Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem

Abstract

This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution for the forward and backward fractional heat problems is expressed in terms of eigen function expansion and Mittag–Leffler function. Due to the instability of determining initial data, a regularized truncated solution is considered. Further, the stability estimate for the exact solution and the convergence estimates for the regularized solution using an á-priori choice rule and an á-posteriori choice rule are derived.

Keywords:

• Inverse problems
• ill-posed problems
• regularization
• truncation method
• heat equation
• fractional derivative
• Mittag–Leffler function

AMS CLASSIFICATIONS:

• 35R30
• 35K05
• 26A33
• 33E12

Acknowledgements

The first author would like to acknowledge the support received from Indian Institute of Technology Madras and also the excellent facilities provided by the Department of Mathematics at Indian Institute of Technology Madras. Further, the authors thank the referees for their comments and suggestions which made the revised version better in many respects.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Indian Institute of Technology Madras through the Institute Woman Post-Doctoral Fellowship and also supported by the National Board for Higher Mathematics (Department of Atomic Energy, India) through the Post-Doctoral Fellowship [Grant No: 0204/15/2017/R&D-II/10380].