A comparative study of the point implicit schemes on solving the 2D time fractional cable equation

Balasim, Alla Tareq; Ali, Norhashidah Hj. Mohd.

doi:10.1063/1.4995882

Cited by 9 publications

(15 citation statements)

References 15 publications

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“…In this section, we study the stability analysis of the WAFD numerical schemes (21) and (23) using the fractional version of the John von Neumann stability technique. First, we start by studying the stability analysis for the resulting numerical scheme of the two-dimensional fractional Cable Equation 21, then we will repeat the same for the two-dimensional fractional reaction-sub-diffusion Equation (23).…”

Section: Stability Analysismentioning

confidence: 99%

“…We are going to get the stability condition for the numerical scheme (21): Theorem 1. The WAFD scheme derived in (21) is stable under the following stability criterion…”

Section: Stability Analysismentioning

confidence: 99%

“…Now, we are going to prove the stability condition for the numerical scheme (23) by using the fractional John von Neumann stability, as we did for the numerical scheme (21), in the following theorem.…”

Section: Stability Analysismentioning

confidence: 99%

“…However, all the above research papers have studied the one‐dimensional fractional Cable equation. Additionally, few researchers have considered the two‐dimensional fractional Cable equation 21–24 . In this paper, we consider the following initial‐boundary value problem of the two‐dimensional fractional Cable equation which is usually written in the following form:

u_{t} false(x, y, t false) = D_{t}^{1 - β} false[u_{x x} false(x, y, t false) + u_{y y} false(x, y, t false) false] - μ D_{t}^{1 - α} u false(x, y, t false), false(x, y false) \in normalΩ, 0 \leq t < T,

where

0 < β, α \leq 1, μ

is a constant and

D_{t}^{1 - γ}

is the fractional derivative defined by the Riemann‐Liouville operator of order

1 - γ

, where

γ = β, α

, with initial condition and boundary condition are given as follow:

u false(x, y, 0 false) = g false(x, y false), false(x, y false) \in normalΩ,

u false(x, y, t false) {false|}_{\partial normalΩ} = 0, 0 \leq t \leq T,

where

normalΩ = false{false(x, y<...>

…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, few researchers have considered the two-dimensional fractional Cable equation. [21][22][23][24] In this paper, we consider the following initial-boundary value problem of the two-dimensional fractional Cable equation which is usually written in the following form:…”

Section: Introductionmentioning

confidence: 99%

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A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

Sweilam

Ahmed

Adel

2020

Math Methods in App Sciences

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Recently, many numerical techniques were presented to solve the fractional anomalous sub‐diffusion equations, and the results were excellent. In this paper, we study a simple numerical technique to solve two important types of fractional anomalous sub‐diffusion equations that appear strongly in chemical reactions and spiny neuronal dendrites, which are the two‐dimensional fractional Cable equation and the two‐dimensional fractional reaction sub‐diffusion equation. The proposed technique is a simple one which is an extension of the weighted average finite difference technique. The stability analysis of the proposed method is studied by means of John von Neumann stability analysis technique. An accurate stability criterion which is valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced. Four numerical examples are presented (two for the Cable equation and two for the reaction sub‐diffusion equation) to demonstrate the effectiveness and the accuracy of the presented method.

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Section: Stability Analysismentioning

confidence: 99%

“…We are going to get the stability condition for the numerical scheme (21): Theorem 1. The WAFD scheme derived in (21) is stable under the following stability criterion…”

Section: Stability Analysismentioning

confidence: 99%

Section: Stability Analysismentioning

confidence: 99%

u_{t} false(x, y, t false) = D_{t}^{1 - β} false[u_{x x} false(x, y, t false) + u_{y y} false(x, y, t false) false] - μ D_{t}^{1 - α} u false(x, y, t false), false(x, y false) \in normalΩ, 0 \leq t < T,

where

0 < β, α \leq 1, μ

is a constant and

D_{t}^{1 - γ}

is the fractional derivative defined by the Riemann‐Liouville operator of order

1 - γ

, where

γ = β, α

, with initial condition and boundary condition are given as follow:

u false(x, y, 0 false) = g false(x, y false), false(x, y false) \in normalΩ,

u false(x, y, t false) {false|}_{\partial normalΩ} = 0, 0 \leq t \leq T,

where

normalΩ = false{false(x, y<...>

…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

Sweilam

Ahmed

Adel

2020

Math Methods in App Sciences

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The design of new high-order group iterative method in the solution of two-dimensional fractional cable equation

Khan

Ali

Hamid

2021

Alexandria Engineering Journal

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A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

Khan

Alias

Khan

et al. 2023

Sci Rep

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In this article, we developed a new higher-order implicit finite difference iterative scheme (FDIS) for the solution of the two dimension (2-D) time fractional Cable equation (FCE). In the new proposed FDIS, the time fractional and space derivatives are discretized using the Caputo fractional derivative and fourth-order implicit scheme, respectively. Moreover, the proposed scheme theoretical analysis (convergence and stability) is also discussed using the Fourier analysis method. Finally, some numerical test problems are presented to show the effectiveness of the proposed method.

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A comparative study of the point implicit schemes on solving the 2D time fractional cable equation

Cited by 9 publications

References 15 publications

A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

The design of new high-order group iterative method in the solution of two-dimensional fractional cable equation

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

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