High-order central difference scheme for Caputo fractional derivative

Ying, Yuping; Lian, Yuan; Tang, Shaoqiang; Liu, Wing Kam

doi:10.1016/j.cma.2016.12.008

Cited by 26 publications

(4 citation statements)

References 19 publications

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“…FADEs with fractional derivative in time describe motion of particles with memory in time (Meerschaert and Tadjeran, 2004;Schumer et al, 2009).In recent years, much work has been done on the equation (3.1) when the fractional order derivative is considered as Caputo derivative. Many authors presented their approaches to approximate Caputo derivative with a high order of convergence (Aguilar and Hernández, 2014;Li and Zeng, 2015;Duan et al, 2016;Ying et al, 2017;Kumar et al, 2017Kumar et al, , 2018. Here, we use time fractional derivative as generalized Caputo derivative and then equation (3.1) becomes GFADE:…”

Section: Resultsmentioning

confidence: 99%

High-order approximation for generalized fractional derivative and its application

Yadav

Pandey

Shukla

et al. 2019

HFF

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Purpose This paper aims to present a high-order scheme to approximate generalized derivative of Caputo type for μ ∈ (0,1). The scheme is used to find the numerical solution of generalized fractional advection-diffusion equation define in terms of the generalized derivative. Design/methodology/approach The Taylor expansion and the finite difference method are used for achieving the high order of convergence which is numerically demonstrated. The stability of the scheme is proved with the help of Von Neumann analysis. Findings Generalization of fractional derivatives using scale function and weight function is useful in modeling of many complex phenomena occurring in particle transportation. The numerical scheme provided in this paper enlarges the possibility of solving such problems. Originality/value The Taylor expansion has not been used before for the approximation of generalized derivative. The order of convergence obtained in solving generalized fractional advection-diffusion equation using the proposed scheme is higher than that of the schemes introduced earlier.

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Section: Resultsmentioning

confidence: 99%

High-order approximation for generalized fractional derivative and its application

Yadav

Pandey

Shukla

et al. 2019

HFF

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“…In particular, there have been substantial developments in the theory and application of numerical methods for fractional partial differential equations. For example, from a theoretical point of view, theoretical analyses of conservative finitedifference schemes to solve the Riesz space-fractional Gross-Pitaevskii system have been proposed in the literature [4], along with convergent three-step numerical methods to solve double-fractional condensates, explicit dissipation-preserving methods for Riesz space-fractional nonlinear wave equations in multiple dimensions [5], energy conservative difference schemes for nonlinear fractional Schrödinger equations [6], conservative difference schemes for the Riesz space-fractional sine-Gordon equation [7], high-order central difference schemes for Caputo fractional derivatives [8], among other examples.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

2021

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In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

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“…The non-integer derivative helps in understanding the complete memory effect of the system. A broad literature of models with fractional derivatives can be found in [13][14][15][16][17]. Therefore, motivated by our ongoing research work into this special branch of mathematics (namely, fractional calculus), we study non-integer KGE by changing integer order derivative in both time and space using the Liouville-Caputo derivative of fractional order in the following manner:…”

Section: Introductionmentioning

confidence: 99%

An Efficient Computational Method for the Time-Space Fractional Klein-Gordon Equation

Singh¹,

Kumar

Pandey

2020

Front. Phys.

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In this paper, we present a computational method to solve the fractional Klein-Gordon equation (FKGE). The proposed technique is the grouping of orthogonal polynomial matrices and collocation method. The benefit of the computational method is that it reduces the FKGE into a system of algebraic equations which makes the problem straightforward and easy to solve. The main reason for using this technique is its high accuracy and low computational cost compared to other methods. The main solution behaviors of these equations are due to fractional orders, which are explained graphically. Numerical results obtained by the proposed computational method are also compared with the exact solution. The results obtained by the suggested technique reveals that the method is very useful for solving FKGE.

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High-order central difference scheme for Caputo fractional derivative

Cited by 26 publications

References 19 publications

High-order approximation for generalized fractional derivative and its application

High-order approximation for generalized fractional derivative and its application

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

An Efficient Computational Method for the Time-Space Fractional Klein-Gordon Equation

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