Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone, Angelamaria; Conte, Dajana; D’Ambrosio, Raffaele; Paternoster, Beatrice

doi:10.3390/axioms7030045

Cited by 22 publications

(7 citation statements)

References 60 publications

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“…Solving directly such a nonlinear Volterra integral equation requires involved numerical methods which are beyond the scope of the present paper [115,116]. Yet, noting that the range of positions explored by the sphere during creep is very limited [see Fig.…”

Section: Discussionmentioning

confidence: 99%

A mesoscale study of creep in a microgel using the acoustic radiation force

2019

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We study the motion of a sphere of diameter 330 µm embedded in a Carbopol microgel under the effect of the acoustic radiation pressure exerted by a focused ultrasonic field. The sphere motion within the microgel is tracked using videomicroscopy and compared to conventional creep and recovery measurements performed with a rheometer. We find that under moderate ultrasonic intensities, the sphere creeps as a power law of time with an exponent α 0.2 that is significantly smaller than the one inferred from global creep measurements below the yield stress of the microgel (α 0.4). Moreover, the sphere relaxation motion after creep and the global recovery are respectively consistent with these two different exponents. By allowing a rheological characterization at the scale of the sphere with forces of the order of micronewtons, the present experiments pave the way for acoustic "mesorheology" which probes volumes and forces intermediate between standard macroscopic rheology and classical microrheology. They also open new questions about the effects of the geometry of the deformation field and of the sphere size and surface properties on the creep behaviour of soft materials.

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Section: Discussionmentioning

confidence: 99%

A mesoscale study of creep in a microgel using the acoustic radiation force

2019

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“…We start with the construction of DIMSIMs of type 1 with p = q = r = s = 2 and c = [0 1 2 ] T . By the order conditions (1.4), we obtain a two-parameter family of the methods depending on the parameters a 21 and v 1 . The coefficient matrices of these methods are…”

Section: Methods Of Ordermentioning

confidence: 99%

“…Special Runge-Kutta methods for VIDEs have been presented by Wolfe and Phillips [36], Lubich [30], and Brunner [7]. Furthermore, collocation and spectral collocation methods have been studied in [8,10,20,21,22,26,27,35]. Also, an elegant numerical method based on linear barycentric rational interpolation has been introduced by Abdi and Hosseini [1].…”

Section: Introductionmentioning

confidence: 99%

Explicit General Linear Methods With a Large Stability Region for Volterra Integro-Differential Equations

Mahdi

Hojjati

Abdi

2019

Mathematical Modelling and Analysis

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In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods. with z(t) := t t0 K t, s, y(s) ds. Here, the functions f : I × R m × R m → R m and K : D × R m → R m , with D := {(t, s) : t 0 ≤ s ≤ t ≤ T } are continuous on

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“…In [5], the authors review recent findings on the use of collocation methods for numerically solving Volterra integral and integro-differential equations. Both one-step and multi-step methods are considered, studying their convergence and providing comparisons in terms of efficiency and accuracy.…”

Section: Numerical Solution Of Differential Equationsmentioning

confidence: 99%

Advanced Numerical Methods in Applied Sciences

Brugnano

Iavernaro

2019

Axioms

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The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

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Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cited by 22 publications

References 60 publications

A mesoscale study of creep in a microgel using the acoustic radiation force

A mesoscale study of creep in a microgel using the acoustic radiation force

Explicit General Linear Methods With a Large Stability Region for Volterra Integro-Differential Equations

Advanced Numerical Methods in Applied Sciences

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