A higher order non-polynomial spline method for fractional sub-diffusion problems

Li, Xuhao; Wong, Patricia J. Y.

doi:10.1016/j.jcp.2016.10.006

Cited by 34 publications

(29 citation statements)

References 30 publications

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“…Note that

O (h^{4})

has been the best spatial convergence until the present work. The numerical experiments in the next section will further demonstrate the outperformance of the parametric quintic spline scheme. (b)In our earlier paper we have considered a different fractional sub‐diffusion problem – it involves a second‐order space derivative

u_{x x}

instead of the fourth‐order space derivative in the current work. By using a different parametric quintic spline operator with different properties from H , we have derived a scheme that is better than other finite difference type methods for the second‐order fractional sub‐diffusion problem.…”

Section: Solvability and Convergence Of Parametric Quintic Spline Schemementioning

confidence: 99%

“…To achieve our objectives, we shall apply a parametric quintic spline method in the spatial dimension, and use a higher order approximation for fractional derivatives, namely

L 2 - 1_{σ}

scheme, to increase the efficiency further. It is noted that the parametric quintic spline method has been effective in solving boundary value problems of integer‐order differential equations, and subsequently parametric cubic splines have been applied to fractional differential equations, while the first application of parametric quintic spline to a second‐order fractional sub‐diffusion problem appears in Li and Wong . The fourth‐order fractional sub‐diffusion problem considered in the present work is as follows:

({, \begin{matrix} 0true_{0}^{C} D_{t}^{γ} u (x, t) + b^{2} \frac{\partial^{4} u}{\partial x^{4}} = f (x, t), x \in [0, L], t \in [0, T] \\ u (x, 0) = ϕ (x), x \in [0, L] \\ 0true u (0, t) = g_{1} (t), u (L, t) = g_{2} (t), \frac{\partial^{2} u (0, t)}{\partial x^{2}} = g_{3} (t), \frac{\partial^{2} u (L, t)}{\partial x^{2}} = g_{4} (t), t \in [0, T] \end{matrix})

where

…”

Section: Introductionmentioning

confidence: 99%

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Numerical solutions of fourth‐order fractional sub‐diffusion problems via parametric quintic spline

Wong

2019

Z Angew Math Mech

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In this paper, we develop a numerical scheme for a fourth-order fractional subdiffusion problem using parametric quintic spline and a non-uniform approximation for Caputo fractional derivatives. The solvability, convergence and stability of the scheme are established in maximum norm, and it is shown that the convergence order is higher than some earlier work done. Four numerical experiments are further carried out to demonstrate the efficiency of the proposed scheme as well as to compare with other methods. K E Y W O R D Sfractional differential equation, numerical solution, parametric spline, quintic spline, sub-diffusion equation

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“…Note that

O (h^{4})

u_{x x}

Section: Solvability and Convergence Of Parametric Quintic Spline Schemementioning

confidence: 99%

“…To achieve our objectives, we shall apply a parametric quintic spline method in the spatial dimension, and use a higher order approximation for fractional derivatives, namely

L 2 - 1_{σ}

({, \begin{matrix} 0true_{0}^{C} D_{t}^{γ} u (x, t) + b^{2} \frac{\partial^{4} u}{\partial x^{4}} = f (x, t), x \in [0, L], t \in [0, T] \\ u (x, 0) = ϕ (x), x \in [0, L] \\ 0true u (0, t) = g_{1} (t), u (L, t) = g_{2} (t), \frac{\partial^{2} u (0, t)}{\partial x^{2}} = g_{3} (t), \frac{\partial^{2} u (L, t)}{\partial x^{2}} = g_{4} (t), t \in [0, T] \end{matrix})

where

…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of fourth‐order fractional sub‐diffusion problems via parametric quintic spline

Wong

2019

Z Angew Math Mech

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“…In this subsection, we shall introduce the parametric quintic spline operator

H

which plays a critical role in obtaining high accuracy in the spatial dimension. We begin with the definition of parametric quintic spline which is similarly defined in . Definition A function

Q (x; ξ) \in C^{(4)} [0, L]

is called the parametric quintic spline with parameter

normalξ

(with respect to mesh

{normalΩ}_{h}

) if its restriction

Q_{j} (x; ξ) \equiv Q_{j} (x)

[x_{j - 1}, x_{j}], 1 \leq j \leq M

satisfies

Q_{j} (x_{i}) = u_{i}, i = j - 1, j

and

Q_{j}^{(4)} (x) + {normalξ}^{2} Q_{j}'' (x) = true(F_{j} + {normalξ}^{2} W_{j} true) \frac{x - x_{j - 1}}{h} + true(F_{j - 1} + {normalξ}^{2} W_{j - 1} true) \frac{x_{j} - x}{h},

where

Q_{j}'' (x_{i}) = W_{i}, ...

…”

Section: Pqs‐wsgl Numerical Schemementioning

confidence: 99%

“…In this subsection, we shall introduce the parametric quintic spline operator H which plays a critical role in obtaining high accuracy in the spatial dimension. We begin with the definition of parametric quintic spline which is similarly defined in [18,21].…”

Section: Parametric Quintic Splinementioning

confidence: 99%

An efficient numerical treatment of fourth‐order fractional diffusion‐wave problems

Wong

2018

Numerical Methods Partial

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In this paper, we consider the numerical treatment of a fourth‐order fractional diffusion‐wave problem. Our proposed method includes the use of parametric quintic spline in the spatial dimension and the weighted shifted Grünwald‐Letnikov approximation of fractional integral. The solvability, stability, and convergence of the numerical scheme are rigorously proved. It is shown that the theoretical convergence order improves those of earlier work. Simulation is further carried out to demonstrate the numerical efficiency of the proposed scheme and to compare with other methods.

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“…The analytical solutions of such equations are usually difficult to obtain, so seeking numerical solutions becomes more important and emergent. These numerical methods mainly covers compact difference methods [3][4][5], finite element methods [6,7], spectral methods [8,9], meshless methods [10,11], the homotopy analysis method [12], the Legendre operational matrix method [13], and spline collocation methods [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A high-order numerical scheme using orthogonal spline collocation for solving the two-dimensional fractional reaction–subdiffusion equation

2019

Adv Differ Equ

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In this paper, a high-order numerical scheme is proposed for solving the two-dimensional fractional reaction-subdiffusion equation. The method is based on adopting a third-order weighted and shifted Grünwald difference (WSGD) operator to approximate the time Caputo fractional derivative and applying the orthogonal spline collocation (OSC) method to approximate the spatial derivative. Stability and convergence analysis of the proposed method are rigorously proved. Several numerical examples in one variable and in two space variables are presented to validate our theoretical analysis.Recently, Tian et al. [28] proposed second-and third-order approximations for the Riemann-Liouville fractional derivative with order α (0 < α < 1) by using the weighted and shifted Grünwald difference (WSGD) operators. Thereafter, some related discussion problems covering the WSGD idea were addressed by many scholars. Following the idea of the WSGD operator, Wang and Vong [29] used compact finite difference schemes for the modified time sub-diffusion equation with Riemann-Liouville fractional derivative and the temporal Caputo fractional diffusion-wave equation. Ji and Sun [30] applied a thirdorder approximation for the Caputo fractional derivative and used compact difference scheme to solve fractional sub-diffusion equation. In [31], Liu et al. developed a highorder local discontinuous Galerkin method combined with WSGD approximation for a Caputo time-fractional sub-diffusion equation. In [32], a temporal second-order fully discrete two-grid finite element (FE) scheme is presented to solve nonlinear fractional Cable equation, in which the spatial direction is approximated by two-grid FE method and Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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A higher order non-polynomial spline method for fractional sub-diffusion problems

Cited by 34 publications

References 30 publications

Numerical solutions of fourth‐order fractional sub‐diffusion problems via parametric quintic spline

Numerical solutions of fourth‐order fractional sub‐diffusion problems via parametric quintic spline

An efficient numerical treatment of fourth‐order fractional diffusion‐wave problems

A high-order numerical scheme using orthogonal spline collocation for solving the two-dimensional fractional reaction–subdiffusion equation

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