A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term

Mohebbi, Akbar; Abbaszadeh, Mostafa; Dehghan, Mehdi

doi:10.1016/j.jcp.2012.11.052

Cited by 105 publications

(64 citation statements)

References 27 publications

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“…The stability and convergence conditions of the difference Table 1: The comparison of the difference scheme (13) with difference scheme in [20] for h = 1/1000.…”

Section: Resultsmentioning

confidence: 99%

“…(α, β) h Finite difference scheme (13) Finite difference scheme in [20] e∞(τ, h) S-order CPU time (s) e∞(τ, h) S-order CPU time (s) (0.4,0.1) (14) with difference scheme in [20] for h = 1/500.…”

Section: Resultsmentioning

confidence: 99%

“…(α, β) h Finite difference scheme (14) Finite difference scheme in [20] e∞(τ, h) S-order CPU time (s) e∞(τ, h) S-order CPU time (s) (0.2,0.1) schemes are given by using the Fourier method. Finally, numerical experiments have been carried out to support the theoretical claims.…”

Section: Resultsmentioning

confidence: 99%

“…In particular the crossover between more and less anomalous behavior has been observed in the volatility of some share prices [28]. Here, we compare the numerical results of the finite difference schemes (13), (14) with those of the numerical scheme in [20]. The maximum-norm error, temporal and spatial convergence orders, and CPU time for these finite difference schemes are listed in Tables 1-4 for different α, β.…”

Section: Respectivelymentioning

confidence: 99%

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High‐order compact difference schemes for the modified anomalous subdiffusion equation

Ding

2015

Numerical Methods Partial

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In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for a Riemann-Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are O(τ 2 + h 6 ) and O(τ 2 + h 8 ), respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.

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“…The stability and convergence conditions of the difference Table 1: The comparison of the difference scheme (13) with difference scheme in [20] for h = 1/1000.…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Respectivelymentioning

confidence: 99%

See 2 more Smart Citations

High‐order compact difference schemes for the modified anomalous subdiffusion equation

Ding

2015

Numerical Methods Partial

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“…Diethelm et al [14] propose a predictor-corrector-type approach for the numerical solution of both linear and nonlinear fractional differential equations including systems of equations. Mohebbi et al [15] study a high-order accurate numerical method for obtaining the solution to the subdiffusion equation with a nonlinear source term. Cui [16] goes on to solve the two-dimensional time-fractional diffusion equation using a high-order compact alternating direction implicit method.…”

Section: Introductionmentioning

confidence: 99%

A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme

Jacobs

2015

Abstract and Applied Analysis

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A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.

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Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Izadkhah

Saberi‐Nadjafi

2014

Math Methods in App Sciences

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In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. IntroductionRecently, fractional differential operators are indisputably found to play a fundamental role in the modeling of a considerable number of phenomena. Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diffusion [1][2][3][4]. They have been used in modeling turbulent flow [5,6], chaotic dynamics of classical conservation systems [7], and even finance [8,9] (see [1] for more information).Also, fractional calculus emerged as an important and efficient tool for the study of dynamical systems where classical methods reveal strong limitations. In [10], a fractional advection-dispersion equation is derived by extending Fick's first law from isotropic media to heterogeneous media and is particularly suitable for description of the highly skewed and heavy-tailed dispersion processes observed in rivers and other natural media.In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11][12][13], finite element methods [14,15], and spectral methods [16]. In [17][18][19], the authors used their proposed numerical schemes to solve Bagley-Torvik equation and other ordinary fractional differential equations. Gegenbauer polynomials have received much attention for their fundamental properties as well as for their use in applied mathematics. The described functions are a key ingredient to the implementation of spectral and pseudo-spectral methods to solve certain types of differential equations. Gegenbauer polynomials are a convenient basis for polynomial approximations because they are eigenfunctions of corresponding differential operators. For numerical methods, it is usually as convenient as efficient to convert between representations of a polynomial by expansion coefficients or by function values, respectively.Spectral approximations, such as the Fourier approximation based upon trigonometric polynomials for periodic problems, and the Chebyshev, Legendre, or the general Gegenbauer approximation based upon polynomials for nonperiodic problems are exponentially accurate for analytic functions [20][21][22][23]. In [24], the authors provided collocation method for natural convection heat transfer equations embedded in porous medium by using the rational Gegenbauer polynomials. Gottlieb and Shu in [25] used the Geg...

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A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term

Cited by 105 publications

References 27 publications

High‐order compact difference schemes for the modified anomalous subdiffusion equation

High‐order compact difference schemes for the modified anomalous subdiffusion equation

A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

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