A unified spectral method for FPDEs with two-sided derivatives; Part II: Stability, and error analysis

Samiee, Mehdi; Zayernouri, Mohsen; Meerschaert, Mark M.

doi:10.1016/j.jcp.2018.07.041

Cited by 27 publications

(24 citation statements)

References 43 publications

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“…The L 2 -error decays linearly in the log-log scale plot as temporal expansion order N increases in both cases, indicating the spectral convergence of PG method. In [36], we obtain the theoretical convergence rate of e L 2 and compare with the corresponding practical ones.…”

Section: Numerical Test (I)mentioning

confidence: 99%

“…Furthermore, exponential convergence is observed for a sinusoidal smooth function in a spatial p-refinement. To check the stability and spectral convergence of the PG method, we carried out the corresponding discrete stability and error analysis of the method for (3.26) in [36]. Despite the high accuracy and the efficiency of the method especially in higher-dimensional problems, treatment of FPDEs in complex geometries and FPDEs with variable coefficients will be studies in our future works.…”

Section: Numerical Test (V)mentioning

confidence: 99%

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A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

Samiee

Zayernouri

Meerschaert

2019

Journal of Computational Physics

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We develop a unified Petrov-Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form 0 D 2τu]+ f , where 2τ ∈ (0, 2), 2µ i ∈ (0, 1) and 2ν j ∈ (1, 2), in a (1+d)-dimensional space-time hypercube, d = 1, 2, 3, · · · , subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigenfunctions of the fractional Sturm-Liouville eigen-problems of the first kind in [49], called Jacobi poly-fractonomials, as temporal bases, and the eigen-functions of the boundary-value problem of the second kind as temporal test functions. Next, we construct our spatial basis/test functions using Legendre polynomials, yielding mass matrices being independent of the spatial fractional orders (µ i , ν j , i, j = 1, 2, · · · , d). Furthermore, we formulate a novel unified fast linear solver for the resulting high-dimensional linear system based on the solution of generalized eigen-problem of spatial mass matrices with respect to the corresponding stiffness matrices, hence, making the complexity of the problem optimal, i.e., O(N d+2 ). We carry out several numerical test cases to examine the CPU time and convergence rate of the method. The corresponding stability and error analysis of the Petrov-Galerkin method are carried out in [36].

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Section: Numerical Test (I)mentioning

confidence: 99%

Section: Numerical Test (V)mentioning

confidence: 99%

A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

Samiee

Zayernouri

Meerschaert

2019

Journal of Computational Physics

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“…We also formulate a fast solver for the corresponding weak form of (1), following [50], which significantly reduces the computational expenses in high-dimensional problems. -We establish the well-posedness of the weak form of the problem in the underlying distributed Sobolev spaces respecting the analysis in [51] and prove the stability of proposed numerical scheme. We additionally perform the corresponding error analysis, where the distributed Sobolev spaces enable us to obtain accurate error estimate of the scheme.…”

Section: Introductionmentioning

confidence: 99%

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

Samiee

Kharazmi

Meerschaert

et al. 2020

Commun. Appl. Math. Comput.

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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.

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“…These eignefunctions are comprised of smooth and fractional parts, where the latter can be tunned to capture singularities of true solution. They are successfully employed in constructing discrete solution/test function spaces and developing a series of high-order and efficient Petrov-Galerkin spectral methods, see [39,64,59,56,58,57,32,31,30].…”

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Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

Kharazmi

Zayernouri

2019

Journal of Verification, Validation and Uncertainty Quantification

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Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov-Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure.Key words. Stochastic fractional PDEs, forward uncertainty quantification, Monte Carlo simulation, probabilistic collocation method, Smolyak sparse grid, Petrov-Galerkin spectral method.1. Introduction. Fractional models construct a tractable mathematical framework to describe and predict the behavior of multi-scales multi-physics complex phenomena. Particularly, fractional differential equations, as a well-structured generalization of their integer order counterparts, provide a rigorous mathematical tool to develop models, describing anomalous behavior in complex physical systems [79,25,27,17,24,45,46,8,43], where the anomalies manifest in heavy tailed distribution of corresponding underlying stochastic processes, moreover, exhibiting sharp peaks, intermittency, and asymmetry in the underlying distributions. Significant approximations as inherent part of assumptions upon which the model is built, lack of information about true values of parameters (incomplete data), and random nature of quantities being modeled pervade uncertainty in the corresponding mathematical formulations [16,55]. In this work, we develop an uncertainty quantification (UQ) framework in the context of stochastic fractional partial differential equations (SFPDEs), in which we characterize different sources of uncertainties and further propagate the associated randomness to the system response quantity of interest (QoI).Types and Sources of Uncertainty. The model uncertainties are in general being classified as aleatory, epistemic, and mixed, according to their fundamental essence. They can also be broadly characterized as occurring in model inputs, numerical approximations, and model form. Model inputs encompass all model parameters coming from geometry, constitutive laws, and fields equation, while also pertaining surrounding interactions, such as boundary conditions and random forcing sources (noise). Numerical approximations, which are an essence of differential equations since they generally do not lend themselves to analytical solutions, introduce uncertainty by imposing different sources of discretization error, iterative convergence error, and round off error. The fractional derivatives introduce derivative ord...

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A unified spectral method for FPDEs with two-sided derivatives; Part II: Stability, and error analysis

Cited by 27 publications

References 43 publications

A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

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

Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

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