Derivatives Of Complex Order Research Articles

Global optimization of a nonlinear system of differential equations involving $$\psi $$-Hilfer fractional derivatives of complex order

Global optimization of a nonlinear system of differential equations involving $$\psi $$-Hilfer fractional derivatives of complex order

Restrictions in a distributed complex fractional order linear constitutive equations of viscoelasticity

We analyze a viscoelastic body in a linear stress state by the use of the distributed complex order fractional derivative in the constitutive equation. A model is formulated so that it takes into account all complex order derivatives of the form t0Dtα+iβ(α), α∈[α1,α2]⊂(0,1), β∈C1([α1,α2]). Using a new procedure, we derive restrictions on a model that follows from the Clausius–Duhem inequality under isothermal conditions. We apply our analysis to the creep test, as well as to generalized wave equation on R, with a specific constitutive equation of Kelvin–Voigt type.

State of charge estimation for lithium-ion batteries based on a novel complex-order model

The accuracy of the battery model is decisive in model-based state of charge (SOC) estimation. In this paper, complex-order derivatives (CDs) are applied in the scope of battery modeling, parameter identification, and SOC estimation. Firstly, a novel complex-order equivalent circuit model (Co-ECM) for lithium-ion batteries, which considers an innovative complex-order constant phase element, is proposed. Secondly, the structure characteristics of the Co-ECM are analyzed, and a complex-order particle swarm optimization algorithm is developed to identify the Co-ECM parameters. Finally, a novel complex-order unscented Kalman filter is designed to estimate the battery SOC, while CDs capture the system past behavior and tackle the nonlinearities of the constant phase element. Also, the proposed Co-ECM is compared with two other alternatives (i.e., integer-order and fractional-order ECM) based on data from two battery test cycles at different temperatures. The results show that the new Co-ECM leads to SOC estimation accuracy higher than the traditional models over a wide range of temperature (0∘C, 25∘C and 45∘C), with root-mean-squared error (RMSE) and mean absolute error (MAE) less than 0.47% and 0.41%, respectively. Moreover, experiments with data polluted with artificial noise revealed that the proposed model has superior robustness against noisy information. The new Co-ECM is, thus, shown to be a prime option for battery SOC estimation.

Moving point load on a beam with viscoelastic foundation containing fractional derivatives of complex order

Moving point load on a beam with viscoelastic foundation containing fractional derivatives of complex order

Spontaneous activation under atrial fibrosis: A model using complex order derivatives

The computational modeling of the cardiac electrophysiology allows assertive and quantitative study of the atrial fibrosis under fibrillation conditions. The cardiac electrical propagation is described by the so-called monodomain model, that consists of a nonlinear parabolic reaction-diffusion equation. Fibroblast proliferation, which is an essential component of the fibrotic process, can be modeled by considering the membrane ionic kinetics as a reactive component. However, such a mathematical description does not account the structural feature of fibroblasts. In this work, the electrophysiological properties of fibroblast proliferation and coupling with cardiomyocytes are investigated, using mathematical and computational modelling. The study is focused on the conditions under which spontaneous activations occur in a fibrotic domain. The proposed fibrosis model takes account the electrical and structural interactions of fibroblasts within the myocardium. The electrical component is described through an ionic kinetics formalism, while the structural component is obtained by means of a triplet of complex order derivatives that constructs the diffusion operator. A theoretical analysis determines the model parameters that generate unstable solutions, and numerical simulations illustrate and validate the analytical outcomes. The results evince a strong modulation of the stability conditions of the fibrotic model by the real and imaginary part of the fractional derivative order. The fibrosis structural complexity, controlled by the fractional order, determines the extent of the parameter space that generates spontaneous activation. Moreover, not all the unstable parameter configurations generate electrical propagation. In the cases of electrical conduction after spontaneous activation, the conduction velocity in the fibrotic domain is significantly slower than the one observed in healthy atrial tissue. The results give a new perspective for the development of atrial fibrosis models including the ectopic activity as an initiation factor for fibrillation activity. Indeed, the proposed design exploits the complex order fractional derivatives, to generate a wide set of electrophysiological scenarios.

A COMPLEX ORDER MODEL OF ATRIAL ELECTRICAL PROPAGATION FROM FRACTAL POROUS CELL MEMBRANE

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard diffusion mathematical equation, that considers the myocardium as a continuum. However, the modeling of structural properties and their influence on electrical propagation still reveal several limitations. In this paper, a model of cardiac electrical propagation is proposed based on complex order derivatives. By assuming that the myocardium has an underlying fractal process, the complex order dynamics emerges as an important modeling option. In this perspective, the real part of the order corresponds to the fractal dimension, while the imaginary part represents the log-periodic corrections of the fractal dimension. Indeed, the imaginary part in the derivative implies characteristic scales within the cardiac tissue. The analytical and numerical procedures for solving the related equation are presented. The sinus rhythm and persAF conditions are implemented using the Courtemanche formalism. The electrophysiological properties are measured and analyzed on different scales of observation. The results indicate that the complex order modulates the electrophysiology of the atrial system, through the variation of its real and imaginary parts. The combined effect of the two components yields a broad range of electrophysiological conditions. Therefore, the proposed model can be a useful tool for modeling electrical and structural properties during cardiac conduction.

Complex-order particle swarm optimization

In this paper, the generalization of the Particle Swarm Optimization (PSO) algorithm is proposed. The new algorithm involves the adoption of complex-order derivatives (CD). Since the CD produce complex-valued results, conjugate pairs of CD are considered for designing the Complex-Order PSO (CoPSO). First, an extensive sensitivity analysis is carried out for studying the influence of the control parameters on the performance of CoPSO. Then, a set of classical benchmark functions are tested to verify the performance of CoPSO. Both valued- and ranked-based methods are conducted to compare the performance of the algorithm on the whole test suite. The Friedman test is applied to determine the average ranking of the algorithms based on their performances. Additionally, the mean and the standard deviation of the best results are examined in each experiment. The results indicate that the CoPSO has outstanding performance in comparison with previous algorithms, including the standard PSO, the fractional order PSO and the linear and nonlinear decreasing inertia weight PSO. The experimental results indicate the feasibility and efficiency of the CoPSO.

Tuning of complex coefficient PI/PD/PID controllers for a universal plant structure

Controllers are tuned either analytically or through optimisation techniques for integer, fractional and fractional complex order plants with dead time by considering only the positive frequency information of the plant. This is applicable only for integer / fractional order plants containing real coefficients which have an even symmetrical magnitude and odd symmetrical phase behaviour in frequency responses. The frequency response for (i) integer / fractional order plants containing complex coefficients and (ii) fractional complex order plants containing real / complex coefficients have unsymmetrical magnitude and phase behaviour. Tuning of controllers for such plants by considering only its positive frequency response produces reduced stability margins and in turn deteriorates its time response. Hence, tuning of controllers for such plants requires both positive and negative frequency information which in turn demands complex coefficient controllers. To address the problem, complex coefficient PI/PD/PID controllers are proposed in this paper. Unified expressions are derived for complex coefficient PI/PD/PID controllers by considering both positive and negative frequency information of plant with complex coefficients and complex order derivatives plus dead time. Numerical simulations are performed for three different case studies with the proposed complex coefficient integer order controllers and results are compared with real coefficient PI/PD/PID controllers under various operating conditions.

Reflection properties of zeta related functions in terms of fractional derivatives

Abstract We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ζ(n, a) of integer first argument can be extended to the more general case of ζ(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ζ(s, a) with ℜ(s) > 2 is essentially the Weyl (s − 2)-derivative of ζ(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch’s transcendent.

Atrial Rotor Dynamics Under Complex Fractional Order Diffusion.

The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order γ = α + jβ. A set of variations for γ is tested. The real part α takes values ranging from 1.1 to 2 and the imaginary part β from 0 to 0.28. Under this scheme, the standard diffusion is recovered when α = 2 and β = 0. The effect of γ on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by γ, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.

Fractional Controller Tuning Expressions for a Universal Plant Structure

In general, researchers use numerous methods to tune controllers for the given generalized plant structure. This structure deals with integer/fractional order plant plus dead time with real coefficient. However, controller tuning methods for complex coefficient plus fractional complex order plant with dead time are not reported in the existing literature. In this letter, a unified fractional and fractional complex order controller expressions are derived for the proposed universal structure. This structure comprises complex coefficient plus fractional complex order derivatives with dead time. To demonstrate the controller tuning for the defined structure, two different case studies are simulated to meet the desired specifications.

Non-linear boundary value problems involving Caputo derivatives of complex fractional order

We study approximate solutions of CDtβy(t)=f(t,y(t)), separately, for β ∈ (0, 1) and β ∈ (1, 2) with different boundary data, where CDtβ is the Caputo fractional derivative of complex-order. For this purpose we use the expansion formula for such fractional derivatives and prove the existence and the uniqueness of approximate solutions under certain conditions and their convergence to the original solutions.

On a constitutive equation of heat conduction with fractional derivatives of complex order

We study the heat conduction with a general form of a constitutive equation containing fractional derivatives of real and complex order. Using the entropy inequality in a weak form, we derive sufficient conditions on the coefficients of a constitutive equation that guarantee that the second law of thermodynamics is satisfied. This equation, in special cases, reduces to known ones. Moreover, we present a solution of a temperature distribution problem in a semi-infinite rod with the proposed constitutive equation.

On a New Class of Constitutive Equations for Linear Viscoelastic Body

Abstract We study a viscoelastic body involving a constitutive equation with distributed order fractional derivatives of complex order. Using a dissipation inequality in a weak form, we derive a sufficient conditions on coefficients of a model that guarantee that the Second law of thermodynamics under isothermal conditions is satisfied. Several known constitutive equations follow from our model as special cases. As an application, a new constitutive equation is related to an equation of motion of a generalized linear oscillator.

Wave equation for generalized Zener model containing complex order fractional derivatives

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.

Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Two variational problems of finding the Euler---Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler---Lagrange equations is obtained. It is shown that the sequence of approximated Euler---Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.

Fractional complex-order model for HIV infection with drug resistance during therapy

We propose a fractional complex-order model for drug resistance in HIV infection. We consider three distinct growth rates for the CD4+ T helper cells. We simulate the model for different values of the fractional derivative of complex order Dα±jβ, where α,β ∈ R +, and for distinct growth rates. The fractional derivative of complex order is a generalization of the integer-order derivative where α = 1 and β = 0. The fractional complex-order system reveals rich dynamics and variation of the value of the complex-order derivative sheds new light on the modeling of the intracellular delay. Additionally, fractional patterns are characterized by time responses with faster transients and slower evolutions towards the steady state.

Effect of drug-resistance in a fractional complex-order model for HIV infection

We study a fractional complex-order (FO) model for drug-resistance in HIV infection during therapy. We simulate the model for different values of the fractional derivative of complex order (FD) Dα ±ȷβ, where α,β ∈ R+. The FD is a generalization of the integer order derivative where α = 1 and β = 0. The FO system reveals rich dynamics. The novelty of the paper is attributed to the dynamics of the model promoted by the variation of the complex-order derivative. The obtained interesting dynamics may point in other directions to model the intracellular delay.

Advanced fault diagnosis by means of complex order derivatives

Advanced fault diagnosis by means of complex order derivatives

Optimal Controllers with Complex Order Derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complex-order control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.