Research in Applied Mathematics vol. 1 (2017), Article ID 101264, 16 pages doi:10.11131/2017/101264 AgiAl Publishing House http://www.agialpress.com/ Research Article A Finite Difference Scheme for the Time-Fractional Landau-Lifshitz-Bloch Equation C. Ayouch 1 , E. H. Essoufi 1 , and M. Tilioua 2 1 Laboratoire MISI, FST Settat, Univ. Hassan I, 26000 Settat, Morocco 2 M2I Laboratory, MAMCS Group, FST Errachidia, Univ. Moulay Ismaïl, P.O. Box 509, Boutalamine, 52000 Errachidia, Morocco Abstract. In this paper, we study a finite difference scheme for temporal discretization of the time-fractional Landau-Lifshitz-Bloch equation, such as the fractional time derivative of order  is taken in the sense of Caputo. An existence result is established for the semi-discrete problem by Schaefer’s fixed point theorem. Stability and error analysis are then provided, showing that the temporal accuracy is of order 2−. Keywords: Landau-Lifshitz-Bloch equation, finite difference method, fractional differential equations, Schaefer’s fixed point, existence, stability, error analysis Mathematics Subject Classification: 78A25, 35Q60, 35B40 1. Introduction The conventional Landau-Lifshitz (LL) or Landau-Lifshitz-Gilbert (LLG) equation is a grand master equation that provides quantitative predictions on magnetic structure and magnetiza- tion dynamics of ferromagnets at low temperatures. At high temperatures, however, the LLG equation fails to describe the longitudinal relaxation and thus it is necessary to extend the LL equation. In particular, it is desirable to obtain a similarly effective equation, which is capable of quantitatively computing magnetization dynamics at high temperatures in real time. Such extension is not only theoretically interesting but also technologically relevant. For high temperatures the LLG equation must be replaced by a more thermodynamically consistent approach such as the Landau-Lifshitz-Bloch (LLB) equation [3]. The LLB equation essentially interpolates between the LLG equation at low temperatures and the Ginzburg-Landau theory of phase transitions. It is valid not only below but also above the Curie temperature   . An impor- tant property of the LLB equation is that the magnetization magnitude is no longer conserved but is a dynamical variable [4]. The spin polarization vector (, ), where =   0  , and  is magnetization vector and  0  is the saturation magnetization value at =0. For Ω⊂ℝ  ,≥1,  satisfies the following LLB equation   =  ×  eff () +  1 || 2 ( ⋅ eff () ) −  2 || 2 × ( × eff () ) , (1) where >0 is the gyromagnetic ratio, the symbol × denotes the vector cross product in ℝ 3 ,  1 and  2 are the longitudinal and transverse damping parameters, respectively. How to cite this article: C. Ayouch, E. H. Essoufi, and M. Tilioua, “A Finite Difference Scheme for the Time-Fractional Landau- Lifshitz-Bloch Equation,” Research in Applied Mathematics, vol. 1, Article ID 101264, 16 pages, 2017. doi:10.11131/2017/101264 Page 1 Corresponding Author M. Tilioua tilioua@melix.org Editor Jianlong Qiu Dates Received 14 November 2016 Accepted 24 February 2017 Copyright © 2017 C. Ayouch et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.