Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

Gao, Guang‐Hua; Sun, Zhi‐zhong

doi:10.1007/s11075-016-0167-y

Cited by 24 publications

(14 citation statements)

References 26 publications

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“…The Newton–Cotes quadrature scheme can be divided into closed and open approaches, depending on whether the function values at the end points are included. Following the closed approach, different quadrature rules used for DO derivatives include the trapezoid rule [ 56 , 87 , 106 , 108 , 109 , 110 , 111 , 112 , 113 , 114 , 115 , 116 , 117 ], the Simpson’s rule [ 87 , 106 , 111 , 112 , 116 , 117 , 118 , 119 , 120 , 121 ], and the Boole’s rule [ 122 ]. All these schemes are also associated with different orders of convergence.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…Li [ 179 ] developed a numerical scheme with high spatial accuracy (

) by combining WSGLM and the parametric quintic spline method. Another scheme capable of delivering high spatial accuracy (

) was proposed by using the WSGLM for temporal approximation and high-order compact difference scheme for spatial approximation [ 117 ]. Yang [ 180 ] also proposed a similar composite method based on WSGLM in time and orthogonal spline collocation method in space.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

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Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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Section: Mathematical Backgroundmentioning

confidence: 99%

“…Li [ 179 ] developed a numerical scheme with high spatial accuracy (

) by combining WSGLM and the parametric quintic spline method. Another scheme capable of delivering high spatial accuracy (

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…Numerical experiments on the space and time convergence orders of the C-N scheme and the PASE-I scheme are given below. We define L 2 as the error, order 1 as the spaceconvergent order, and order 2 as the time-convergent order [30,31]:…”

Section: Numerical Experimentsmentioning

confidence: 99%

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

Pan

Yang

2020

Mathematical Problems in Engineering

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This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

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“…In [15], Li et al studied Galerkin finite element methods with non-uniform temporal meshes for multi-term Caputo-type FDEs, and proved that the corresponding finite element schemes were unconditionally convergent and stable. Gao et al [16] constructed two difference schemes to solve 1D time distributed-order fractional wave equations and derived them from a weighted and shifted Grünwald formula with second-order accuracy.…”

Section: Introductionmentioning

confidence: 99%

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

Jian

Huang

et al. 2020

Preprint

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In this paper, fast numerical methods are established for solving a class of time distributedorder and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Grünwald formula in time and the fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing the preconditioned Krylov subspace method is developed to solve the symmetric positive definite Toeplitz linear systems derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the discretized Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around one, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other numerical methods.

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Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

Cited by 24 publications

References 26 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

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