Propagation of singularities for integrodifferential equations

Desch, Gertrud; Grimmer, Ronald

doi:10.1016/0022-0396(86)90027-6

Cited by 20 publications

(22 citation statements)

References 26 publications

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“…In this class of viscoelastic media wavefronts propagate with finite speeds but the stress, strain and their derivatives of arbitrary order are continuous except perhaps at the source [5,10,18,22]. In time domain a positive relaxation spectrum in the generalized formulation (4) or (6) is equivalent to the following property of the relaxation modulus g: g is infinitely differentiable on ]0, ∞[ and its derivatives g (n) satisfy the inequalities…”

Section: Non-negative Relaxation Spectrum In Anisotropic Viscoelasticitymentioning

confidence: 99%

Relations Between Relaxation Modulus and Creep Compliance in Anisotropic Linear Viscoelasticity

Hanyga

Seredyńska

2007

J Elasticity

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The results obtained previously for scalar and class P completely monotone relaxation moduli are extended to arbitrary anisotropy. It is shown for general anisotropic viscoelastic media that, if the relaxation modulus is a locally integrable completely monotone function, then the creep compliance is a Bernstein function and conversely. The elastic and equilibrium limits of the two material functions are related to each other. The relaxation modulus or its derivative can be singular at 0. A rigorous general formulation of the relaxation spectrum in an anisotropic viscoelastic medium is given. The effect of Newtonian viscosity on creep compliance is examined. Notation d space dimension; D = d(d + 1)/2; S the linear space of symmetric d × d matrices; G the set of symmetric operators on S; T d the set of linear transformations of a d-dimensional vector space V d ; A. Hanyga (B) 42 A. Hanyga, M. Seredyńska I unit operator on S; A transpose of A; v A w = v k A kl w l ; v, F w := v kl F klpq w pq ; R + =]0, ∞[, R + = [0, ∞[; C + := {z ∈ C | Im z > 0}; C − := C\ ] − ∞, 0]; L p (A; V) the space of measurable functions f : A → V with | f | p ∈ L 1 (A; R + ); M(A; V) the set of positive Radon measures on A with values in V; M + (A; T d ) the set of Radon measures on A with values in T d ; |H|(E) total variation of a Radon measure H ∈ M(A; V); σ, σ1D stress, stress tensor; e, e scalar strain, strain tensor; g, j scalar relaxation modulus and creep compliance; G, J tensorial relaxation modulus and creep compliance.

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Section: Non-negative Relaxation Spectrum In Anisotropic Viscoelasticitymentioning

confidence: 99%

Relations Between Relaxation Modulus and Creep Compliance in Anisotropic Linear Viscoelasticity

Hanyga

Seredyńska

2007

J Elasticity

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“…From (2.3), (2.6), and (2.7), and similar to (3.15), we obtain 18) and from (2.3) and (2.4), we obtain…”

Section: S(t − S) F (S)ds (310) 6 Journal Of Inequalities and Applimentioning

confidence: 66%

Convergence for Hyperbolic Singular Perturbation of Integrodifferential Equations

2007

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“…An impulsive differential equation with singularity perturbation is quite complicated and is difficult to find an exact solution, so there are not many researches in this field. Most of the researches were done by a few groups of mathematicians before year 2000 (see [3,6,9,[17][18][19] A proportional-integral-derivative controller (PID controller) is, today, found in several areas where control has to be exerted. It is the most common form of feedback and is an important component of a distributed control system.…”

Section: Introductionmentioning

confidence: 99%

PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

Witayakiattilerd

2017

Advances in Mathematical Physics

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We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result.

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Propagation of singularities for integrodifferential equations

Cited by 20 publications

References 26 publications

Relations Between Relaxation Modulus and Creep Compliance in Anisotropic Linear Viscoelasticity

Relations Between Relaxation Modulus and Creep Compliance in Anisotropic Linear Viscoelasticity

Convergence for Hyperbolic Singular Perturbation of Integrodifferential Equations

PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

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