Improving a Tikhonov regularization method with a fractional-order differential operator for the inverse black body radiation problem

Abstract

Tikhonov regularization is an usual method to solve an ill-posed problem, recommended when the input data are contaminated with noise. However, in some cases, the use of this technique is not sufficient to provide good solutions. In this work, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black body radiation problem. The method proposed consisted in including the norm of the fractional-order derivative of the solution in the original functional proposed by Tikhonov. In the present framework, the regularized solution depends on the regularization parameter, λ, and on the fractional derivative order, α. For α assuming real value between 0 and 2, the solution obtained is more precise than those from the usual Tikhonov regularization method.

Keywords:

• Fractional calculus
• ill-posed problem
• Grünwald–Letnikov derivative
• Tikhonov regularization method

2010 Mathematics Subject Classifications:

• 26A33
• 35R30

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Nelson H. T. Lemes http://orcid.org/0000-0002-1309-542X

José P. C. dos Santos http://orcid.org/0000-0002-4161-1786

João P. Braga http://orcid.org/0000-0002-2869-7079

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001 and FAPEMIG.