Extended Algorithms for Approximating Variable Order Fractional Derivatives with Applications

Moghaddam, Behrouz Parsa; Machado, J. A. Tenreiro

doi:10.1007/s10915-016-0343-1

Cited by 72 publications

(40 citation statements)

References 43 publications

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“…Fractional integrals and derivatives have played an important role in analysing the behaviour of the physical phenomena through different domains in science and engineering. The major contributions can be seen in biology, bioengineering and viscoelasticity (for more information the readers can see []). Numerical integration is a tool used by scientists and engineers to obtain approximate values of integrals that can not be solved analytically.…”

Section: Numerical Approximation Of Fractional Integrals and Caputo Fmentioning

confidence: 99%

On algebraic trigonometric integro splines

Eddargani

Lamnii

2019

Z Angew Math Mech

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In this paper, we present a new kind of quadratic approximation operator reproducing of both algebraic and trigonometric functions. It is called integro quadratic splines interpolant, which agree with the given integral values of a univariate real-valued function over the same intervals, rather than the functional values at the knots. Efficient approximations of fractional integrals and fractional Caputo derivatives based on this interpolant, are constructed and well studied. The general approximation error is studied too, and the super convergence property is also derived when the interval is equally partitioned. Numerical examples illustrate that our method is very effective and our quadratic algebraic trigonometric integro spline has higher approximation ability than others. K E Y W O R D Salgebraic trigonometric splines, error bound, fractional caputo derivatives, fractional integrals, integro spline quasi-interpolant

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Section: Numerical Approximation Of Fractional Integrals and Caputo Fmentioning

confidence: 99%

On algebraic trigonometric integro splines

Eddargani

Lamnii

2019

Z Angew Math Mech

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“…In this example, we consider two cases, ̺(x) = 1.5 and ̺(x) = 1 + 0.5 |sin x|, x ∈ [0, π 2 ]. Table 4 displays a comparison between the M-and IM-algorithms [13,14] by means of the maximum AEs.…”

Section: Example 2 Consider the Following Vo-f Bagley-torvik Equationmentioning

confidence: 99%

New Recursive Approximations for Variable-Order Fractional Operators With Applications

Zaky

Doha

Taha

et al. 2018

Mathematical Modelling and Analysis

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To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.

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“…Zhuang et al [60] presented explicit and implicit finite-difference methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. Moghaddam and Machado [27] studied a finite-difference scheme for approximating variable-order fractional derivatives of arbitrary order, and introduced a new formulation of experimental convergence to compare the performance of algorithms. Tavares [37] presented a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order.…”

Section: Introductionmentioning

confidence: 99%

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Wei

Zhai

Zhang

2020

Commun. Appl. Math. Comput.

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The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L 2-convergence of the scheme are proved for all variable-order (t) ∈ (0, 1). The proposed method is of accuracyorder O(3 + h k+1) , where , h, and k are the temporal step size, the spatial step size, and the degree of piecewise P k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

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Extended Algorithms for Approximating Variable Order Fractional Derivatives with Applications

Cited by 72 publications

References 43 publications

On algebraic trigonometric integro splines

On algebraic trigonometric integro splines

New Recursive Approximations for Variable-Order Fractional Operators With Applications

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

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