A higher order numerical scheme for generalized fractional diffusion equations

Ding, Qinxu; Wong, Patricia J. Y.

doi:10.1002/fld.4852

Cited by 8 publications

(13 citation statements)

References 34 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The next lemma relates the generalized Caputo fractional derivative and the generalized Riemann–Liouville fractional integral. This result generalizes the corresponding result for the classical case.Lemma [39]. Let 0 < α < 1 and x ∈ ( a , b ) be fixed .…”

Section: Derivation Of Numerical Schemesupporting

confidence: 78%

“…The next result gives the relation between the generalized Caputo fractional derivative (1.4) and the generalized Riemann–Liouville fractional derivative (2.1). This relation generalizes the corresponding relation for the classical fractional derivatives.Lemma [39, 40]. Let 0 < α < 1.…”

Section: Derivation Of Numerical Schemementioning

confidence: 56%

“…This relation generalizes the corresponding relation for the classical fractional derivatives. [39,40]. Let 0 < < 1.…”

Section: Fourth Order Approximation For Generalized Fractional Derivamentioning

confidence: 99%

See 2 more Smart Citations

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

Ding

Wong

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t). Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.

show abstract

Section: Derivation Of Numerical Schemesupporting

confidence: 78%

Section: Derivation Of Numerical Schemementioning

confidence: 56%

See 1 more Smart Citation

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

Ding

Wong

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is observed that the errors ‖ E ‖ are small, however the convergence order is reduced to around (2 − α ) due to the weak singularity of the solution at

x = 0

. This order reduction due to the presence of weak singularity of the solution is also observed in the work of previous studies 52–56 …”

Section: Numerical Examplessupporting

confidence: 68%

A higher order numerical scheme for solving fractional Bagley‐Torvik equation

Ding

Wong

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this paper, we develop a higher order numerical method for the fractional Bagley‐Torvik equation. The main tools used include a new fourth‐order approximation for the fractional derivative based on the weighted shifted Grünwald‐Letnikov difference operator and a discrete cubic spline approach. We show that the theoretical convergence order improves those of previous work. Five examples are further presented to illustrate the efficiency of our method and to compare with other methods in the literature.

show abstract

“…Lemma 3. 53,55,[57][58][59][60] Let u C 3þα ℝ ð Þ and all derivatives of u L 1 ℝ ð Þ, then we have the following form of the third-order WSGD (3-WSGD):…”

Section: Preliminariesmentioning

confidence: 99%

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Khalaf

Elsaid

Hammad

et al. 2023

Numer Methods Biomed Eng

View full text Add to dashboard Cite

Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.

show abstract

A higher order numerical scheme for generalized fractional diffusion equations

Cited by 8 publications

References 34 publications

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

A higher order numerical scheme for solving fractional Bagley‐Torvik equation

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Contact Info

Product

Resources

About