Discontinuous Galerkin Methods for Problems with Fractional Derivatives

在这项工作中，研究者和他的学生将考虑适用于求解分数阶微分方程的不连续伽辽金方法的发展。在一系列初步但非常鼓舞人心的计算实验的激励下，将考虑分数阶常微分方程和偏微分方程的局部不连续伽辽金方法。这些初步的计算观测支持了一些猜想，这些猜想将指导所提出方法的发展和分析。虽然没有详细考虑具体的目标应用程序，但更实际的问题，如计算效率、多维问题和替代公式将在后一部分中讨论。虽然分数阶微积分的概念与牛顿引入的经典微积分一样古老，但分数阶微积分和分数阶方程的发展和分析却远没有那么成熟。然而，在过去的几十年里，分数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。例子可以发现异常传输过程、亚扩散和由记忆效应主导的问题。这些模型的应用范围很广，如多孔、粘弹性或生物流动、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。计划中的活动旨在开发高效和准确的计算技术，使应用科学家和工程师能够更有效地解决这类重要的模型。

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.