Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach

An integrated online adaptive state of charge estimation approach of high-power lithium-ion battery packs.

A novel online adaptive state of charge (SOC) estimation method is proposed, aiming to characterize the capacity state of all the connected cells in lithium-ion battery (LIB) packs. This method is realized using the extended Kalman filter (EKF) combined with Ampere-hour (Ah) integration and open circuit voltage (OCV) methods, in which the time-scale implementation is designed to reduce the computational cost and accommodate uncertain or time-varying parameters. The working principle of power LIBs and their basic characteristics are analysed by using the combined equivalent circuit model (ECM), which takes the discharging current rates and temperature as the core impacts, to realize the estimation. The original estimation value is initialized by using the Ah integral method, and then corrected by measuring the cell voltage to obtain the optimal estimation effect. Experiments under dynamic current conditions are performed to verify the accuracy and the real-time performance of this proposed method, the analysed result of which indicates that its good performance is in line with the estimation accuracy and real-time requirement of high-power LIB packs. The proposed multimodel SOC estimation method may be used in the real-time monitoring of the high-power LIB pack dynamic applications for working state measurement and control

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure

Low-Intensity Pulsed Ultrasound Stimulation Modulates the Nonlinear Dynamics of Local Field Potentials in Temporal Lobe Epilepsy

Low-intensity pulsed ultrasound stimulation (LIPUS) can inhibit seizures associated with temporal lobe epilepsy (TLE), which is the most common epileptic syndrome in adults and accounts for more than half of the cases of intractable epilepsy. Electroencephalography (EEG) signal analysis is an important method for studying epilepsy. The nonlinear dynamics of epileptic EEG signals can be used as biomarkers for the prediction and diagnosis of epilepsy. However, how ultrasound modulates the nonlinear dynamic characteristics of EEG signals in TLE is still unclear. Here, we used low-intensity pulsed ultrasound to stimulate the CA3 region of kainite (KA)-induced TLE mice, simultaneously recorded local field potentials (LFP) in the stimulation regions before, during, and after LIPUS. The nonlinear characteristics, including complexity, approximate entropy of different frequency bands, and Lyapunov exponent of the LFP, were calculated. Compared with the control group, the experimental group showed that LIPUS inhibited TLE seizure and the complexity, approximate entropy of the delta (0.5–4 Hz) and theta (4–8 Hz) frequency bands, and Lyapunov exponent of the LFP significantly increased in response to ultrasound stimulation. The values before ultrasound stimulation were higher ∼1.87 (complexity), ∼1.39 (approximate entropy of delta frequency bands), ∼1.13 (approximate entropy of theta frequency bands) and ∼1.46 times (Lyapunov exponent) than that after ultrasound stimulation (p < 0.05). The above results demonstrated that LIPUS can alter nonlinear dynamic characteristics and provide a basis for the application of ultrasound stimulation in the treatment of epilepsy

Association Between Serum Vitamin D Levels and Parkinson's Disease: A Systematic Review and Meta-Analysis

Background: Vitamin D is an important secosteroid which is involved the development and regulation of brain activity. Several studies have focused on exploring the relationship between serum vitamin D levels and Parkinson's disease (PD), but the conclusion remains ambiguous.Methods: We searched observational studies that explored the association between serum vitamin D levels and PD based on PubMed, EMBASE and Cochrane library from inception through to January 2018. The quality of included studies was evaluated by using Newcastle-Ottawa Scale (NOS). Statistical analysis of this meta-analysis was performed by Stata version 12.0 and R software.Results: Twenty studies with a total of 2,866 PD patients and 2,734 controls were included. Compared with controls, PD patients had lower serum vitamin D levels (WMD −3.96, 95%CI −5.00, −2.92), especially in higher latitude regions (WMD −4.20, 95%CI −5.66, −2.75). Assay methods contributed significantly to high heterogeneity. Furthermore, PD patients with deficient vitamin D levels had advanced risk (OR 2.08, 95%CI 1.35, 3.19) than those patients with insufficient ones (OR = 1.73, 95%CI 1.48, 2.03). In addition, serum vitamin D levels were also related to the severity of PD (WMD −5.27, 95%CI −8.14, −2.39) and the summary correlation coefficient showed strongly negative correlation (r = −0.55, 95%CI −0.73, −0.29). Moreover, the pooled correlation coefficient revealed that serum vitamin D levels were also negatively correlated to the Unified Parkinson's Disease Rating Scale III (UPDRS III) (r = −0.36, 95%CI −0.53, −0.16), but did not correlate with the duration of PD (P = 0.37) and age of patients (P = 0.49).Conclusion: Serum vitamin D levels are inversely associated with the risk and severity of PD. Our results provided an updated evidence of association between low vitamin D levels and PD and prompt the adjunctive therapeutic decisions about vitamin D replacement in PD

Extra-Cerebellar Signs and Non-motor Features in Chinese Patients With Spinocerebellar Ataxia Type 3

Objectives: Our study attempted to systematically explore the prevalence of extra-cerebellar signs and non-motor symptoms, such as anxiety, depression, fatigue, excessive daytime sleepiness (EDS) and sleep disturbances in a cohort of Chinese patients with spinocerebellar ataxia type 3 (SCA3), and further investigated the correlations between non-motor symptoms and clinical characteristics in SCA3 patients.Methods: This study included 68 molecular-proven SCA3 patients. Extra-cerebellar signs were evaluated with the Inventory of Non-Ataxia Symptoms (INAS). The INAS count indicated the number of non-ataxia signs in each patient. The severity of ataxia, fatigue, EDS, sleep quality, anxiety, and depression were assessed using the Scale for the assessment and rating of ataxia (SARA), Fatigue Severity Scale (FSS), Epworth Sleepiness Scale (ESS), Pittsburgh Sleep Quality Index (PSQI), Hamilton Anxiety Rating Scale (HAMA), and the Hamilton Depression Rating Scale (HAMD) (24 items), respectively.Results: Extra-cerebellar signs were detected in 91.2% of all SCA3 patients and the mean total INAS count was 2.72 ± 1.88. Rigidity was the most frequent extra-cerebellar sign (47.1%, N = 32). Sensory symptoms (2.9%, N = 2) and chorea (5.9%, N = 4) were rare, and myoclonus (0%) was not found in this cohort. High frequencies of sleep disturbances (64.7%), fatigue (52.9%), depression (48.5%), and anxiety (42.6%) were detected in SCA3 patients. The Spearman correlation indicated that the HAMD score was associated with the CAG repeat length and HAMA score, while the PSQI score was correlated with the SARA and FSS score. In addition, multivariate linear regression analysis showed that the CAG repeat length, age of onset, sleep disturbances and depression were significant predictors of fatigue in SCA3 patients.Conclusions: Our study indicates that the vast majority of SCA3 patients display extra-cerebellar signs. Except for EDS, anxiety, depression, fatigue and impaired sleep quality are present in SCA3 patients. The CAG repeat length, age of onset, sleep disturbances and depression are predictors of fatigue in SCA3 patients