A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

Abstract

The ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple regularization method based on Dirichlet kernel mollification techniques is introduced. We also proposea priorianda posterioriparameter choice rules and get the corresponding error estimate between the exact solution and its regularized approximation. Moreover, a numerical example is provided to verify our theoretical results.

A Directly Numerical Algorithm for a Backward Time-Fractional Diffusion Equation Based on the Finite Element Method

“…Inverse source problems for time-fractional diffusion equations were studied by using the method of the eigenfunction expansion [14], the integral equation method [15], and the separation of variables method [16], respectively, for recovering the space-dependent or time-dependent source term. In [17], the authors recovered the temperature function from one measured temperature at one interior point of a one-dimensional semi-infinite fractional diffusion equation based on Dirichlet kernel mollification techniques. The authors studied an inverse problem of identifying a spatially varying potential term in a onedimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources in [18].…”

A New Method for TSVD Regularization Truncated Parameter Selection

“…Ill-posed problem is widespread in the field of geophysical survey, such as GNSS rapid positioning, precise orbit solution of spacecraft, and downward continuation of airborne gravity [1][2][3][4][5][6][7]. The least square estimation is not stable in the ill-posed problem and other approaches have to be utilized.…”