Application of numerical inverse Laplace transform algorithms in fractional calculus

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo-Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag-Leffler function; for the Caputo-Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.

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“…applying the numerical inverse Laplace transform algorithm [42] to (48), we obtain the time response.…”

Section: Damper-spring Systemmentioning

confidence: 99%

“…Applying the numerical inverse Laplace transform algorithm [42] to (55), we obtain the time response. The plots for different values of the fractional order γ are shown in Figure 11.…”

Section: Mass-spring-damper Systemmentioning

confidence: 99%

Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

Gómez-Aguilar

Yépez-Martínez

Calderón-Ramón

et al. 2015

Entropy

134

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“…It is observed that the ML function calculation is time consuming and may not give proper results in all the cases. In such cases they can also be plotted using invlap.m subroutine (numerical ILT) [22], [23]. in the given system function.…”

Section: Polesmentioning

confidence: 99%

Time and Frequency Domain Analysis of the Linear Fractional-order Systems

Bhole¹,

Patil²,

Vyawahare³

2012

SpecialIssue

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Abstract-Recent years have seen a tremendous upsurge inthe area related to the use of Fractional-order (FO) differential equations in modeling and control. FO differential equations are found to provide a more realistic, faithful, and compact representations of many real world, natural and manmade systems. FO controllers, on the other hand, have been able to achieve a better closed-loop performance and robustness, than their integer-order counterparts. In this paper, we provide a systematic and rigorous time and frequency domain analysis of linear FO systems. Various concepts like stability, step response, frequency response are discussed in detail for a variety of linear FO systems. We also give the state space representations for these systems and comment on the controllability and observability. The exercise presented here conveys the fact that the time and frequency domain analysis of FO linear systems are very similar to that of the integer-order linear systems.

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“…If both of the poles lie on the right of line l in the complex plane, we have lim t→∞ f i1 (t) = ∞, and then based on (3.3) and (3.4), for f i2 (t), we have It is difficult to plot the exact analytical solution of g 2 (t) from the inverse Laplace transform, so we utilize the numerical inverse Laplace transform [29,30] technique to obtain g 2 (t) in numerical examples.…”

Section: Case 1: α =mentioning

confidence: 99%

“…The analytical impulse response of a fractional second order filter with form s 2 + as + b −γ is derived in [29], and its impulse response invariant discretization is obtained accordingly. Motivated by [29], validated by a numerical inverse Laplace transform technique [30], the impulse response of the generalized fractional second order filter…”

Section: Introductionmentioning

confidence: 99%