A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation

Pandit, Sapna; Mittal, R. C.

doi:10.1108/ec-01-2020-0013

Cited by 15 publications

(4 citation statements)

References 36 publications

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“…Nevertheless, the intrinsic complexity of fractional calculus, caused partially by non‐local properties of fractional derivatives and integrals, makes it rather difficult to find efficient numerical methods in this field. Despite this fact, however, the literature exhibits a growing interest and improving achievements in numerical methods for fractional calculus in general [5–14]. It seems enough to mention here that currently the numerical methods commonly used in fractional calculus included finite difference method and finite element method.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the intrinsic complexity of fractional calculus, caused partially by non-local properties of fractional derivatives and integrals, makes it rather difficult to find efficient numerical methods in this field. Despite this fact, however, the literature exhibits a growing interest and improving achievements in numerical methods for fractional calculus in general [5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

Liu

2023

Numerical Methods Partial

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A novel Hermite radial basis function-based differential quadrature (H-RBF-DQ) method is presented in this paper based on 2D variable order time fractional advection-diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF-DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

Liu

2023

Numerical Methods Partial

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“…The generalized complex Ginzburg-Landau (GCGL) model is considered in Hosseini et al (2021), and its 1-soliton solutions involving different wave structures are retrieved through a series of newly well-organized methods. The main aim of Pandit and Mittal (2021) is to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation. A stable and efficient numerical technique based on modified trigonometric cubic B-spline functions is proposed in Dhiman et al (2021) for solving the time-fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

Abbaszadeh,

Bagheri Salec,

Kamel Abd Al-Khafaji

2023

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PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

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“…These analytical schemes provide the excellent agreement with the exact solution of various PDEs than other approaches. Many researchers and scientist developed several numerical and analytical schemes such as He’s iteration scheme (Khan, 2021), fractional complex transform (Ain et al , 2020), differential transform method (Arikoglu and Ozkol, 2007), scale-3 Haar wavelets (Mittal and Pandit, 2018; Pandit and Mittal, 2020), Adomian decomposition method (Duan et al , 2012), Homotopy perturbation scheme (He et al , 2021), Modified Laplace variational iteration method (Nadeem et al , 2019), two-scale approach (Ain et al , 2021), finite difference approach (Li and Zeng, 2013), high-order finite element scheme (Jiang and Ma, 2011), sub-equation method (Zheng and Wen, 2013) and differential quadrature method (Verma and Jiwari, 2015) to investigate the solution of these PDEs.…”

Section: Introductionmentioning

confidence: 99%

A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative

Nadeem

Li²

2022

HFF

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Purpose This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order. Design/methodology/approach Mohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense. Findings The results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme. Research limitations/implications This paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution. Practical implications This study presents the consistency and authenticity of the graphical solution with the exact solutions. Social implications This study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions. Originality/value A new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations.

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A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation

Cited by 15 publications

References 36 publications

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative

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