Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

Salama, Fouad Mohammad; Ali, Umair; Ali, Ajmal

doi:10.1007/s40819-022-01408-z

Cited by 7 publications

(2 citation statements)

References 48 publications

(51 reference statements)

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“…In summary, explicit group methods have some salient advantages, such as stability, being easy to implement, forming sparse algebraic systems, accelerating the rate of convergence, reducing arithmetic computations per iteration, and extension to multi-dimensional problems. In [57][58][59][60][61][62], different types of explicit group methods were developed based on standard and skewed difference schemes for solving a variety of CO FPDEs.…”

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

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“…For further applications and analysis of explicit group methods, we refer the reader to Salama et al. ( 2022c ), Abdi et al. ( 2021b ), Ali et al.…”

Section: Introductionmentioning

confidence: 99%

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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An efficient high‐order two‐level explicit/implicit numerical scheme for two‐dimensional time fractional mobile/immobile advection‐dispersion model

Ngondiep

2024

Numerical Methods in Fluids

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This article constructs a new two‐level explicit/implicit numerical scheme in an approximate solution for the two‐dimensional time fractional mobile/immobile advection‐dispersion problem. The stability and error estimates of the proposed technique are deeply analyzed in the ‐norm. The developed approach is less time consuming, fourth‐order in space and temporal accurate of order , where is the time step and denotes a positive parameter less than 1. This result shows that the two‐level explicit/implicit formulation is faster and more efficient than a large class of numerical schemes widely discussed in the literature for the considered problem. Numerical experiments are performed to verify the theoretical studies and to demonstrate the efficiency of the new numerical method.

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Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

Cited by 7 publications

References 48 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

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

An efficient high‐order two‐level explicit/implicit numerical scheme for two‐dimensional time fractional mobile/immobile advection‐dispersion model

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