A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

Mardani, Arman; Hooshmandasl, Mohammad Reza; Heydari, Mohammad Hossein; Cattani, Carlo

doi:10.1016/j.camwa.2017.08.038

Cited by 59 publications

(22 citation statements)

References 40 publications

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“…Further research is needed in future. In addition to that, our future work will consider the applications of the present equivalent theory of the fractional oscillators to fractional noise in communication systems (Levy and Pinchas [84], Pinchas [85]), partial differential equations, such as transient phenomena of complex systems or fractional diffusion equations (Toma [86], Bakhoum and Toma [87], Cattani [88], Mardani et al [89]). …”

Section: Discussionmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

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Section: Discussionmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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“…Ding and Jiang considered the fractional Laplacian operator for analytical solutions of multiterm time space fractional advection-diffusion equation with mixed boundary condi-tions on a finite domain in [11]. Recently, researchers have used several numerical methods to solve approximate fractional advection-diffusion equation involving the Kansa method, the finite difference method, a moving least squares meshless, Laplace transform, Bernstein dual Petrov-Galerkin method, the finite volume method, and the local discontinuous Galerkin method [12][13][14][15][16][17][18][19]. Moreover, Cartaladea et al introduced an approximation for the fractional advection-diffusion equation according to lattice Boltzmann method by Bhatnagar-Gross-Krook or multiple-relaxation time collision operators [20].…”

Section: Introductionmentioning

confidence: 99%

A hybrid method for solving time fractional advection–diffusion equation on unbounded space domain

2020

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In this article, a hybrid method is developed for solving the time fractional advection–diffusion equation on an unbounded space domain. More precisely, the Chebyshev cardinal functions are used to approximate the solution of the problem over a bounded time domain, and the modified Legendre functions are utilized to approximate the solution on an unbounded space domain with vanishing boundary conditions. The presented method converts solving this equation into solving a system of algebraic equations by employing the fractional derivative matrix of the Chebyshev cardinal functions and the classical derivative matrix of the modified Legendre functions together with the collocation technique. The accuracy of the presented hybrid approach is investigated on some test problems.

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“…On other hand, the TFAD model given as

\partial_{t}^{α} u - l (x) \partial_{italicxx} u + m (x) \partial_{x} u = g (t, x),

describes many anomalous diffusion phenomena. It has been extensively used to model anomalous diffusion in transport processes through complex and disordered systems with/without fractal media [59]. Numerical solution of some TFAD models has been reported by FDM [46], RBFs‐based methods [39, 58, 60–62], MLS method [59], and Sinc‐Legendre collocation method [63].…”

Section: Introductionmentioning

confidence: 99%

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

Hussain¹,

Haq

2020

Numerical Methods Partial

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In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection-diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter () that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black-Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L ∞ , L 2 , L rms , and L rel error norms as well as number of nodes N over space domain and time-step t. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.

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A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

Cited by 59 publications

References 40 publications

Three Classes of Fractional Oscillators

Three Classes of Fractional Oscillators

A hybrid method for solving time fractional advection–diffusion equation on unbounded space domain

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

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