Discontinuous Galerkin Methods for Problems with Fractional Derivatives

在这方面的努力，调查员和他的学生将考虑发展的不连续Galerkin方法适合于解决分数阶微分方程。受一系列初步但非常令人鼓舞的计算实验的启发，分数阶常微分方程和偏微分方程的局部间断Galerkin方法将被考虑。这些初步的计算观测支持了一些假设，这些将指导所提出的方法的发展和分析。虽然具体的目标应用不被认为是细节，更实际的问题，如计算效率，多维问题，和替代配方将在后期的努力解决。虽然分数微积分的概念是一样古老的经典微积分介绍了牛顿，分数微积分和分数方程的发展和分析是不太成熟。然而，在过去的几十年中，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和重要的描述。例子可以发现异常的传输过程，子扩散，和记忆效应为主的问题。这些模型的应用被发现在广泛的领域，如多孔，粘弹性，或生物流动，原油在油藏中的流动，聚变等离子体问题，复杂材料的性能建模，金融市场等计划的活动寻求开发高效和准确的计算技术，使应用科学家和工程师更有效地解决这一类重要的模型。

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.