An Operator-Based Approach for the Construction of Closed-Form Solutions to Fractional Differential Equations

Navickas, Zenonas; Telksnys, Tadas; Timofejeva, Inga; Marcinkevicius, Romas; Ragulskis, Minvydas

doi:10.3846/mma.2018.040

Cited by 5 publications

(1 citation statement)

References 26 publications

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“…For the fundamental theory of fractional derivatives and equations containing them we refer the reader to [8,11,21,22], see also [25]. Since it is rarely possible to find the solution of a given fractional differential equation in a closed form [16,21], the analysis and development of numerical methods for fractional differential equations has become a very active area of research. In particular, a number of studies have used collocation based methods, see, for example, [9,14,15,18,27].…”

Section: Introductionmentioning

confidence: 99%

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

Soots

Lätt

Pedas

2023

Mathematical Modelling and Analysis

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A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.

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Section: Introductionmentioning

confidence: 99%

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

Soots

Lätt

Pedas

2023

Mathematical Modelling and Analysis

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Multiclass motor imagery classification with Riemannian geometry and temporal-spectral selection

Li,

Tan,

et al. 2024

Med Biol Eng Comput

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An Operator-Based Scheme for the Numerical Integration of FDEs

et al. 2021

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An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The approximate numerical solution is constructed by truncating the power series, and by changing the point of the expansion. The developed adaptive integration step selection strategy is based on the controlled error of approximation induced by the truncation. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

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An Operator-Based Approach for the Construction of Closed-Form Solutions to Fractional Differential Equations

Cited by 5 publications

References 26 publications

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

Multiclass motor imagery classification with Riemannian geometry and temporal-spectral selection

An Operator-Based Scheme for the Numerical Integration of FDEs

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