Detailed Error Analysis for a Fractional Adams Method

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.

doi:10.1023/b:numa.0000027736.85078.be

Cited by 803 publications

(478 citation statements)

References 27 publications

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“…More recently, applications have included classes of nonlinear fractional differential equations and this motivates us to consider their effective numerical methods for the solution of nonlinear fractional differential equations. Diethelm et al have done a lot of excellent works on numerical methods for fractional order ordinary differential equations [4,5]. In this paper, we propose fractional high order numerical methods for the nonlinear fractional ordinary differential equation.…”

Section: Introductionmentioning

confidence: 99%

Fractional high order methods for the nonlinear fractional ordinary differential equation

Lin

Liu

2007

Nonlinear Analysis: Theory, Methods & Applications

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In this paper, we consider the nonlinear fractional order ordinary differential equa-Fractional order linear multiple step methods are introduced. The high order (2-6) approximations of the fractional order ordinary differential equation with initial value are proposed. The consistence, convergence and stability of the fractional high order methods are proved. Finally, some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods.

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Section: Introductionmentioning

confidence: 99%

Fractional high order methods for the nonlinear fractional ordinary differential equation

Lin

Liu

2007

Nonlinear Analysis: Theory, Methods & Applications

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“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”

Section: Correction Of the Lubich Fractional Multistep Methodsmentioning

confidence: 99%

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Ford

Yan

2017

FCAA

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In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

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“…The numerical method used for solving system (1) is described in [21]. The initial value is set as (0.1, 0.1, 0.1, 0.1) T − .…”

Section: Hyperchaotic Systemmentioning

confidence: 99%

A Color Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System

Huang

Sun

et al. 2014

Entropy

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Abstract:In this paper, a new color image encryption algorithm based on a fractional-order hyperchaotic system is proposed. Firstly, four chaotic sequences are generated by a fractional-order hyperchaotic system. The parameters of such a system, together with the initial value, are regarded as the secret keys and the plain image is encrypted by performing the XOR and shuffling operations simultaneously. The proposed encryption scheme is described in detail with security analyses, including correlation analysis, histogram analysis, differential attacks, and key sensitivity analysis. Experimental results show that the proposed encryption scheme has big key space, and high sensitivity to keys properties, and resists statistical analysis and differential attacks, so it has high security and is suitable for color image encryption.

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Detailed Error Analysis for a Fractional Adams Method

Cited by 803 publications

References 27 publications

Fractional high order methods for the nonlinear fractional ordinary differential equation

Fractional high order methods for the nonlinear fractional ordinary differential equation

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

A Color Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System

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