Spline collocation methods for linear multi-term fractional differential equations

Pedas, Arvet; Tamme, Enn

doi:10.1016/j.cam.2011.06.015

Cited by 65 publications

(21 citation statements)

References 12 publications

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“…Several numerical methods are used for fractional-order systems, such as generalizations of predictor-corrector methods [7,11,15], p-fractional linear multi-step methods [14,18] or the Adomian decomposition method [6,12,20]. These numerical schemes have a major drawback due to the non-locality of the fractional differential operators which reflects the hereditary nature of the problem: in order to obtain a reliable estimation of the solution, at every iteration step, all previous iterations have to be taken into account.…”

Section: Introductionmentioning

confidence: 99%

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

2018

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A parallel numerical simulation algorithm is presented for fractionalorder systems involving Caputo-type derivatives, based on the Adams-Bashforth-Moulton (ABM) predictor-corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractionalorder version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior.

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Section: Introductionmentioning

confidence: 99%

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

2018

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“…If at least one of the α i is irrational number, the method of solving equation (1.1) is quite different. For classical and distributional solutions in the case when irrational derivatives appear in (1.1), see for example, [1], [6], [7], [13], [17], [21], and [22]. Our analysis is motivated by the problem of linear fractionally damped oscillator, see, for example [4], and [24].…”

Section: Introductionmentioning

confidence: 99%

Cauchy problems for some classes of linear fractional differential equations

Atanacković

Dolićanin

Pilipović

et al. 2014

Fract Calc Appl Anal

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Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright's function for the case of equations of rational order, is presented. An example is treated in detail.MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

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“…The case where a = 0, that is, the case where (1)- (2) is an initial value problem, has been studied extensively (see for example [3,5,7,8,12,13,23] and the references therein). Here we are only concerned with the case where a = 0, that is, the case where (1)- (2) is a terminal (or boundary) value problem, and we will seek solutions of this problem over a finite interval [0, T ] where 0 < a < T .…”

Section: Introductionmentioning

confidence: 99%

High Order Numerical Methods for Fractional Terminal Value Problems

Ford

Morgado

Rebelo³

2014

Computational Methods in Applied Mathematics

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-In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method for the approximation of the solution to fractional initial value problems. Here we show that the method can be adapted for the effective approximation of the solution of terminal value problems. Moreover, we compare the efficiency of this numerical scheme against other existing methods.2010 Mathematical subject classification: Primary 65L05; secondary 34A08, 26Axx.

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Spline collocation methods for linear multi-term fractional differential equations

Cited by 65 publications

References 12 publications

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

Cauchy problems for some classes of linear fractional differential equations

High Order Numerical Methods for Fractional Terminal Value Problems

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