An inverse time-dependent source problem for a time-space fractional diffusion equation

This paper is devoted to identify a time-dependent source term in a time–space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point. The existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous Dirichlet boundary condition are proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Based on the separation of variables, we transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the fist kind. The generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time-dependent source term. Numerical experiments for six examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.

[1]  Salih Tatar,et al.  An inverse source problem for a one-dimensional space–time fractional diffusion equation , 2015 .

[2]  Fan Yang,et al.  Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion equation , 2017 .

[3]  Harry Pollard,et al.  The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$ , 1948 .

[4]  Mark M. Meerschaert,et al.  Space-time fractional diffusion on bounded domains , 2012 .

[5]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[6]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[7]  Masahiro Yamamoto,et al.  On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement , 2015 .

[8]  Jungang Wang,et al.  A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation , 2014 .

[9]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[10]  T. Wei,et al.  An inverse time-dependent source problem for a time-fractional diffusion equation , 2016 .

[11]  Fangfang Dou,et al.  Fundamental kernel-based method for backward space-time fractional diffusion problem , 2016, Comput. Math. Appl..

[12]  Masahiro Yamamoto,et al.  Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems , 2011 .

[13]  Ying Zhang,et al.  Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate , 2012 .

[14]  Salih Tatar,et al.  Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation , 2016 .

[15]  Per Christian Hansen,et al.  Regularization methods for large-scale problems , 1993 .

[16]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

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

[18]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[19]  Junxiong Jia,et al.  Maximum principles for a time-space fractional diffusion equation , 2016, Appl. Math. Lett..

[20]  Hans-Peter Scheffler,et al.  Stochastic solution of space-time fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Yaolin Jiang,et al.  Analytical solutions for the multi-term time–space fractional advection–diffusion equations with mixed boundary conditions , 2013 .

[22]  Ting Wei,et al.  Reconstruction of a time-dependent source term in a time-fractional diffusion equation , 2013 .

[23]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[24]  V. Kolokoltsov,et al.  Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations , 2014, 1402.6735.