On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

Burrage, Kevin; Burrage, Pamela; Turner, Ian; Zeng, Fanhai

doi:10.3390/axioms7020025

Cited by 3 publications

(3 citation statements)

References 34 publications

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“…where ũ(t) is uniformly bounded for t ∈ [0, T ]. This assumption holds in real applications; see for example, [29,30,31,32,33]. If f (u(t), t) is smooth for t ∈ [0, T ], then γ k ∈ {i+jα, i = 0, 1, .…”

Section: Semi-implicit Time-stepping Methodsmentioning

confidence: 99%

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

Zeng

Turner

Burrage

et al. 2019

Journal of Computational Physics

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In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain.

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Section: Semi-implicit Time-stepping Methodsmentioning

confidence: 99%

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

Zeng

Turner

Burrage

et al. 2019

Journal of Computational Physics

Self Cite

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“…The authors in [6] study systems of fractional differential equations, in which different equations may have a different fractional time derivative at the left-hand side term of the equation. The linear case is completely worked out, providing a theory which collapses to the well-known Mittag-Leffler solution in the case where the indices are the same.…”

Section: Numerical Solution Of Differential Equationsmentioning

confidence: 99%

Advanced Numerical Methods in Applied Sciences

Brugnano

Iavernaro

2019

Axioms

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The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

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“…There exists a wide variety of numerical methods which deal with space and/or fractional differential equations [24][25][26][27][28][29][30][31]. The coupled space fractional Ginzburg-Landau system was numerically investigated in [32].…”

Section: Introductionmentioning

confidence: 99%

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

2021

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A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

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On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

Cited by 3 publications

References 34 publications

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

Advanced Numerical Methods in Applied Sciences

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

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