On numerical techniques for solving the fractional logistic differential equation

Noupoue, Yves Yannick Yameni; Tandoğdu, Yücel; Awadalla, Muath

doi:10.1186/s13662-019-2055-y

Cited by 23 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…as a generalization of the factorial to the case of non-integer arguments. To avoid excessive numerical errors when evaluating the coefficients c i , they are typically computed in a recursive manner according to [39]…”

Section: Influence Of the Initialization Of Fde Modelsmentioning

confidence: 99%

“…In contrast to the case of integer-order models, the time responses of fractional systems significantly depend on the initialization of the pseudo state. This is shown exemplarily in this paper with the help of the Grünwald-Letnikov definition of fractional derivatives [23,32,34,39] to illustrate further that the Caputo initialization corresponds to the special case that an FDE model is initialized with an initial condition that also represents a perfectly constant, infinitely long history of the pseudo states in the past. Although this may hold (at least in good approximation) for the initialization of a dynamic system which is fully in rest, this is obviously not true when resetting integrators after a finitely long time interval.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

Rauh

Malti

2022

Acta Cybern

View full text Add to dashboard Cite

In recent years, fractional differential equations have received a significant increase in their use for modeling a wide range of engineering applications. In such cases, they are mostly employed to represent non-standard dynamics that involve long-term memory effects or to represent the dynamics of system models that are identified from measured frequency response data in which magnitude and phase variations are observed that could be captured either by low-order fractional models or high-order rational ones. Fractional models arise also when synthesizing CRONE (Commande Robuste d'Ordre Non Entier) and/or fractional PID controllers for rational or fractional systems. In all these applications, it is frequently required to transform the frequency domain representation into time domain. When doing so, it is necessary to carefully address the issue of the initialization of the pseudo state variables of the time domain system model. This issue is discussed in this article for the reinitialization of fractional integrators which arises among others when solving state estimation tasks for continuous-time systems with discrete-time measurements. To quantify the arising time-domain truncation errors due to integrator resets, a novel interval observer-based approach is presented and, finally, visualized for a simplified battery model.

show abstract

Section: Influence Of the Initialization Of Fde Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

Rauh

Malti

2022

Acta Cybern

View full text Add to dashboard Cite

show abstract

“…Zabidi et al [31] have proposed an Adams-type multistep method for the numerical solution of FODEs. In [32], the authors studied the numerical solution of FODEs using the Hadamard derivative and integral. In [33], the authors studied the numerical solution of fractional order Fredholm integrodiferential equations with the Atangana-Baleanu derivative.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

Kamran,

Subhan,

Shah

et al. 2023

Complexity

View full text Add to dashboard Cite

In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is considered. We study the exact and numerical solutions of the said problem with a fractal-fractional differential operator. The abovementioned operator is arising widely in the mathematical modeling of various physical and biological problems. In our scheme, first, the integrodifferential equation with the fractal-fractional differential operator is converted to an integrodifferential equation with the Riemann–Liouville differential operator. Additionally, the obtained integrodifferential equation is then converted to the equivalent integrodifferential equation involving the Caputo differential operator. Then, we convert the integrodifferential equation under the Caputo differential operator using the Laplace transform to an algebraic equation in the Laplace space. Finally, we convert the obtained solution from the Laplace space into the real domain. Moreover, we utilize two techniques which include analytic inversion and numerical inversion. For numerical inversion of the Laplace transforms, we have to evaluate five methods. Extensive results are presented. Furthermore, for numerical illustration of the abovementioned methods, we consider three problems to demonstrate our results.

show abstract

“…Stability analysis with existence and uniqueness conditions for the fractional logistic model in the sense of Caputo operator was thoroughly discussed by El-Sayed et al in [31]. Fractional order logistic equation was studied by Noupoue et al in [32] using numerical schemes called generalized Euler's method, power series expansion (Grunwald Letnikov) technique, and CF approach while existence and uniqueness for the nonlinear logistic equation were obtained via Hadamard fractional derivative and integral formulae. They proved that the minimum rate of error (3.2%) based upon the real data is obtained with the CF approach when the order of the differential equation is α = 1.005.…”

Section: Introductionmentioning

confidence: 99%

Fractional numerical dynamics for the logistic population growth model under Conformable Caputo: a case study with real observations

2021

View full text Add to dashboard Cite

Models of population dynamics are substantially non-Markovian in nature and exhibit behavior for memory effects. This research study investigates the logistic growth model from population dynamics under both classical and non-classical (fractional) differential operators using actual statistical data. For the non-classical differential operator, the operator called conformable fractional derivative having order β in the sense of Liouville-Caputo (LC) of order α is employed while taking care of its dimensional consistency. Working parameters including conformable fractional derivative order β, LC having order α, growth rate ζ, and carrying capacity Φ have been optimized, and the results are compared with those of classical one. Error rate suggests the superiority of the fractional conformable logistic model whose solutions are investigated for uniqueness via fixed point theory. Numerical simulations using Adams' iterative technique recently proposed for the conformable fractional derivative are illustrated in figures for a thorough understanding of the model's behavior, along with a statistical summary for the operators. An attractive chaotic attractor is observed in the fractional logistic model when ζ 4.023. Cumulative effects of growth rate and carrying capacity on the population's change in the logistic model are also investigated with 3D graphics.

show abstract

On numerical techniques for solving the fractional logistic differential equation

Cited by 23 publications

References 29 publications

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

Fractional numerical dynamics for the logistic population growth model under Conformable Caputo: a case study with real observations

Contact Info

Product

Resources

About