High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Brzeziński, Dariusz W.; Ostalczyk, Piotr

doi:10.2478/bpasts-2014-0078

Cited by 10 publications

(8 citation statements)

References 19 publications

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“…The following approximations utilising the modified trapezoidal rule can be [57] (for different formulations cf. [66,67]). For the left sided derivatives we The obtained results are shown in Figs (2), (3) and (4), where the compariso sureẼ X against the anisotropy of non-locality, the order of derivative α, and l 15…”

Section: Examplementioning

confidence: 99%

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Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In this paper, a generalisation of previous author's formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction. NotationSection 2and finallywhere] denotes the interval of non-local interaction, and Γ is the Euler gamma functionNext, based on non-standard definition of motion by Eq. (4) the authors define the deformation gradient F α aswhere e a and E A are base vectors in coordinate systems {x a } and {X A }, respectively.

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Section: Examplementioning

confidence: 99%

“…The following approximations utilising the modified trapezoidal rule can b [57] (for different formulations cf. [66,67] where f (n) (t j ) denotes classical n-th derivative at t = t j .…”

Section: Examplementioning

confidence: 99%

Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…Classical Jacobi orthogonal polynomials are applied in many important scientific areas that include functions' approximation in collocation points method for solutions of ordinary differential equations known as Sturm-Liouville problem and lately -fractional order derivatives and integrals computations by applying Gauss-Jacobi Quadrature [2], [3].…”

Section: Introductionmentioning

confidence: 99%

Computation of Gauss-Jacobi Quadrature Nodes and Weights with Arbitrary Precision

Brzeziński

2018

Annals of Computer Science and Information Systems

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In the paper there are presented efficient and accurate methods of Gauss-Jacobi nodes and weights computation. They include an enhancement for standard iteration method for Jacobi polynomials zeros finding, weight function formula transformation for increased accuracy of fractional derivatives computation and arbitrary precision application for mitigation of double precision arithmetic flaws. The results of numerical experiments presented in the paper prove high accuracy and efficiency of developed methods for computation of quadratures' nodes and weights, decreased amount of required iterations for polynomials zeros finding and elimination of truncation errors during weights computation. Accuracy of computations depends on height of precision applied for it, which is limited only by accessible hardware.

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“…Double Exponential Quadrature (denoted as DE) [16], [17] on the other hand involves hyperbolic functions substitution in independent variable transformation in integrand and trapezoidal rule applied to the transformed integrand. Application of the methods enables to obtain high accuracy computations results of fractional order derivatives and integrals [6], [7].…”

Section: B Applied New Efficient High-accuracy Methods Of Integratiomentioning

confidence: 99%

Accuracy Evaluation of Classical Integer Order Based and Direct Non-integer Order Numerical Algorithms of Non-integer Order Derivatives and Integrals Computations

Brzeziński

Ostalczyk

2014

Annals of Computer Science and Information Systems

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Abstract-In this paper the authors evaluate in context of numerical calculations accuracy classical integer order and direct non-integer based order numerical algorithms of non-integer orders derivatives and integrals computations. Classical integer order based algorithm involves integer and fractional order differentiation and integration operators concatenation to obtain non-integer order. Riemann-Liouville and Caputo formulas are applied to obtain directly derivatives and integrals of non-integer orders. The following accuracy comparison analysis enables to answer the question, which algorithm of the two is burdened with lower computational error. The accuracy is estimated applying non-integer order derivatives and integrals computational formulas of some elementary functions available in the literature of the subject.

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High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Cited by 10 publications

References 19 publications

Fractional calculus for continuum mechanics – anisotropic non-locality

Fractional calculus for continuum mechanics – anisotropic non-locality

Computation of Gauss-Jacobi Quadrature Nodes and Weights with Arbitrary Precision

Accuracy Evaluation of Classical Integer Order Based and Direct Non-integer Order Numerical Algorithms of Non-integer Order Derivatives and Integrals Computations

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