oil Exploration , 2002 .

[11]  Y. Hon,et al.  A fundamental solution method for inverse heat conduction problem , 2004 .

[12]  Lin Jianzhong,et al.  Numerical research on the coherent structure in the viscoelastic second-order mixing layers , 1998 .

[13]  Y. C. Hon,et al.  A computational method for inverse free boundary determination problem , 2008 .

[14]  R. Gorenflo,et al.  Discrete random walk models for space-time fractional diffusion , 2002, cond-mat/0702072.

[15]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[16]  Masahiro Yamamoto,et al.  Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation , 2009 .

[17]  Y. Hon,et al.  The method of fundamental solution for solving multidimensional inverse heat conduction problems , 2005 .

[18]  I. Podlubny.  Fractional differential equations , 1998 .

[19]  R. Vilela Mendes.  A fractional calculus interpretation of the fractional volatility model , 2008 .

[20]  Diego A. Murio,et al.  Stable numerical solution of a fractional-diffusion inverse heat conduction problem , 2007, Comput. Math. Appl..

[21]  Thomas I. Seidman,et al.  An 'optimal filtering' method for the sideways heat equation , 1990 .

[22]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .