Singular Memory Kernels Research Articles

Nonlinear acoustic equations of fractional higher order at the singular limit

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform a singular limit analysis and determine their behavior as the relaxation time tends to zero. We show that, depending on the nonlinearities and assumptions on the data, these models can be seen as approximations of the Westervelt, Blackstock, or Kuznetsov wave equations in nonlinear acoustics. We furthermore establish the convergence rates and thus determine the error one makes when exchanging local and nonlocal models. The analysis rests upon the uniform bounds for the solutions of the acoustic equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernel.

The vanishing relaxation time behavior of multi-term nonlocal Jordan–Moore–Gibson–Thompson equations

The family of Jordan–Moore–Gibson–Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.

Retraction Note to: Singular vs. non-singular memorised relaxation for basic relaxation current of the capacitor

We express the basic relaxation current of capacitor dynamics by using convolution operation of memory kernal function (that we choose as singular and non-singular) with rate of change of applied voltage. We have studied singular and non-singular types of memory kernels in this convolution expression. With these, we form constitutive equations for capacitor dynamics. We conclude that, mathematically, we can use the non-singular kernel now although this does not give much useful practically or physically realisable results and interpretations. It may be that we are unable to interpret these constitutive expressions of capacitor relaxation with non-singular memory kernel. Therefore, we have a question: Do natural relaxation dynamics for dielectrics have a singular memory kernel and is the relaxation current function singular in nature? Is the singular relaxation function for capacitor dynamics with singular memory kernel remains the universal law for dielectric relaxation? However, we are not questioning the researchers modelling the relaxation of dielectric via non-singular functions, but we are hinting about the complexity of basic constituent equation of the capacitor dynamics thus obtained by considering non-singular relaxations.

Memorized relaxation with singular and non-singular memory kernels for basic relaxation of dielectric vis-à-vis Curie-von Schweidler & Kohlrausch relaxation laws

We have constructed the basic dielectric relaxations evolution expression for relaxing current as convolution operation of the chosen memory kernel and rate of change of applied voltage. We have studied types of memory kernels singular, non-singular and combination of singular and non-singular (mixed) decaying functions. With these, we form constitutive equations for relaxation dynamics of dielectrics; i.e. capacitor. We observe that though mathematically we can use non-singular kernels yet this does not give presently much useful practical or physically realizable results and interpretations. We relate our observations to relaxation currents given via Curie-von Schweidler and Kohlraush laws. The Curie-von Schweidler law gives singular function power law as basic relaxation current in dielectric relaxation; whereas the Kohlraush law is Electric field relaxation in dielectric as non-Debye function taken as stretched exponential i.e. non-singular function. These two laws are used since late nineteenth century for various dielectric relaxation and characterization studies. Here we arrive at general constitutive equation for capacitor and with each type of memory kernel we give corresponding impedance function and in some cases equivalent circuit representation for the capacitor element. We classify these systems as Curie-von Shweidler type for system with singular memory kernel function and Kohlraush type for system evolved via using non-singular function or mixed functions as memory kernel. We note that use of singular memory kernel gives constituent relations and impedance functions that are experimentally verified in large number of cases of dielectric studies. Therefore, we have a question, does natural relaxation dynamics for dielectrics have a singular memory kernel, and the relaxation current function is singular in nature? Is it the singular relaxation function for capacitor dynamics with singular memory kernel remains universal law for dielectric relaxation? However, we are not questioning researchers modeling relaxation of dielectric via non-singular functions, yet we are hinting about complexity and lack of interpretability of basic constituent equation of dielectric relaxation dynamics thus obtained via considering non-singular and mixed memory kernels; perhaps due to insufficient experimental evidences presently. However, the method employed in this study is general method. This method can be used to form memorized constituent equations for other systems (say Radioactive Decay/Growth, Diffusion and Wave phenomena) from basic evolution equation, i.e. effect function as convolution of memory kernel with cause function.

Response functions in linear viscoelastic constitutive equations and related fractional operators

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Carleman estimates for integro-differential parabolic equations with singular memory kernels

On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator $$\partial _t-\triangle +\int _0^t K(x,t,r)\triangle \ dr$$ where the integral kernel has a behaviour like a weakly singular one. In the proof we consider the integral term as a perturbation. The crucial point is a special choice of the time factor of the weight function.

A magneto-viscoelasticity problem with a singular memory kernel

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analysed; it is obtained coupling an integro-differential equation modelling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modelling the presence of a magnetic field. The case under investigation generalizes a previous study since the relaxation function is allowed to be unbounded at the origin, provided it belongs to L1; the magnetic model equation adopted, as in the previous results (Carillo et al., 2011, 2012; Chipot et al. 2008, 2009) is the penalized Ginzburg–Landau magnetic evolution equation.

A linear viscoelasticity problem with a singular memory kernel: an existence and uniqueness result

The existence and uniqueness of solution to a one-dimensional hyperbolic problem arising in viscoelasticity is here considered. Specifically, the case of a singular kernel is analyzed. This choice is suggested by applications according to the literature to model a wider variety of materials.

Diffusion of high-frequency energy in fluid-saturated porous media

The modern mathematical theory of microlocal analysis shows that the energy associated with the high-frequency solutions of hyperbolic partial differential equations (such as the wave or the Navier equations) satisfy Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous media. For long propagation times the latter can be approached by diffusion equations. Some classical results of the structural acoustics literature about the heat conduction analogy and the statistical energy analysis of structural dynamics at higher frequencies are recovered in this process. The purpose of this communication is to focus on such a diffusive regime for isotropic, fluid-saturated porous media. More specifically, we have derived the diffusion parameters (transport mean-free path and diffusion constant) for such media. Our model considers Biot’s equations of poroelasticity, where thermal and viscous effects are modelized by dynamic tortuosity and compressibility with singular memory kernels. The macroscopic bulk modulus and density of the dry solid phase are assumed to be homogeneous random processes, while tortuosity and porosity remain constant.

Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels

A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration.

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Inverse problems for the identification of memory kernels in the linear theory of viscoelasticity with constitutive stress-strain-relation of Boltzmann type are dealt with in the case of weakly singular kernels in the space L p , and of continuous kernels with power singularity at zero. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence, uniqueness, and stability of solutions are proved.

Time-Domain Finite Element Analysis of Viscoelastic Structures with Fractional Derivatives Constitutive Relations

Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional calculus operators are presented. To avoid difficulties concerning fractional-order initial conditions, a form of the fractlonal calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Lefler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A simplification of the fractional derivative of the memory kernel is used, in connection with Grunwald's definition of fractional differentiation and a backward Euler rule, for the time evolution of the convolution term. A desirable feature of this process is that no actual evaluation of the memory kernel is needed. This, together with the Newmark method for time integration, enables the direct calculation of the time evolution of the nodal degrees of freedom. To illustrate the ability of the numerical procedure a few numerical examples are presented. In one example the numerically obtained solution is compared with a time series expansion of the analytical solution.

Time-domain finite element analysis of viscoelastic structures with fractional derivatives constitutive relations

Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional calculus operators are presented. To avoid difficulties concerning fractional-order initial conditions, a form of the fractlonal calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Lefler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A simplification of the fractional derivative of the memory kernel is used, in connection with Grunwald's definition of fractional differentiation and a backward Euler rule, for the time evolution of the convolution term. A desirable feature of this process is that no actual evaluation of the memory kernel is needed. This, together with the Newmark method for time integration, enables the direct calculation of the time evolution of the nodal degrees of freedom. To illustrate the ability of the numerical procedure a few numerical examples are presented. In one example the numerically obtained solution is compared with a time series expansion of the analytical solution.

Identification of Weakly Singular Memory Kernels in Heat Conduction

AbstractInverse problems of identification of memory kernels in linear heat conduction are dealt with in case of weakly singular kernels in the space Lp and of continuous kernels with power singularity. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence and stability of solutions are proved.

Regularity and stability for a viscoelastic material with a singular memory kernel

We consider a linear viscoelastic material whose relaxation function may exhibit an initial singularity. We show that the Laplace transform method is still applicable in order to study existence, uniqueness and asymptotic behaviour of the solution to the dynamic problem. In order to provide these results, we impose on the relaxation function only restrictions deriving from Thermodynamics. Moreover, by using energy estimates, we establish a stability theorem. Finally, for a class of singular kernels, we obtain a regularity result which ensures the asymptotic stability of the solution.

On wave propagation in linear viscoelasticity

We discuss the initial value problem in one-dimensional linear visco-elasticity with a step-jump in the initial data. If the memory kernel is sufficiently smooth on [ 0 , ∞ ) \left [ {0,\infty } \right ) , the solution exhibits discontinuities propagating along characteristics and a (higher order) stationary discontinuity at the position of the original step-jump. For a singular memory kernel, the propagating waves are smoothed in a manner depending on the nature of the singularity in the kernel, but the stationary discontinuity remains. We also discuss the effects of these phenomena on the regularity of solutions with arbitrary initial data.