We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-$τ$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered.
A nonstandard fractional Jacobi pseudospectral method for weakly regular right-sided Caputo terminal problems
Al-Azhar University（爱资哈尔大学）；Beni-Suef University（贝尼苏韦夫大学）；Mississippi State University（密西西比州立大学）；Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊玛目穆罕默德·本·沙特伊斯兰大学）
A finite difference method for space fractional differential equations with variable diffusivity coefficient
King Fahd University of Petroleum and Minerals（法赫德国王石油与矿产大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；CNRS, LIMSI, Université Paris-Scalay（法国国家科学研究中心，LIMSI，巴黎萨克雷大学）
A New $L2-1_σ$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface
School of Arts and Sciences, Guangzhou Maritime University（广州航海学院人文与科学学部）；School of Information Engineering, Zhejiang Ocean University（浙江海洋大学信息工程学院）；Fuzhou Vocational Technical College（抚州职业技术学院）
A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation
College of Mathematics and Computer Science, Fuzhou University（福州大学数学与计算机科学学院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method
University of Insubria（因苏布里亚大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；University of Uppsala（乌普萨拉大学）；University of Milano - Bicocca（米兰比可卡大学）
Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
Guangzhou Maritime University（广州航海学院）；Nanjing University of Information Science and Technology（南京信息工程大学）；Kyoto University（京都大学）；East China University of Technology（东华理工大学）
