A Unified Petrov-Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations  

作　　者：Mehdi Samiee Ehsan Kharazmi Mark M.Meerschaert Mohsen Zayernouri 

机构地区：[1]Department of Mechanical Engineering,Michigan State University,East Lansing,MI 48824,USA [2]Department of Computational Mathematics,Science and Engineering,Michigan State University,East Lansing,MI 48824,USA [3]Department of Probability and Statistics,Michigan State University,East Lansing,MI 48824,USA

出　　处：《Communications on Applied Mathematics and Computation》2021年第1期61-90,共30页应用数学与计算数学学报（英文）

基　　金：This work was supported by the AFOSR Young Investigator Program(YIP)award(FA9550-17-1-0150),the MURI/ARO(W911NF-15-1-0562);tthe National Science Foundation Award(DMS-1923201);the ARO Young Investigator Program Award(W911NF-19-1-0444)。

摘　　要：Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings.In complex multi-fractal anomalous transport phenomena,distributed-order partial differential equations appear as tractable mathematical models,where the underlying derivative orders are distributed over a range of values,hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics.We develop a unified,fast,and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions,respectively.By defining the proper underlying distributed Sobolev spaces and their equivalent norms,we rigorously prove the well-posedness of the weak formulation,and thereby,we carry out the corresponding stability and error analysis.We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

关 键 词：Distributed Sobolev space Well-posedness analysis Discrete inf-sup condition Spectral convergence Jacobi poly-fractonomials Legendre polynomials 

分 类 号：O17[理学—数学]