Analysis and Implementation of Numerical Scheme for the Variable-Order Fractional Modified Sub-Diffusion Equation

Ali, Umair; Abdullah, Farah Aini; WANG, MIAO-KUN; Salama, Fouad Mohammad

doi:10.1142/s0218348x22402538

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…Consequently, VO FPDEs have attracted the attention of numerous researchers and scholars as accurate models for describing a large variety of phenomena in various branches of science and engineering, such as mechanics, viscoelasticity, anomalous diffusion, wave propagation, control theory, ecology, and many others. Similar to the CO FPDEs, it is difficult to solve VO FPDEs analytically, and numerical techniques are very often resorted to, for instance, see [26][27][28][29]. A profound discussion of the definitions, applications, and numerical simulations of the VO fractional operators can be seen in the insightful review papers [30,31].…”

Section: Introductionmentioning

confidence: 99%

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

Salama

2024

Fractal Fract

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In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.

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Section: Introductionmentioning

confidence: 99%

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

Salama

2024

Fractal Fract

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“…Mehandiratta et al [ 19 – 22 ] extended the method to TFDE on metric star graph. Ali et al [ 1 ] consider the variable-order fractional modified sub-diffusion equation. Moreover, other discretization of time-fractional derivative such as generalized Jacobi spectral method [ 4 , 18 ] has also been used for solving TFDE.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Bi-yun

Xiao

2023

Numer Algor

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In this paper, we focus on developing a high efficient algorithm for solving d -dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

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Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations.

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Analysis and Implementation of Numerical Scheme for the Variable-Order Fractional Modified Sub-Diffusion Equation

Cited by 5 publications

References 28 publications

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

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