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A Petrov-Galerkin Spectral Element Method for Fractional Elliptic Problems
Ehsan Kharazmi, Mohsen Zayernouri, George Em Karniadakis
We develop a new -continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_{0}{\mathcal{D}}_{x}^α u(x) - λu(x) =…
Implicit-Explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions
Wanrong Cao, Fanhai Zeng, Zhongqiang Zhang +1
We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order . From the known structure of th…
A Tunably-Accurate Laguerre Petrov-Galerkin Spectral Method for Multi-Term Fractional Differential Equations on the Half Line
Anna Lischke, Mohsen Zayernouri, George Em Karniadakis
We present a new tunably-accurate Laguerre Petrov-Galerkin spectral method for solving linear multi-term fractional initial value problems with derivative orders at most one and co…
Petrov-Galerkin and Spectral Collocation Methods for distributed Order Differential Equations
Ehsan Kharazmi, Mohsen Zayernouri, George Em Karniadakis
Distributed order fractional operators offer a rigorous tool for mathematical modelling of multi-physics phenomena, where the differential orders are distributed over a range of va…
Adaptive Finite Element Method for fractional differential equations using Hierarchical Matrices
Xuan Zhao, Xiaozhe Hu, Wei Cai +1
A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-unifo…