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The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both L 1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on α -difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions. [ABSTRACT FROM AUTHOR]

Sex estimation from fragmentary bone remain is still challenge for forensic pathologist. Ulna has been reported useful for sex estimation by metric analysis. This study generated sex estimation function for fragment and complete of ulnar bone in a Thai population. The function was generated from 200 pairs of ulnar bone, and others 20 pair of ulnar bone were used for test the accuracy of the functions. Olecranon width was the best single variable for sex predicting of proximal part of ulna, which right olecranon width could be classified the sex 90.5 %. While distal end width of ulna was the variable for predicting the sex of distal part, which left distal end width could be classified the sex with 83.0 %. Stepwise discriminant function analysis was applied to proximal part. For proximal part of right ulna 4 measurements were selected (inferior-medial trochlear notch length, olecranon width, olecranon-coronoid process length, and maximum proximal ulnar width), while the left side, superior trochlear notch width, olecranon width, and maximum proximal ulnar width were chosen, and their functions could be predicted the sex with 91.0 % and 90.0 %, respectively. Our results indicated the ulnar bone had high ability for estimating the sex in a Thai population. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop and analyze high-order time stepping schemes for approximately solving semilinear subdiffusion equations. We apply the convolution quadrature generated by k-step backward differentiation formula (BDFk) to discretize the time-fractional derivative with order α ∈ (0, 1) and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li, and Zhou [SIAM J. Sci. Comput., 39 (2017), pp. A3129-A3152], while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part and using the generating function technique, we prove that the convergence order of the corrected BDFk scheme is O(τmin(k,1+2α-ε)) without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

An implicit-explicit multistep method based on the backward difference formulae (BDF) is proposed for time discretization of parabolic equations with a non-self-adjoint operator. Implicit and explicit schemes are used for the self-adjoint and anti-self-adjoint parts of the operator, respectively. For a k-step method, some correction terms are added to the starting k - 1 steps to maintain kth-order convergence without imposing further compatibility conditions at the initial time. Long-time kth-order convergence for the numerical method is proved under the assumptions that the operator is coercive and that the non-self-adjoint part is low order. Such an operator often appears in practical computation (such as the Stokes--Darcy system) but may violate the standard sectorial angle condition used in the literature for analysis of BDF. In particular, the proposed method and analysis in this paper extend the long-time energy error analysis of the Stokes--Darcy system in Chen et al. [SIAM J. Numer. Anal., 51 (2013), pp. 2563--2584; Numer. Math., 134 (2016), pp. 857--879] to general symmetrized and decoupled BDF methods up to order 6 by using the generating function technique. [ABSTRACT FROM AUTHOR]