Fractional Differential Equations

Prefaces Acknowledgements 1. Mountains and their climatological study 2. Geographical controls of mountain meteorological elements 3. Circulation systems related to orography 4. Climatic characteristics of mountains 5. Regional case studies 6. Mountain bioclimatology 7. Changes in mountain climates Appendix General index Author index.

Mountain Weather and Climate

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

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Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

BACKGROUND: The Coronavirus Disease 2019 (COVID-19) outbreak has escalated the burden of psychological distress. We aimed to evaluate factors associated with psychological distress among the predominantly general population during the COVID-19 pandemic. METHODS: We searched PubMed, EMBASE, Scopus, Cochrane Library, PsycINFO, and World Health Organization COVID-19 databases (Dec 2019-15 July 2020). We included cross-sectional studies that reported factors associated with psychological distress during the COVID-19 pandemic. Primary outcomes were self-reported symptoms of anxiety and depression. Random-effects models were used to pool odds ratios (OR) and 95% confidence intervals (CI). The protocol was registered in PROSPERO (#CRD42020186735). FINDINGS: We included 68 studies comprising 288,830 participants from 19 countries. The prevalence of anxiety and depression was 33% (95% CI: 28%-39%) and 30% (26%-36%). Women versus men (OR: 1.48 [95% CI: 1.29-1.71; I2 = 90.8%]), younger versus older (< versus ≥35 years) adults (1.20 [1.13-1.26]; I2 = 91.7%), living in rural versus urban areas (1.13 [1.00-1.29]; I2 = 82.9%), lower versus higher socioeconomic status (e.g. lower versus higher income: 1.45 [1.24-1.69; I2 = 82.3%]) were associated with higher anxiety odds. These factors (except for residential area) were also associated with higher depression odds. Furthermore, higher COVID-19 infection risk (suspected/confirmed cases, living in hard-hit areas, having pre-existing physical or mental conditions) and longer media exposure were associated with higher odds of anxiety and depression. INTERPRETATION: One in three adults in the predominantly general population have COVID-19 related psychological distress. Concerted efforts are urgently needed for interventions in high-risk populations to reduce urban-rural, socioeconomic and gender disparities in COVID-19 related psychological distress.

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Factors associated with psychological distress during the coronavirus disease 2019 (COVID-19) pandemic on the predominantly general population: A systematic review and meta-analysis.

A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and fractional integrals of order $2-\alpha$, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O($h^{k+1}$) for subdiffusion, and an order of convergence of ${\cal O}(h^{k+1/2})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

Discontinuous Galerkin method for fractional convection-diffusion equations

We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and a fractional integral of order $2-\alpha$. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method scheme is proposed for the equations. We prove stability and optimal order of convergence ${\cal O}(h^{k+1})$ for the fractional diffusion problem, and an order of convergence of ${\cal O}(h^{k+\frac{1}{2}})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Discontinuous%20Galerkin%20method.pdf

Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Stable%20Multi-domain.pdf

Qinwu Xu

Papers

Discontinuous Galerkin method for fractional convection-diffusion equations

Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations

Stable multi-domain spectral penalty methods for fractional partial differential equations

Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient

A multi-domain spectral method for time-fractional differential equations