A unified difference-spectral method for time–space fractional diffusion equations

In this paper, we aim to analyze the complicated dynamical motion of a quarter-car suspension system with a sinusoidal road excitation force. First, we consider a new mathematical model in the form of fractional-order differential equations. In the proposed model, we apply the Caputo–Fabrizio fractional operator with exponential kernel. Then to solve the related equations, we suggest a quadratic numerical method and prove its stability and convergence. A deep investigation in the framework of time-domain response and phase-portrait shows that both the chaotic and nonchaotic behaviors of the considered system can be identified by the fractional-order mathematical model. Finally, we present a state-feedback controller and a chaos optimal control to overcome the system chaotic oscillations. Simulation results demonstrate the effectiveness of the proposed modeling and control strategies.

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“…Eventually, from Lemma 3.3 in [36] and applying the Grönwall inequality, we get F m+1 ≤ C 1 F 0 , in where C 1 is a general constant.…”

Section: Quadratic Numerical Methodsmentioning

confidence: 93%

On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control

Băleanu¹,

Sajjadi²,

Jajarmi³

et al. 2021

Adv Differ Equ

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“…In previous studies, the time‐space fractional diffusion equations are studied with the Dirichlet boundary conditions. Recently, Baeumer et al considered the following two‐dimensional fractional diffusion equation with the fractional reflecting boundary condition

{(|, \frac{\partial^{β - 1} u false(x, y, t false)}{\partial {false(- y false)}^{β - 1}})}_{y = 0^{+}} = 0, t > 0

where

\frac{\partial^{β - 1}}{\partial {false(- y false)}^{β - 1}}

is right‐sided Riemann‐Liouville fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…Pang et al derived a fourth order finite difference schemes for one-and two-dimensional time-space fractional sub-diffusion equations in Pang and Sun. 17 In previous studies, [10][11][12][13][14][15][16][17] the time-space fractional diffusion equations are studied with the Dirichlet boundary conditions. Recently, Baeumer et al considered the following two-dimensional fractional diffusion equation with the fractional reflecting boundary condition 18,19 −1 u(x, , t)…”

Section: Introductionmentioning

confidence: 99%

“…Hanert, proposed a flexible numerical scheme for the time‐space fractional diffusion equation, which can accommodate either low‐order finite‐difference and finite‐element discretizations or high‐order pseudo‐spectral discretizations in the space. Two finite difference schemes were developed for two types of time‐space fractional diffusion equations with the weighted and shifted Grünwald operator approximation to the space fractional derivative in Cao and Li., Hanert and Piret, derived a pseudo‐spectral scheme to solve time‐space fractional diffusion equation, which has exponential tempering in both time and space direction. In Arshad et al, a trapezoidal scheme was proposed to discretize time‐space fractional diffusion equation with second‐order accuracy in both space and time.…”

Section: Introductionmentioning

confidence: 99%

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Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions

Xie

Fang

2019

Math Methods in App Sciences

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In this paper, the finite difference scheme is developed for the time‐space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann‐Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ2−α+h2). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.

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“…This method performs better than other existing methods. Huang and Yang [14] combined the spectral Galerkin method in space and the fractional trapezoid method in time having the spectral accuracy in space.…”

Section: Introductionmentioning

confidence: 99%

Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations

Huang¹

2019

EAJAM

Self Cite

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Two second-order convolution quadrature methods for fractional nonlinear diffusion-wave equations with Caputo derivative in time and Riesz derivative in space are constructed. To improve the numerical stability, the fractional diffusion-wave equations are firstly transformed into equivalent partial integro-differential equations. Then, a second-order convolution quadrature is applied to approximate the Riemann-Liouville integral. This deduced convolution quadrature method can handle solutions with low regularity in time. In addition, another second-order convolution quadrature method based on a new second-order approximation for discretising the Riemann-Liouville integral at time t k−1/2 is constructed. This method reduces computational complexity if Crank-Nicolson technique is used. The stability and convergence of the methods are rigorously proved. Numerical experiments support the theoretical results.

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A unified difference-spectral method for time–space fractional diffusion equations

Cited by 23 publications

References 34 publications

On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control

On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control

Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions

Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations

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