An integro quadratic spline approach for a class of variable-order fractional initial value problems

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

doi:10.1016/j.chaos.2017.03.065

Cited by 45 publications

(13 citation statements)

References 25 publications

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“…[7,23,138]. Especially, more and more mathematical physical equations have been solved by using computationally efficient numerical methods [73,95,124].…”

Section: Numerical Methods For Vo-fdesmentioning

confidence: 99%

“…For instance, Bhrawy and Zaky [13] explored a high-order numerical scheme for soving the multidimensional VO fractional Schrödinger equations. Moghaddam et al [73] developed a technique for the approximate solution in regard to the VO fractional Bagley-Torvik and Basset differential equations in the area of fluid dynamics; meanwhile, the accuracy of the proposed algorithm was properly verified. Due to many parameters involved in the existing physiological models for bone remodeling, Neto et al [78] presented a new approach with VO derivative to simplify its structure and provide more compact models that lead to similar results.…”

Section: Vo Fractional Control Model Fractional Derivative Model Hasmentioning

confidence: 99%

See 1 more Smart Citation

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

261

138

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Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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“…[7,23,138]. Especially, more and more mathematical physical equations have been solved by using computationally efficient numerical methods [73,95,124].…”

Section: Numerical Methods For Vo-fdesmentioning

confidence: 99%

Section: Vo Fractional Control Model Fractional Derivative Model Hasmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

261

138

View full text Add to dashboard Cite

show abstract

“…Recently, some authors have considered the applications of derivatives of variable order in various sciences such as anomalous diffusion modeling, mechanical applications, multi-fractional Gaussian noises. Among these, there have been many works dealing with numerical methods for some class of variable order fractional differential equations, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

Zhang

2019

Mathematics

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The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

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“…Due to the variable-order fractional derivatives, a few papers deal with the existence of exact solutions to such equations (Zhang 2018). Recently, several numerical methods are proposed for solving variable-order fractional differential and integral equations based on the finite difference (Moghaddam and Machado 2016), B-linear spline (Machado and Moghaddam 2018), Cubic spline (Moghaddam and Machado 2017a, b;Yaghoobi et al 2017), and integro quadratic spline interpolations (Moghaddam et al 2017;Keshi et al 2018). Moghaddam and Machado developed finite-difference approach schemes for variableorder fractional operators, and applied the method to approximate variable-order fractional Ricatti and variable-order fractional Emden-Fowler equations (Moghaddam and Machado 2016).…”

Section: Introductionmentioning

confidence: 99%

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

Yan

Han

et al. 2019

Comp. Appl. Math.

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In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach's fixed-point theorem and the Gronwall-Bellman lemma. Third, we employ the Legendre-Gauss and Jacobi-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces L 2 [0, 1] and L ∞ [0, 1]. Finally, numerical results are given to support the theoretical conclusions. Keywords Spectral collocation method • Variable fractional order • Initial value problem • Convergence analysis Mathematics Subject Classification 65L60 • 41A05 • 41A10 • 41A25 Communicated by José Tenreiro Machado.

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An integro quadratic spline approach for a class of variable-order fractional initial value problems

Cited by 45 publications

References 25 publications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

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