A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

Khan, Muhammad Asim; Ali, Norhashidah Hj. Mohd.; Hamid, Nur Nadiah Abd

doi:10.1186/s13662-020-03061-6

Cited by 6 publications

(3 citation statements)

References 34 publications

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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

Fractal Fract

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“…( 2021 ), Khan et al. ( 2020 ). In this work, we focus on a general class of time FPDEs (i.e., TFADRE) that constitutes previously researched classes as a special case.…”

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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“…This is useful for long time simulations, especially when attempting to solve multi-dimensional fractional problems [33][34][35][36]. It is well known that explicit group methods can diminish the computational complexity and reduce the computational time of numerical algorithms effectively [37][38][39][40][41][42][43][44]. However, numerical approximations based on explicit group methods for fractional mobile/immobile equations are still at an early stage of development.…”

Section: Introductionmentioning

confidence: 99%

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

Salama

Ali

2022

Int. J. Appl. Comput. Math

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In this paper, we shall present the development of two explicit group schemes, namely, fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods for solving the time fractional mobile/immobile equation in two space dimensions. The presented methods are formulated based on two Crank-Nicolson (C-N) finite difference schemes established at two different grid spacings. The stability and convergence of order O(τ 2−α + h 2 ) are rigorously proven using Fourier analysis. Several numerical experiments are conducted to verify the efficiency of the proposed methods. Meanwhile, numerical results show that the FEG and MFEG algorithms are able to reduce the computational times and iterations effectively while preserving good accuracy in comparison to the C-N finite difference method.

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A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

Cited by 6 publications

References 34 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

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

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