Abstract In this contribution, fractionalâorder controllers of the type PD ÎŒ and PI λ are applied to a class of irrational transfer function models that appear in largeâscale systems, such âŠ
Abstract In this contribution, fractionalâorder controllers of the type PD ÎŒ and PI λ are applied to a class of irrational transfer function models that appear in largeâscale systems, such as networks of mechanical/electrical elements and distributed parameter systems. More precisely, by considering the fractionalâorder controller in the Laplace domain with , a stability analysis in the parameterâspace is presented. Furthermore, as a way to measure the controller robustness, the controller's fragility analysis using the parameterâspace is derived. Finally, several applications that demonstrate the utility of our results are included.
In this work, we present numerical calculations of the acoustic scattering properties of inclusions characterized by fractional order behavior. Fundamental quantities, such as the differential and the total scattering cross âŠ
In this work, we present numerical calculations of the acoustic scattering properties of inclusions characterized by fractional order behavior. Fundamental quantities, such as the differential and the total scattering cross sections, are calculated for a wide range of forcing frequencies, and their characteristics are analyzed in the perspective of remote sensing and material characterization applications. The numerical simulations show the occurrence of resonance frequencies related to the fractional order mismatch and suggest the ability of fractional inclusion to induce lensing effects similar to those observed in materials with a negative index of refraction. The tools developed in this framework offer a novel approach for modeling and predicting the transport of field quantities throughout complex inhomogeneous and highly scattering media.
This work presents a simple procedure for designing fractional PD <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ÎŒ</sup> controllers for a type of implicit operators, which have recently been studied to describe large-scale systems. The âŠ
This work presents a simple procedure for designing fractional PD <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ό</sup> controllers for a type of implicit operators, which have recently been studied to describe large-scale systems. The methodology developed proposes a geometrical approach that allows characterizing the parameter-space of the PD <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ό</sup> controller into stable and unstable regions. Several numerical examples illustrate the effectiveness of the proposed results.
Thermoacoustic oscillations have been one of the most exciting discoveries of the physics of fluids in the 19th century. Since its inception, scientists have formulated a comprehensive theoretical explanation of âŠ
Thermoacoustic oscillations have been one of the most exciting discoveries of the physics of fluids in the 19th century. Since its inception, scientists have formulated a comprehensive theoretical explanation of the basic phenomenon which has later found several practical applications to engineering devices. To date, all studies have concentrated on the thermoacoustics of fluid media where this fascinating mechanism was exclusively believed to exist. Our study shows theoretical and numerical evidence of the existence of thermoacoustic instabilities in solid media. Although the underlying physical mechanism exhibits some interesting similarities with its counterpart in fluids, the theoretical framework highlights relevant differences that have important implications on the ability to trigger and sustain the thermoacoustic response. This mechanism could pave the way to the development of highly robust and reliable solid-state thermoacoustic engines and refrigerators.
In this paper, we present a system identification (SI) procedure that enables building linear timeâdependent fractionalâorder differential equation (FDE) models able to accurately describe timeâdependent behavior of complex systems. The âŠ
In this paper, we present a system identification (SI) procedure that enables building linear timeâdependent fractionalâorder differential equation (FDE) models able to accurately describe timeâdependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the MittagâLeffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an errorâcost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shellâandâtube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractionalâorder differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.
This study explores the use of fractional differential equations to model the vibration of single (SDOF) and multiple degree of freedom (MDOF) discrete parameter systems. In particular, we explore methodologies âŠ
This study explores the use of fractional differential equations to model the vibration of single (SDOF) and multiple degree of freedom (MDOF) discrete parameter systems. In particular, we explore methodologies to simulate the dynamic response of discrete systems having non-uniform coefficients (that is, distribution of mass, damping, and stiffness) by using fractional order models. Transfer functions are used to convert a traditional integer order model into a fractional order model able to match, often times exactly, the dynamic response of the active degree in the initial integer order system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting differential models have both frequency-dependent and complex fractional order. The presented methodology is practically equivalent to a model order reduction technique that is able to match the response of non-uniform MDOF systems to a simple fractional single degree of freedom (F-SDOF) systems. The implications of this type of modeling approach will be discussed.
This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional âŠ
This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.
We explore the use of fractional continuum models to perform 2D tomographic imaging of interest for applications to structural damage detection. Fractional models allow a more flexible approach to field âŠ
We explore the use of fractional continuum models to perform 2D tomographic imaging of interest for applications to structural damage detection. Fractional models allow a more flexible approach to field transport simulations in inhomogeneous media and, under certain conditions, enable capturing physical phenomena that cannot be accounted for by integer order models. This study addresses the specific example of heat conduction in a two- dimensional inhomogeneous domain and the reconstruction of the internal parameters based on temperature boundary measurements. The field evolution is assumed to be governed by a fractional diffusion equation while the parameter identification problem is formulated in inverse form. The reconstruction is performed with respect to the characteristic parameters of the fractional model, with particular attention to the order of the derivative. Numerical results show that the inverse procedure correctly identifies the spatial location of the inhomogeneity and, to some extent, the order of the fractional model.
Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the âŠ
Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an implicit integro-differential operator even though the individual components themselves satisfy equations that use explicit operators of integer order. The networks considered here are infinite trees and ladders, and each is composed only of two types of integer-order components with potential-driven flows that are repeated ad infinitum. In special cases the equivalent operator for the system is a fractional-order derivative, but in general it is implicit and can only be expressed as a solution of an operator equation. These implicit operators, which are a generalization of fractional-order derivatives, play an important role in the analysis and modeling of complex systems.
This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional âŠ
This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.
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In this chapter, we consider the time-like evolution of sets of state variables, a subject often called dynamical systems. We begin with a brief consideration of discrete dynamical systems known âŠ
In this chapter, we consider the time-like evolution of sets of state variables, a subject often called dynamical systems. We begin with a brief consideration of discrete dynamical systems known as iterated maps, which are a nonlinear extension of the difference equations posed in Section 4.9 or the finite difference method posed in Example 4.24. We also show some of the striking geometries that result from iterated maps known as fractals. Generally, however, we are concerned with systems that can be described by sets of ordinary differential equations, both linear and nonlinear. In that solution to nonlinear differential equations is usually done in discrete form, there is a strong connection to iterated maps. And we shall see that discrete and continuous dynamical systems share many features.
Our overarching aim in writing this book is to build a bridge to enable engineers to better traverse the domains of the mathematical and physical worlds. Our focus is on âŠ
Our overarching aim in writing this book is to build a bridge to enable engineers to better traverse the domains of the mathematical and physical worlds. Our focus is on neither the nuances of pure mathematics nor the phenomenology of physical devices but instead is on the mathematical tools used today in many engineering environments. We often compromise strict formalism for the sake of efficient exposition of mathematical tools. Whereas some results are fully derived, others are simply asserted, especially when detailed proofs would significantly lengthen the presentation. Thus, the book emphasizes method and technique over rigor and completeness; readers who require more of the latter can and should turn to many of the foundational works cited in the extensive bibliography.
Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced âŠ
Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the networkâs structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.
Particle motion in an unsteady peristaltic fluid flow is analyzed. The fluid is incompressible and Newtonian in a two-dimensional planar geometry. A perturbation method based on a small ratio of âŠ
Particle motion in an unsteady peristaltic fluid flow is analyzed. The fluid is incompressible and Newtonian in a two-dimensional planar geometry. A perturbation method based on a small ratio of wave height to wavelength is used to obtain a closed-form solution for the fluid velocity field. This analytical solution is used in conjunction with an equation of motion for a small rigid sphere in nonuniform flow taking Stokes drag, virtual mass, Fax\'en, Basset, and gravity forces into account. Fluid streamlines and velocity profiles are calculated. Theoretical values for pumping rates are compared with available experimental data. An application to ureteral peristaltic flow is considered since fluid flow in the ureter is sometimes accompanied by particles such as stones or bacteriuria. Particle trajectories for parameters that correspond to calcium oxalates for calculosis and Escherichia coli type for bacteria are analyzed. The findings show that retrograde or reflux motion of the particles is possible and bacterial transport can occur in the upper urinary tract when there is a partial occlusion of the wave. Dilute particle mixing is also investigated, and it is found that some of the particles participate in the formation of a recirculating bolus, and some of them are delayed in transit and eventually reach the walls. This can explain the failure of clearing residuals from the upper urinary tract calculi after successful extracorporeal shock wave lithotripsy. The results may also be relevant to the transport of other physiological fluids and industrial applications in which peristaltic pumping is used.
The fluid mechanics and operating characteristics of bearings that support rotating surfaces in micromachines are different from their larger cousins. The present study analyzes microbearings represented as an eccentric cylinder âŠ
The fluid mechanics and operating characteristics of bearings that support rotating surfaces in micromachines are different from their larger cousins. The present study analyzes microbearings represented as an eccentric cylinder rotating in a stationary housing. The flow Reynolds number is assumed small, the clearance between shaft and housing is not small relative to the overall bearing dimensions, and there is slip at the walls due to non-continuum effects. The two-dimensional governing equations are written in terms of the streamfunction in bipolar coordinates and an infinite-series solution is obtained. For high values of the eccentricity and low slip factors the flow may develop a recirculation region. The force and torque on the load-bearing inner cylinder increase with increasing eccentricity and decrease with increasing slip.
Technical Briefs On the Steady-State Velocity of the Inclined Toroidal Thermosyphon M. Sen, M. Sen Departamento de Fluidos y Termica, UNAM, 04510 Mexico, D.F., Mexico; Departamento de Energia Solar, Instituto âŠ
Technical Briefs On the Steady-State Velocity of the Inclined Toroidal Thermosyphon M. Sen, M. Sen Departamento de Fluidos y Termica, UNAM, 04510 Mexico, D.F., Mexico; Departamento de Energia Solar, Instituto de Investigaciones en Materials, UNAM, 04510 MeÂŽxico, D.F., Mexico Search for other works by this author on: This Site PubMed Google Scholar E. Ramos, E. Ramos Departamento de Energia Solar, Instituto de Investigaciones en Materials, UNAM, 04510 MeÂŽxico, D.F., Mexico Search for other works by this author on: This Site PubMed Google Scholar C. TrevinËo C. TrevinËo Departamento de Fluidos y Termica, UNAM, 04510 Mexico, D.F., Mexico Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information M. Sen Departamento de Fluidos y Termica, UNAM, 04510 Mexico, D.F., Mexico; Departamento de Energia Solar, Instituto de Investigaciones en Materials, UNAM, 04510 MeÂŽxico, D.F., Mexico E. Ramos Departamento de Energia Solar, Instituto de Investigaciones en Materials, UNAM, 04510 MeÂŽxico, D.F., Mexico C. TrevinËo Departamento de Fluidos y Termica, UNAM, 04510 Mexico, D.F., Mexico J. Heat Transfer. Nov 1985, 107(4): 974-977 (4 pages) https://doi.org/10.1115/1.3247533 Published Online: November 1, 1985 Article history Received: June 13, 1984 Online: October 20, 2009
This work investigates the effect a fractional derivative may have on the spectrum of relaxation modes of a viscoelastic material. It is shown that the order of the fractional derivative âŠ
This work investigates the effect a fractional derivative may have on the spectrum of relaxation modes of a viscoelastic material. It is shown that the order of the fractional derivative results in a modification to the constitutive relationships that exist within the Rouse model for viscoelasticity. These relationships that are used in engineering analyses have been previously developed from an empirical standpoint. The resulting modification to these constitutive relationships further supports the inclusion of fractional calculus in models of viscoelastic materials and hence increase their level of confidence associated with their usage.
Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced âŠ
Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the networkâs structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.
Generalized constitutive relationships for viscoelastic materials are suggested in which the customary time derivatives of integer order are replaced by derivatives of fractional order. To this point, the justification for âŠ
Generalized constitutive relationships for viscoelastic materials are suggested in which the customary time derivatives of integer order are replaced by derivatives of fractional order. To this point, the justification for such models has resided in the fact that they are effective in describing the behavior of real materials. In this work, the fractional derivative is shown to arise naturally in the description of certain motions of a Newtonian fluid. We claim this provides some justification for the use of ad hoc relationships which include the fractional derivative. An application of such a constitutive relationship to the prediction of the transient response of a frequency-dependent material is included.
The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100MHz. To describe this power law behavior in soft tissue, a hierarchical âŠ
The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the KramersâKronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.
A microscopic fractional calculus theory of viscoelasticity is developed on the basis of lattice dynamics by generalizing the standard model of a chain of coupled simple harmonic oscillators to a âŠ
A microscopic fractional calculus theory of viscoelasticity is developed on the basis of lattice dynamics by generalizing the standard model of a chain of coupled simple harmonic oscillators to a chain of coupled fractional oscillators by generalizing the integral equations of motion of a chain of simple harmonic oscillators into ones involving fractional integrals. This set of integral equations of motion pertaining to the chain of coupled fractional oscillators is solved by using Laplace transforms. In the continuum limit the time-fractional diffusion-wave equation in one dimension is obtained. The response of the system to the sinusoidal forcing in this limit consists of a transient part and an attenuated steady part. Expressions for the absorption coefficient and the specific dissipation function are derived and numerical applications are discussed.
This paper discusses the fractional oscillator equation involving fractional time derivatives of the Riemann-Liouville type. The exact solution of the fractional oscillator equation was obtained. On this basis the correspondence âŠ
This paper discusses the fractional oscillator equation involving fractional time derivatives of the Riemann-Liouville type. The exact solution of the fractional oscillator equation was obtained. On this basis the correspondence between the fractional time derivative and the dissipative properties was established. The relationship between the order of the fractional time derivative and the dissipative constant was derived. In addition to the exact solution, the perturbation approach was developed as well. It was shown that the perturbation method and exact solution lead to similar results. On the basis of the exact solution, the concept of fractional time evolution was illustrated.
We investigate a generalized tomographic imaging framework applicable to a class of inhomogeneous media characterized by non-local diffusive energy transport. Under these conditions, the transport mechanism is well described by âŠ
We investigate a generalized tomographic imaging framework applicable to a class of inhomogeneous media characterized by non-local diffusive energy transport. Under these conditions, the transport mechanism is well described by fractional-order continuum models capable of capturing anomalous diffusion that would otherwise remain undetected when using traditional integer-order models. Although the underlying idea of the proposed framework is applicable to any transport mechanism, the case of fractional heat conduction is presented as a specific example to illustrate the methodology. By using numerical simulations, we show how complex inhomogeneous media involving non-local transport can be successfully imaged if fractional order models are used. In particular, results will show that by properly recognizing and accounting for the fractional character of the host medium not only allows achieving increased resolution but, in case of strong and spatially distributed non-locality, it represents the only viable approach to achieve a successful reconstruction.
We consider the fractional oscillator being a generalization of the conventional linear oscillator in the framework of fractional calculus. It is interpreted as an ensemble average of ordinary harmonic oscillators âŠ
We consider the fractional oscillator being a generalization of the conventional linear oscillator in the framework of fractional calculus. It is interpreted as an ensemble average of ordinary harmonic oscillators governed by stochastic time arrow. The intrinsic absorption of the fractional oscillator results from the full contribution of the harmonic oscillators' ensemble: these oscillators differs a little from each other in frequency so that each response is compensated by an antiphase response of another harmonic oscillator. This allows to draw a parallel in the dispersion analysis for the media described by the fractional oscillator and the ensemble of ordinary harmonic oscillators with damping. The features of this analysis are discussed.
In many important engineering systems, including cyber-physical systems, there are often are many interacting simple components. It is well known that, in some cases, such systems may be modeled with âŠ
In many important engineering systems, including cyber-physical systems, there are often are many interacting simple components. It is well known that, in some cases, such systems may be modeled with fractional-order differential equations, and some of our prior work has shown that in certain cases fractional systems may be considered a subset of a class of systems where the dynamics may only be implicitly defined. In the implicit case, there is no straight-forward corresponding time domain representation. This paper considers series expansion approximations to the implicit operator describing the dynamics of the system to determine time-domain approximations for the implicit non-rational transfer function and validates the results via simulation. The contribution of this paper is to show that such expansions provide good approximations for the types of systems in the range of parameter values we considered, and for the special case of a network of springs and dampers, accurate equivalent spring and damper constants are computed which can provide intuitive insight into the nature of the response of the very high-order system.
We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative âŠ
We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in (0,2] and skewness theta, and the first-order time derivative with a Caputo derivative of order beta in (0,2]. The fundamental solution is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. By using the Mellin transform, we provide a general representation of the solution in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation known for the standard diffusion equation to suitable ranges of the relevant parameters alpha and beta. We derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the corresponding spatial probability densities.
We introduce and investigate a framework for constructing algorithms to invert Laplace transforms numerically. Given a Laplace transform \hat{f} of a complex-valued function of a nonnegative real-variable, f, the function âŠ
We introduce and investigate a framework for constructing algorithms to invert Laplace transforms numerically. Given a Laplace transform \hat{f} of a complex-valued function of a nonnegative real-variable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) \approx f_n (t) \equiv \frac{1}{t} \sum_{k = 0}^{n}\omega_{k}\hat{f}\biggl(\frac{\alpha_{k}}{t}\biggr),\quad 0 < t < \infty, where the weights Ï k and nodes α k are complex numbers, which depend on n, but do not depend on the transform \hat{f} or the time argument t. Many different algorithms can be put into this framework, because it remains to specify the weights and nodes. We examine three one-dimensional inversion routines in this framework: the Gaver-Stehfest algorithm, a version of the Fourier-series method with Euler summation, and a version of the Talbot algorithm, which is based on deforming the contour in the Bromwich inversion integral. We show that these three building blocks can be combined to produce different algorithms for numerically inverting two-dimensional Laplace transforms, again all depending on the single parameter n. We show that it can be advantageous to use different one-dimensional algorithms in the inner and outer loops.
We present a new approach to the statistical study and modelling of number counts of faint point sources in astronomical images, i.e. counts of sources whose flux falls below the âŠ
We present a new approach to the statistical study and modelling of number counts of faint point sources in astronomical images, i.e. counts of sources whose flux falls below the detection limit of a survey. The approach is based on the theory of α-stable distributions. We show that the non-Gaussian distribution of the intensity fluctuations produced by a generic point source population â whose number counts follow a simple power law â belongs to the α-stable family of distributions. Even if source counts do not follow a simple power law, we show that the α-stable model is still useful in many astrophysical scenarios. With the α-stable model it is possible to totally describe the non-Gaussian distribution with a few parameters which are closely related to the parameters describing the source counts, instead of an infinite number of moments. Using statistical tools available in the signal processing literature, we show how to estimate these parameters in an easy and fast way. We demonstrate that the model proves valid when applied to realistic point source number counts at microwave frequencies. In the case of point extragalactic sources observed at CMB frecuencies, our technique is able to successfully fit the distribution of deflections and to precisely determine the main parameters which describe the number counts. In the case of the Planck mission, the relative errors on these parameters are small either at low and at high frequencies. We provide a way to deal with the presence of Gaussian noise in the data using the empirical characteristic function of the . The formalism and methods here presented can be very useful also for experiments in other frequency ranges, e.g. X-ray or radio Astronomy.
The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the âŠ
The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reducing the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.
Recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena, such as stress-strain relationships in polymeric materials. In the rheological constitutive equations this implies the âŠ
Recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena, such as stress-strain relationships in polymeric materials. In the rheological constitutive equations this implies the replacement of the first-order derivatives by fractional derivatives. Here we show that such procedures have hierarchical mechanical analogues. We focus on the generalized dashpot and the generalized Maxwell model and display the corresponding arrangements. Our models allow a transparent interpretation of the parameters which enter the fractional equations, and reveal that the internal dynamics are hierarchically constrained.
Diffusion has been widely used to describe a random walk of particles or waves, and it requires only one parameter -- the diffusion constant. For waves, however, diffusion is an âŠ
Diffusion has been widely used to describe a random walk of particles or waves, and it requires only one parameter -- the diffusion constant. For waves, however, diffusion is an approximation that disregards the possibility of interference. Anderson localization, which manifests itself through a vanishing diffusion coefficient in an infinite system, originates from constructive interference of waves traveling in loop trajectories -- pairs of time-reversed paths returning to the same point. In an open system of finite size, the return probability through such paths is reduced, particularly near the boundary where waves may escape. Based on this argument, the self-consistent theory of localization and the supersymmetric field theory predict that the diffusion coefficient varies spatially inside the system. A direct experimental observation of this effect is a challenge because it requires monitoring wave transport inside the system. Here, we fabricate two-dimensional photonic random media and probe position-dependent diffusion inside the sample from the third dimension. By varying the geometry of the system or the dissipation which also limits the size of loop trajectories, we are able to control the renormalization of the diffusion coefficient. This work shows the possibility of manipulating diffusion via the interplay of localization and dissipation.
This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in âŠ
This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in complex media. Analytical expressions that describe power law attenuation and anomalous dispersion in each direction are derived for these fractional calculus models.
The exact stability condition for certain class of fractionalâorder (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the âŠ
The exact stability condition for certain class of fractionalâorder (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractionalâorder transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.
This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.
This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.