Relevant bibliographies by topics / Free differential calculus

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Free differential calculus'

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Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "Free differential calculus"

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Parisi, Francesco, and Jonathan Klick. "The Differential Calculus of Consent." Journal of Public Finance and Public Choice 20, no. 2 (October 1, 2002): 115–24. http://dx.doi.org/10.1332/251569202x15665366114888.

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Abstract Existing treatments of the choice of an optimal voting rule ignore the effects of the rule on political bargaining. Specifically, more stringent majority requirements reduce intra-coalitional free riding in political compromise, leading to greater gains from political trade. Once this benefit of increasing the vote share necessary to enact a proposal is recognized, we suggest that the optimal voting rule in the presence of transactions costs will actually be closer to unanimity than the optimal majority derived by Buchanan - Tullock [1962].
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Shpil'rain, V. �. "Some applications of free differential calculus in Group theory." Mathematical Notes of the Academy of Sciences of the USSR 49, no. 3 (March 1991): 334–35. http://dx.doi.org/10.1007/bf01158317.

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MORENO, GIOVANNI. "ON FAMILIES IN DIFFERENTIAL GEOMETRY." International Journal of Geometric Methods in Modern Physics 10, no. 09 (August 30, 2013): 1350042. http://dx.doi.org/10.1142/s0219887813500424.

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Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.
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Huang, Zhiyuan, and Shunlong Luo. "Wick Calculus of Generalized Operators and its Applications to Quantum Stochastic Calculus." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 03 (July 1998): 455–66. http://dx.doi.org/10.1142/s0219025798000247.

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A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.
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Karaçuha, Serkan, and Christian Lomp. "Integral calculus on quantum exterior algebras." International Journal of Geometric Methods in Modern Physics 11, no. 04 (April 2014): 1450026. http://dx.doi.org/10.1142/s0219887814500261.

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Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.
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Majid, S. "Free braided differential calculus, braided binomial theorem, and the braided exponential map." Journal of Mathematical Physics 34, no. 10 (October 1993): 4843–56. http://dx.doi.org/10.1063/1.530326.

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MIKHALEV, ALEXANDER A., and ANDREJ A. ZOLOTYKH. "RANK AND PRIMITIVITY OF ELEMENTS OF FREE COLOUR LIE (p-)SUPERALGEBRAS." International Journal of Algebra and Computation 04, no. 04 (December 1994): 617–55. http://dx.doi.org/10.1142/s021819679400018x.

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Using Fox differential calculus we study characteristics of orbits of elements of free colour Lie (p-)superalgebras under action of the automorphism groups of these algebras. In particular, an effective criterion for an element to be primitive and an algorithm for finding the rank of an element are obtained.
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Kenoufi, Abdelouahab. "Linear Algebra and Differential Calculus in Pseudo-Intervals Vector Space." TEMA (São Carlos) 17, no. 3 (December 20, 2016): 283. http://dx.doi.org/10.5540/tema.2016.017.03.0283.

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In this paper one proposes to use a new approach of interval arithmetic, the so-called pseudo- intervals [1, 5, 13]. It uses a construction which is more canonical and based on the semi-group completion into the group, and it allows to build a Banach vector space. This is achieved by embedding the vector space into free algebra of dimensions higher than 4. It permits to perform linear algebra and differential calculus with pseudo-intervals. Some numerical applications for interval matrix eigenmode calculation, inversion and function minimization are exhibited for simple examples.
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Fan, Xiliang, and Yong Ren. "Bismut formulas and applications for stochastic (functional) differential equations driven by fractional Brownian motions." Stochastics and Dynamics 17, no. 04 (May 4, 2017): 1750028. http://dx.doi.org/10.1142/s0219493717500289.

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In this paper by using Malliavin calculus we prove derivative formulas of Bismut type for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, the dimensional-free Harnack type inequalities and the strong Feller property are presented.
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TRAVERSA, FABIO L., and FABRIZIO BONANI. "ASYMPTOTIC STOCHASTIC CHARACTERIZATION OF PHASE AND AMPLITUDE NOISE IN FREE-RUNNING OSCILLATORS." Fluctuation and Noise Letters 10, no. 02 (June 2011): 207–21. http://dx.doi.org/10.1142/s021947751100048x.

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Starting from the definition of the stochastic differential equation for amplitude and phase fluctuations of an oscillator described by an ordinary differential equation, we study the associated Fokker–Planck equation by using tools from stochastic integral calculus, harmonic analysis and Floquet theory. We provide an asymptotic characterization of the relevant correlation functions, showing that within the assumption of a linear perturbative analysis for the amplitude fluctuations phase noise and orbital fluctuations at the same time are asymptotically statistically independent, and therefore the nonlinear perturbative analysis of phase noise recently derived still exactly holds even if orbital noise is taken into account.
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Dissertations / Theses on the topic "Free differential calculus"

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Trey, Baptiste. "Existence et régularité des formes optimales pour des problèmes d'optimisation spectrale Free boundary regularity for a multiphase shape optimization problem. Communications in Partial Dfferential Equations Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form Existence and regularity of optimal shapes for elliptic operators with drift. Calculus of Variations and Partial Differential Equations." Thesis, Université Grenoble Alpes, 2020. http://www.theses.fr/2020GRALM019.

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Dans cette thèse, on étudie l'existence et la régularité des formes optimales pour certains problèmes d'optimisation spectrale qui font intervenir un opérateur elliptique avec condition de Dirichlet.On s'intéresse d'abord au problème de la minimisation de la valeur propre principale d'un opérateur avec un terme de transport borné.Que le terme de transport soit fixé ou non, ce problème admet une solution parmi les quasi-ouverts, et si le terme de transport est en outre le gradient d'une fonction Lipschitzienne, alors les solutions sont des ouverts localement de classe C^{1,alpha} en dehors de points exceptionnels.On étudie ensuite en dimension deux la régularité des solutions à un problème d'optimisation à plusieurs phases pour la première valeur propre du Laplacien de Dirichlet.Enfin, on s'intéresse aux ensembles optimaux pour la somme des k premières valeurs propres d'un opérateur elliptique sous forme divergence. On montre que les k premières fonctions propres sur un ensemble optimal sont lipschitziennes de sorte que les ensembles optimaux sont ouverts, et on étudie ensuite la régularité de la frontière des ensembles optimaux
In this thesis, we study the existence and the regularity of optimal shapes for some spectral optimization problems involving an elliptic operator with Dirichlet boundary condition.First of all, we consider the problem of minimizing the principal eigenvalue of an operator with bounded drift under inclusion and volume constraints.Whether the drift is fixed or not, this problem admits solutions among the class of quasi-open sets, and if the drift is furthermore the gradient of a Lipschitz continuous function, then the solutions are open sets and C^{1,alpha}-regular except on a set of exceptional points.Next, we study in dimension two the regularity of the solutions to a multi-phase optimization problem for the first eigenvalue of the Dirichlet Laplacian.Finally, we focus on the optimal sets for the sum of the first k eigenvalues of an operator in divergence form. We prove that the first k eigenfunctions on an optimal set are Lipschitz continuous so that the optimal sets are open sets, and we then study the regularity of the boundary of the optimal sets
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Rocha, Eugénio Alexandre Miguel. "Uma Abordagem Algébrica à Teoria de Controlo Não Linear." Doctoral thesis, Universidade de Aveiro, 2003. http://hdl.handle.net/10773/21444.

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Doutoramento em Matemática
Nesta tese de Doutoramento desenvolve-se principalmente uma abordagem algébrica à teoria de sistemas de controlo não lineares. No entanto, outros tópicos são também estudados. Os tópicos tratados são os seguidamente enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres, estabilidade de um sistema de corpos rolantes, algoritmos para aritmética digital, e equações integrais de Fredholm não lineares. No primeiro e principal tópico estudam-se representações para as soluções de sistemas de controlo lineares no controlo. As suas trajetórias são representadas pelas chamadas séries de Chen. Estuda-se a representação formal destas séries através da introdução de várias álgebras não associativas e técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo e de segundo tipo. Apresenta-se uma demonstração alternativa para as coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas de primeiro tipo. Estas últimas estão intimamente ligadas ao logaritmo da série de Chen que, por sua vez, tem fortes relações com uma fórmula designada na literatura por “continuous Baker-Campbell- Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia de um campo de vectores não linear e periódico no tempo em relação a uma truncatura do logaritmo de Chen. No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer dois corpos que rolem um sobre o outro sem deslizar ou torcer. Constroem-se controlos fechados e dependentes do tempo que tornam a origem do sistema de dois corpos num sistema localmente assimptoticamente estável. Vários exemplos e algumas implementações em Maple°c são discutidos. No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o valor de várias funções fundamentais na aritmética digital, sendo possível a sua implementação em microprocessadores. São também obtidos os seus domínios de convergência. No último tópico, também em apêndice, demonstra-se a existência e unicidade de solução para uma classe de equações integrais não lineares com atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade no sentido de Fréchet da solução em relação à função de atraso.
In this PhD thesis several subjects are studied regarding the following topics: formulas for nonlinear control systems on free Lie algebras, stabilizability of nonlinear control systems, digital arithmetic algorithms, and nonlinear Fredholm integral equations with delay. The first and principal topic is mainly related with a problem known as the continuous Baker-Campbell-Hausdorff exponents. We propose a calculus to deal with formal nonautonomous ordinary differential equations evolving on the algebra of formal series defined on an alphabet. We introduce and connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel, chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of those notions were also introduced into the universal enveloping algebra of a free Lie algebra. We study Chen series and iterated integrals by relating them with nonlinear control systems linear in control. At the heart of all the theory of Chen series resides a zinbiel and shuffle homomorphism that allows us to construct a purely formal representation of Chen series on algebras of words. It is also given a pre-zinbiel representation of the chronological exponential, introduced by A.Agrachev and R.Gamkrelidze on the context of a tool to deal with nonlinear nonautonomous ordinary differential equations over a manifold, the so-called chronological calculus. An extensive description of that calculus is made, collecting some fragmented results on several publications. It is a fundamental tool of study along the thesis. We also present an alternative demonstration of the result of H.Sussmann about coordinates of second kind using the mentioned tools. This simple and comprehensive proof shows that coordinates of second kind are exactly the image of elements of the dual basis of a Hall basis, under the above discussed homomorphism. We obtain explicit expressions for the logarithm of Chen series and the respective coordinates of first kind, by defining several operations on a forest of leaf-labelled trees. It is the same as saying that we have an explicit formula for the functional coefficients of the Lie brackets on a continuous Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We apply those formulas to relate some noncommutative symmetric functions, and we also connect the monodromy map of a time-periodic nonlinear vector field with a truncation of the Chen logarithm. On the second topic, we study any system of two bodies rolling one over the other without twisting or slipping. By using the Chen logarithm expressions, the monodromy map of a flow and Lyapunov functions, we construct time-variant controls that turn the origin of a control system linear in control into a locally asymptotically stable equilibrium point. Stabilizers for control systems whose vector fields generate a nilpotent Lie algebra with degree of nilpotency · 3 are also given. Some examples are presented and Maple°c were implemented. The third topic, on appendix, concerns the construction of efficient algorithms for Digital Arithmetic, potentially for the implementation in microprocessors. The algorithms are intended for the computation of several functions as the division, square root, sines, cosines, exponential, logarithm, etc. By using redundant number representations and methods of Lyapunov stability for discrete dynamical systems, we obtain several algorithms (that can be glued together into an algorithm for parallel execution) having the same core and selection scheme in each iteration. We also prove their domains of convergence and discuss possible extensions. The last topic, also on appendix, studies the set of solutions of a class of nonlinear Fredholm integral equations with general delay. The delay is of functional character modelled by a continuous lag function. We ensure existence and uniqueness of a continuous (positive) solution of such equation. Moreover, under additional conditions, it is obtained the Fr´echet differentiability of the solution with respect to the lag function.
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Roux, Raphaël. "Étude probabiliste de systèmes de particules en interaction : applications à la simulation moléculaire." Phd thesis, Université Paris-Est, 2010. http://tel.archives-ouvertes.fr/tel-00597479.

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Ce travail présente quelques résultats sur les systèmes de particules en interaction pour l'interprétation probabiliste des équations aux dérivées partielles, avec des applications à des questions de dynamique moléculaire et de chimie quantique. On présente notamment une méthode particulaire permettant d'analyser le processus de la force biaisante adaptative, utilisé en dynamique moléculaire pour le calcul de différences d'énergies libres. On étudie également la sensibilité de dynamiques stochastiques par rapport à un paramètre, en vue du calcul des forces dans l'approximation de Born-Oppenheimer pour rechercher l'état quantique fondamental de molécules. Enfin, on présente un schéma numérique basé sur un système de particules pour résoudre des lois de conservation scalaires, avec un terme de diffusion anormale se traduisant par une dynamique de sauts sur les particules
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Books on the topic "Free differential calculus"

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M, Chadam John, and Rasmussen Henning Ph D, eds. Emerging applications in free boundary problems: Proceedings of the International Colloquium 'Free Boundary Problems, Theory and Applications'. Harlow, Essex, England: Longman Scientific & Technical, 1993.

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Symposium on "Free Boundary Problems: Theory & Applications" (1990 Montréal, Québec). Free boundary problems involving solids: Proceedings of the International Colloquium 'Free Boundary Problems--Theory and Applications'. Edited by Chadam John M and Rasmussen Henning Ph D. Harlow, Essex, England: Longman Scientific & Technical, 1993.

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Roe, John. Winding around: The winding number in topology, geometry, and analysis. Providence, Rhode Island: American Mathematical Society, 2015.

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1974-, Nelson Sam, ed. Quandles: An introduction to the algebra of knots. Providence, Rhode Island: American Mathematical Society, 2015.

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Grossman, Michael. Bigeometric Calculus: A System with a Scale-Free Derivative. BookSurge Publishing, 2006.

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Guionnet, Alice. Free probability. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0003.

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Free probability was introduced by D. Voiculescu as a theory of noncommutative random variables (similar to integration theory) equipped with a notion of freeness very similar to independence. In fact, it is possible in this framework to define the natural ‘free’ counterpart of the central limit theorem, Gaussian distribution, Brownian motion, stochastic differential calculus, entropy, etc. It also appears as the natural setup for studying large random matrices as their size goes to infinity and hence is central in the study of random matrices as their size go to infinity. In this chapter the free probability framework is introduced, and it is shown how it naturally shows up in the random matrices asymptotics via the so-called ‘asymptotic freeness’. The connection with combinatorics and the enumeration of planar maps, including loop models, are discussed.
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Chadam, J. M., and Helen Rasmussen. Emerging Applications in Free Boundary Problems (Research Notes in Mathematics Series). Chapman & Hall/CRC, 1993.

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Chadam, J. M., and Helen Rasmussen. Free Boundary Problems Involving Solids (Research Notes in Mathematics Series). Chapman & Hall/CRC, 1993.

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Book chapters on the topic "Free differential calculus"

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Kaliuzhnyi-Verbovetskyi, Dmitry, and Victor Vinnikov. "NC functions and their difference-differential calculus." In Foundations of Free Noncommutative Function Theory, 15–32. Providence, Rhode Island: American Mathematical Society, 2014. http://dx.doi.org/10.1090/surv/199/02.

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Gupta, Narain. "Chapter I: Magnus Embeddings and Free Differential Calculus." In Contemporary Mathematics, 1–22. Providence, Rhode Island: American Mathematical Society, 1987. http://dx.doi.org/10.1090/conm/066/01.

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Kaliuzhnyi-Verbovetskyi, Dmitry, and Victor Vinnikov. "Higher order nc functions and their difference-differential calculus." In Foundations of Free Noncommutative Function Theory, 33–59. Providence, Rhode Island: American Mathematical Society, 2014. http://dx.doi.org/10.1090/surv/199/03.

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Braides, A. "Free discontinuity problems and their non-local approximation." In Calculus of Variations and Partial Differential Equations, 171–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-57186-2_6.

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Mikhalev, Alexander A., and Andrej A. Zolotykh. "Applications of Fox Differential Calculus to Free Lie Superalgebras." In Non-Associative Algebra and Its Applications, 285–90. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0990-1_47.

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Leaci, A. "Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth." In Calculus of Variations and Partial Differential Equations, 153–69. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-57186-2_5.

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Chen, K. T., R. H. Fox, and R. C. Lyndon. "Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series." In Collected Papers of K.-T. Chen, 104–18. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-2096-1_10.

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Zeidler, Eberhard. "Free Local Extrema of Differentiable Functionals and the Calculus of Variations." In Nonlinear Functional Analysis and its Applications, 189–228. New York, NY: Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4612-5020-3_5.

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"Differential Equations of Second Order." In Free Calculus, 59–72. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812790842_0004.

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"Differential Equations of First Order." In Free Calculus, 51–58. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812790842_0003.

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Conference papers on the topic "Free differential calculus"

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Venkataraman, P. "B-Spline Based Free Form Solutions of Nonlinear Systems." In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/detc2004-57672.

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B-spline parametric curves provide alternative solutions to multipoint nonlinear systems. They solve the problem with less effort than traditional numerical techniques. The new approach demonstrated in this paper is natural and direct. It uses the least squared error technique to identify curves that satisfy the differential relations and boundary conditions. No state space integration is required. No Euler-Lagrange relations are to be satisfied and the Hamiltonian principle is not necessary. In addition, the solution can have properties that are currently not investigated or encouraged. This solution can also be analytically described. The problem set up is simple and is uniform over different classes of problems. The casual nature of problem definition and set up are demonstrated through three examples of increasing complexity: (i) Blassius two-point boundary-value problem (fluids); (ii) Brachistochrone problem (calculus of variation); and (iii) Planar trajectory interception problem (optimal control).
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Catania, Giuseppe, and Silvio Sorrentino. "Experimental Identification of a Fractional Derivative Linear Model for Viscoelastic Materials." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85725.

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Non integer, fractional order derivative rheological models are known to be very effective in describing the linear viscoelastic dynamic behaviour of mechanical structures made of polymers [1]. The application of fractional calculus to viscoelasticity can be physically consistent [2][3][4] and the resulting non integer order differential stress-strain constitutive relation provides good curve fitting properties, requires only a few parameters and leads to causal behaviour [5]. When using such models the solution of direct problems, i.e. the evaluation of time or frequency response from a known excitation can still be obtained from the equations of motion using standard tools such as modal analysis [6]. But regarding the inverse problem, i.e. the identification from measured input-output vibrations, no general technique has so far been established, since the current methods do not seem to easily work with differential operators of non integer order. In this paper a frequency domain method is proposed for the experimental identification of a linear viscoelastic model, namely the Fractional Zener also known as Fractional Standard Linear Solid [5], to compute the frequency dependent complex stress-strain relationship parameters related to the material. The procedure is first checked with respect to numerically generated frequency response functions for testing its accuracy, and then to experimental inertance data from a free-free homogeneous beam made of High Density Polyethylene (HDPE) in plane flexural and axial vibration.
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Praharaj, R. K., and N. Datta. "Application of Fractional Calculus in Modelling Viscoelastic Foundation of Ship Structures for Passive Vibration Control." In ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/omae2020-18674.

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Abstract Viscoelastic materials (VEM) are often used in the ship floors to dampen the noise and vibration induced by machinery (engines, pumps, alternators) and live loads (human traffic and liquid movement in pipelines). Therefore, accurate modeling of such materials is important for the iterative design-analysis process. VEM exhibits both time and frequency-dependent properties. Traditionally, integer-order viscoelastic mechanical models are used to describe the rheological properties of the viscoelastic materials. However, such models find difficulties in predicting the rheological characteristics since their kernel functions are a combination of exponential functions. For instance, the integer-order Maxwell model is good at describing the stress relaxation behavior while is poor in capturing creep, and vice-versa for the integer-order Kelvin-Voigt model. However, the fractional-order mechanical model can overcome the above problem with a fewer number of model parameters. Therefore, in this paper, the fractional derivative-based Kelvin-Voigt mechanical model is employed to describe the time-dependent vibratory behavior of the VEM. To validate the effectiveness of the above model, a thin elastic plate bonded with a thin layer of VEM, subject to a concentrated impact load, is studied. Galerkin’s method and Triangular strip matrix approach are used to solve the partial fractional differential equations of motion. The semi-analytical approach of modal superposition is used to generate the response, whose first dynamic overshoot of displacement is crucial to contain the dynamic stress within the working stress level. This bypasses the computationally expensive Finite Element Method. Additionally, ANSYS is limited to the integer-order damping model only. This analysis gives insights into the efficacy of the VEM chosen for passive vibration control of structural components of ship hulls. A case study for free vibration is done with a commercially available VEM, which is used as surface flooring to cover steel plates of ship hulls, thereby acting as vibration dampers. Free vibration results show that the damping coefficient of the plate foundation system increases with increasing the order of the derivative. In addition, the amplitude of the transient response decreases with the order of the derivative. Thus, the classical integer-order mechanical model overestimates the damping of the viscoelastic materials, which leads to underestimating the displacement and associated stresses. The results are verified with literature and ANSYS.
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