Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation

Liu, Yanqin; Sun, HongGuang; Yin, Xiuling; Feng, Libo

doi:10.1007/s00033-019-1244-6

Cited by 14 publications

(8 citation statements)

References 43 publications

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“…Assume that u ∈ H 3 (Ω), u t , u tt , ∈ H 2 (Ω), u ttt ∈ L 2 (Ω). Then for any integer number 1 ≤ n ≤ N, there hold (18). Similar as (8), and noting that 𝜃 0 t = 0, we can obtain…”

Section: Superclose and Superconvergence Analysis For Fully Discrete ...mentioning

confidence: 94%

“…Equation (2) originates from the generalized Oldroyd‐B fluid model, which is a kind of non‐Newtonian fluids and has been found to approximate the response of many dilute polymeric liquids. Based on finite difference scheme in time and Legendre spectral approximation in space, Reference [18] established a fully discrete scheme for one dimensional case of Equation (2) when

L_{1} = L_{2} = 1

. Reference [6] used two new different finite difference schemes to approximate one dimensional case of Equation (2).…”

Section: Introductionmentioning

confidence: 99%

“…Reference [7] dealt with Equation (2) by the mixed L schemes to approximate the time fractional terms and the finite element to discretize the space. All these literature [6,7,17,18] investigated the stability and convergence. As far as we know, there is no literature concerned about the superclose and superconvergence analysis for Equations (1) and (2).…”

Section: Introductionmentioning

confidence: 99%

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Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Shi

Zhao

Wang

et al. 2021

Numerical Methods Partial

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A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank-Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time-space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

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Section: Superclose and Superconvergence Analysis For Fully Discrete ...mentioning

confidence: 94%

L_{1} = L_{2} = 1

. Reference [6] used two new different finite difference schemes to approximate one dimensional case of Equation (2).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Shi

Zhao

Wang

et al. 2021

Numerical Methods Partial

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show abstract

“…Consequently, the study of efficient numerical algorithms for solving the FPDEs plays a critical role in scientific computing and engineering applications. At present, many numerical methods have been developed for solving the FPDEs, such as finite difference methods [18][19][20][21][22][23][24][25], finite element methods [26][27][28][29][30], and spectral methods [31,32].…”

Section: Introductionmentioning

confidence: 99%

A Finite Difference Method for Solving the Wave Equation with Fractional Damping

Cui,

Ji,

Dai

2023

MCA

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In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a new numerical approximation for the Caputo fractional derivative and apply it for the 1D case to obtain a time-stepping method. We then develop an alternating direction implicit (ADI) scheme for the 2D case. Using the discrete energy method, we prove that the proposed difference schemes are unconditionally stable and convergent in both 1D and 2D cases. Finally, several numerical examples are given to verify the theoretical results.

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“…Parallelly, there are extensive profound literatures concerning theory and computation of the forward problems for the multi-term time-fractional diffusion equations (see [15,24,26,29,32,52] for an incomplete list). On the other hand, the published works were considered from different aspect on the inverse problems for multi-term counterpart, e.g., see [11-13, 23, 25, 27, 28, 43, 44, 47] etc.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

Sun¹,

Li²,

Zhang³

2021

Inverse Problems

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In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion equation from the noisy boundary Cauchy data in the one-dimensional case. The uniqueness for the inverse problem is derived based on the analytic continuation, the Laplace transformation and the Gel’fand–Levitan theory. Finally, the Levenberg–Marquardt regularization method with a regularization parameter chosen by a sigmoid-type function is applied for finding a stable approximate solution. Three numerical examples are provided to show the effectiveness of the proposed method.

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Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation

Cited by 14 publications

References 43 publications

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

A Finite Difference Method for Solving the Wave Equation with Fractional Damping

Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

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