Approximation methods for solving fractional equations

Zeid, Samaneh Soradi

doi:10.1016/j.chaos.2019.05.008

Cited by 31 publications

(13 citation statements)

References 321 publications

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“…For some recent review of this subject, readers may consider the works of Zeid [90], Guo [30], Weilbeer [83] and Podlubny [64]. Approaches in the literature can be categorized by their method of integration, either cumulative or recursive.…”

Section: Current Methods For Numerical Approximation Of the Caputo Dementioning

confidence: 99%

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Zhang

Capilnasiu

Sommer

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelasticity, current biomechanical models in the computation community often assume only hyperelasticity. Fractional viscoelastic constitutive models have been successfully used in literature to capture the material response. However, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is O(N 2 T ) and a storage cost that is O(N T ) (N T denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is O(N ) and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and the error estimate is presented to demonstrate efficacy of the method. The method is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem. Fractional Derivatives and their Approximations Caputo fractional derivativeThe concept of fractional calculus started with questions about about the generalization of integral and differential operators by L'Hospital and Leibniz [68] from the set of integers to the set of real numbers.Subsequently, many prominent mathematicians focused on fractional calculus (for reviews, see, Ross [68] and Machado [51]). Within the field, many different definitions for fractional differential and integral operators of arbitrary order have been introduced [64]. In this work, we focus on the Caputo definition where the fractional derivative, D α t (with α > 0), of a n-times differentiable function f can be written as,

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Section: Current Methods For Numerical Approximation Of the Caputo Dementioning

confidence: 99%

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Zhang

Capilnasiu

Sommer

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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“…To address this challenge, recent studies suggest the use of radial basis functions [ 14 ] and the spectral tau algorithm [ 15 ]. More details about practical approximation techniques for solving FDEs and FOCPs can be found in the fresh review articles in [ 16 , 17 ], respectively.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

Chen

Soradi-Zeid

Jahanshahi

et al. 2020

Entropy

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A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm’s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

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“…To achieve this high level of accuracy and better analysis, various kinds of techniques including, analytical, numerical, soliton and wavelet have been adopted in the past [12][13][14][15][16][17][18][19][20][21]. Literature survey proved that there are many lacks to obtain the better approximation and analysis of fractional ordered problems arising in applied sciences.…”

Section: Introductionmentioning

confidence: 99%

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

et al. 2020

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The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

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Approximation methods for solving fractional equations

Cited by 31 publications

References 321 publications

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

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