Discontinuous Galerkin Methods for Problems with Fractional Derivatives

解决分数阶导数问题的不连续伽辽金方法

• 批准号: 1115416
• 负责人: Johnny Guzman
• 金额: $ 31.09万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115416&HistoricalAwards=false
• 关键词: Discontinuous; Galerkin; Methods; Problems; Fractional

In this effort the investigator and his student will consider developments of discontinuous Galerkin methods suitable for solving fractional differential equations. Motivated by a series of preliminary but very encouraging computational experiments, local discontinuous Galerkin methods for both fractional ordinary and partial differential equations will be considered. These initial computational observations supports a number of conjectures and these will guide the development and analysis of the proposed methods. While specific target applications are not considered detail, problems of a more practical character such as computational efficiency, multi-dimensional problems, and alternative formulations will be addressed in the latter part of the effort.While the notion of fractional calculus is as old as that of classical calculus introduced by Newton, the development and analysis of fractional calculus and fractional equations is not nearly as mature. However, during the last few decades fractional calculus has emerged as a natural and important description for a broad range of non-classical phenomena in the applied sciences and engineering. Examples can be found anomalous transport processes, sub-diffusion, and problems dominated by memory effects. Applications of such models are found in wide range of areas such as porous, visco-elastic, or biological flows, flow of crude oil in reservoirs, fusion plasma problems, modeling of properties of complex materials, financial markets etc. The planned activities seek to develop efficient and accurate computational techniques to allow application scientists and engineers to more effectively solve this important class of models.

在这项工作中，研究者和他的学生将考虑开发适合求解分数阶微分方程的不连续伽辽金方法。在一系列初步但非常令人鼓舞的计算实验的推动下，将考虑分数常微分方程和偏微分方程的局部不连续伽辽金方法。这些最初的计算观察支持了许多猜想，这些猜想将指导所提出方法的开发和分析。虽然没有详细考虑具体的目标应用，但更实际的问题，例如计算效率、多维问题和替代公式，将在后期工作中得到解决。虽然分数阶微积分的概念很古老与牛顿引入的经典微积分相比，分数阶微积分和分数方程的发展和分析还远没有那么成熟。然而，在过去的几十年里，分数阶微积分已经成为应用科学和工程中广泛的非经典现象的自然而重要的描述。例子包括异常传输过程、次扩散和记忆效应主导的问题。此类模型的应用广泛，例如多孔、粘弹性或生物流、油藏中的原油流动、聚变等离子体问题、复杂材料特性建模、金融市场等。计划的活动旨在开发高效、准确的计算技术，使应用科学家和工程师能够更有效地解决这一类重要的模型。