On the convergence of a new reliable algorithm for solving multi-order fractional differential equations

Hesameddini, Esmail; Rahimi, Azam; Asadollahifard, Elham

doi:10.1016/j.cnsns.2015.10.020

Cited by 44 publications

(18 citation statements)

References 29 publications

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“…On the basis of (10), (11), (26), the item in equation can be converted into the matrix, we can deduce concretely them as follows:…”

Section: The Numerical Algorithmmentioning

confidence: 99%

“…, that is to say β 2 = β 3 -1 ∈ (0, 1), according to (10), (11), (26), (28), the item in equation can be converted into the matrix as follows:…”

Section: The Numerical Algorithmmentioning

confidence: 99%

“…Multiorder fractional differential equation [24] is one of the most important types of fractional differential equations. Authors of [25,26] investigated the existence, uniqueness, convergence of the solution for multi-order fractional differential equation. Because there is no exact solution, most different numerical methods, such as stable fractional Chebyshev differentiation matrix [27], fractional-order operational method [28], spectral collocation methods [29], and so on, have been used to investigate the approximate solutions of multiorder fractional differential equation.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis

Han¹,

Chen²,

Liu³

et al. 2018

Adv Differ Equ

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In this article, we investigate numerical solution of a class of multi-order fractional differential equations with error correction and convergence analysis. According to fractional differential definition in Caputo's sense, fractional differential operator matrix is deduced. The problem is reduced to a set of algebraic equations, and we apply MATLAB to solve the equation. In order to improve the precision of numerical solution, the process of error correction for multi-order fractional differential equation is introduced. By constructing the multi-order fractional differential equation of the error function, the approximate error function is obtained so that the numerical solution is corrected. Then, we analyze the convergence of the shifted Chebyshev polynomials approximation function. Numerical experiments are given to demonstrate the applicability of the method and the validity of error correction.

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“…On the basis of (10), (11), (26), the item in equation can be converted into the matrix, we can deduce concretely them as follows:…”

Section: The Numerical Algorithmmentioning

confidence: 99%

“…, that is to say β 2 = β 3 -1 ∈ (0, 1), according to (10), (11), (26), (28), the item in equation can be converted into the matrix as follows:…”

Section: The Numerical Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis

Han¹,

Chen²,

Liu³

et al. 2018

Adv Differ Equ

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show abstract

“…All the above mentioned works deal with fourth-order fractional diffusion-wave equations containing a single time-fractional derivative term. It is worth noting that multi-term fractional derivatives are used in visco-elastic damping [9], frequency-dependent loss and dispersion [19]. As far as numerical methods are concerned, Liu et al [19] reduced a multiterm fractional differential equation to a system with several single-term equations, employing then a fractional predictor and corrector method.…”

Section: Introductionmentioning

confidence: 99%

A Compact Difference Scheme for Fourth-Order Temporal Multi-Term Fractional Wave Equations and Maximum Error Estimates

Liu¹

2019

EAJAM

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A spatial compact difference scheme for a class of fourth-order temporal multi-term fractional wave equations is developed. The original problem is reduced to a lower order system and the corresponding time fractional derivatives are approximated by the L1-formula. The unconditional stability and convergence of the difference scheme are proved by the energy method. Numerical experiments support theoretical results.

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“…Many numerical solution techniques were developed to obtain analytical, semianalytical, and/or numerical solutions of fractional dynamical systems, for example, the finite difference method [8,9], predictor-corrector approach [10,11], operational matrix method [12,13], variational iteration method [14,15], homotopy perturbation method [16,17], Adomian's decomposition method [18,19], to mention a few. Due to the nonlocal character of the fractional derivative, storing the past responses requires a large amount of computer memory.…”

Section: Introductionmentioning

confidence: 99%

Solving Fractional Dynamical System with Freeplay by Combining Memory-Free Approach and Precise Integration Method

Liu

et al. 2016

Mathematical Problems in Engineering

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The Yuan-Agrawal (YA) memory-free approach is employed to study fractional dynamical systems with freeplay nonlinearities subjected to a harmonic excitation, by combining it with the precise integration method (PIM). By the YA method, the original equations are transformed into a set of first-order piecewise-linear ordinary differential equations (ODEs). These ODEs are further separated as three linear inhomogeneous subsystems, which are solved by PIM together with a predictor-corrector process. Numerical examples show that the results by the presented method agree well with the solutions obtained by the Runge-Kutta method and a modified fractional predictor-corrector algorithm. More importantly, the presented method has higher computational efficiency.

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On the convergence of a new reliable algorithm for solving multi-order fractional differential equations

Cited by 44 publications

References 29 publications

Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis

Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis

A Compact Difference Scheme for Fourth-Order Temporal Multi-Term Fractional Wave Equations and Maximum Error Estimates

Solving Fractional Dynamical System with Freeplay by Combining Memory-Free Approach and Precise Integration Method

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