Search results for "Fractional calculus"

showing 10 items of 128 documents

Path integral solution by fractional calculus

2008

In this paper, the Path Integral solution is developed in terms of complex moments. The method is applied to nonlinear systems excited by normal white noise. Crucial point of the proposed procedure is the representation of the probability density of a random variable in terms of complex moments, recently proposed by the first two authors. Advantage of this procedure is that complex moments do not exhibit hierarchy. Extension of the proposed method to the study of multi degree of freedom systems is also discussed.

HistoryComplex momentsHierarchy (mathematics)Mathematical analysisProbability density functionNon-linear Random VibrationWhite noisePath integral methodComputer Science ApplicationsEducationFractional calculusNonlinear systemPath integral formulationRepresentation (mathematics)Settore ICAR/08 - Scienza Delle CostruzioniRandom variableMathematics
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A Wiener Path Integral Technique for Non-Stationary Response Determination of Nonlinear Oscillators with Fractional Derivative Elements

2014

In this paper a novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators with fractional derivative elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes. In this manner, the analytical Wi…

Hybrid Monte CarloMathematical analysisMonte Carlo methodAnalytical techniquePath integral formulationfractional derivativeProbability density functionFunctional integrationstochastic responseClosed-form expressionWiener path integralMathematicsFractional calculusVulnerability, Uncertainty, and Risk
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Advanced materials modelling via fractional calculus: challenges and perspectives.

2020

Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced mat…

IntroductionComputer scienceGeneral MathematicsGeneral EngineeringCalculusGeneral Physics and AstronomyAdvanced materialsFractional calculusPhilosophical transactions. Series A, Mathematical, physical, and engineering sciences
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An Existence Result for Fractional Kirchhoff-Type Equations

2016

The aim of this paper is to study a class of nonlocal fractional Laplacian equations of Kirchhoff-type. More precisely, by using an appropriate analytical context on fractional Sobolev spaces, we establish the existence of one non-trivial weak solution for nonlocal fractional problems exploiting suitable variational methods.

Kirchhoff typeApplied MathematicsFractional equations010102 general mathematicsMathematical analysisvariational methodsVariational methodAnalysiCritical point result01 natural sciencesFractional equationsFractional equationFractional calculus010101 applied mathematicscritical point resultsSimultaneous equations0101 mathematicsFractional equations variational methods critical point resultsAnalysisMathematicsZeitschrift für Analysis und ihre Anwendungen
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Fractional-order nonlinear hereditariness of tendons and ligaments of the human knee

2020

In this paper the authors introduce a nonlinear model of fractional-order hereditariness used to capture experimental data obtained on human tendons of the knee. Creep and relaxation data on fibrous tissues have been obtained and fitted with logarithmic relations that correspond to power-laws with nonlinear dependence of the coefficients. The use of a proper nonlinear transform allows one to use Boltzmann superposition in the transformed variables yielding a fractional-order model for the nonlinear material hereditariness. The fundamental relations among the nonlinear creep and relaxation functions have been established, and the results from the equivalence relations have been contrasted wi…

LogarithmQuasi-linear viscoelasticityGeneral MathematicsGeneral Physics and AstronomyHarmonic (mathematics)02 engineering and technology01 natural sciences010305 fluids & plasmasTendonsSuperposition principle0203 mechanical engineeringtendons and ligaments0103 physical sciencesHumansEquivalence relationnonlinear hereditarinessKneesingle-integralMechanical PhenomenaMathematicsPolynomial (hyperelastic model)LigamentsMathematical analysisGeneral EngineeringRelaxation (iterative method)Biomechanical PhenomenaFractional calculusNonlinear system020303 mechanical engineering & transportsNonlinear DynamicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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MECHANICAL RESPONSE OF BERNULLI EULER BEAMS ON FRACTIONAL ORDER ELASTIC FOUNDATION

2014

Long-range interactions non-local foundations elastic beams fractional calculusSettore ICAR/08 - Scienza Delle Costruzioni
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Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

2012

Abstract A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis becomes very straightforward and simple since, based on stand…

Markov chainDiscretizationStochastic processMechanical EngineeringMathematical analysisDegrees of freedom (statistics)Stochastic calculusAerospace EngineeringOcean EngineeringStatistical and Nonlinear PhysicsViscoelasticity Fractional calculus Stochastic input Non-stationary responseCondensed Matter PhysicsFractional calculusOperator (computer programming)Nuclear Energy and EngineeringSettore ICAR/08 - Scienza Delle CostruzioniLinear equationCivil and Structural EngineeringMathematics
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On the dynamics of non-local fractional viscoelastic beams under stochastic agencies

2018

Abstract Non-local viscoelasticity is a subject of great interest in the context of non-local theories. In a recent study, the authors have proposed a non-local fractional beam model where non-local effects are represented as viscoelastic long-range volume forces and moments, exchanged by non-adjacent beam segments depending on their relative motion, while local effects are modelled by elastic classical stress resultants. Long-range interactions have been given a fractional constitutive law, involving the Caputo's fractional derivative. This paper introduces a comprehensive numerical approach to calculate the stochastic response of the non-local fractional beam model under Gaussian white no…

Materials scienceDiscretization02 engineering and technologyWhite noiseIndustrial and Manufacturing Engineering0203 mechanical engineeringFractional viscoelasticityComposite materialImpulse responseNon local Timoshenko beamMechanical EngineeringMathematical analysisEquations of motionWhite noise021001 nanoscience & nanotechnologyPhysics::History of PhysicsNon local Timoshenko beam; Fractional viscoelasticity; White noise; State variable expansionFractional calculusNumerical integration020303 mechanical engineering & transportsMechanics of MaterialsStress resultantsFrequency domainCeramics and CompositesState variable expansionSettore ICAR/08 - Scienza Delle CostruzioniFractional viscoelasticity Non local Timoshenko beam State variable expansion White noise0210 nano-technologyNon local Timoshenko beam Fractional viscoelasticity White noise State variable expansionComposites Part B: Engineering
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Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures

2016

The aim of this paper is to propose a fractional viscoelastic and viscoplastic model of asphalt mixtures using experimental data of several tests such as creep and creep recovery performed at different temperatures and at different stress levels. From a best fitting procedure it is shown that both the creep one and recovery curve follow a power law model. It is shown that the suitable model for asphalt mixtures is a dashpot and a fractional element arranged in series. The proposed model is also available outside of the linear domain but in this case the parameters of the model depend on the stress level.

Materials scienceasphalt mixtureStrategy and Managementcreep test0211 other engineering and technologies02 engineering and technologyfractional calculusPower lawcreep test.ViscoelasticityDashpot0203 mechanical engineering021105 building & constructionSettore ICAR/04 - Strade Ferrovie Ed AeroportiComposite materialviscoplasticityviscoelasticityCivil and Structural EngineeringBuilding constructionmechanical modelsViscoplasticityMechanicsmechanical modelFractional calculusfractional calculus asphalt mixture viscoelasticity viscoplasticity rheology mechanical models creep testfractional calculuNonlinear system020303 mechanical engineering & transportsCreepAsphaltrheologyTH1-9745
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Prediction of Dissipative Properties of Flax Fibers Reinforced Laminates by Vibration Analysis

2016

This study proposes an experimental-numeric method to identify the viscoelastic properties of flax fibres reinforced composite laminate (flax/epoxide). The used method consists in identifying the evolutions of both loss factor and stiffness when vibrational frequency changes. In this way, several free-free symmetrically guided beams are excited on a dynamic range of 10 to 4000 Hz with sweep sine excitation focused around the 4-first’s modes. Fractional derivative Zener model is used to identify the on-axis ply complex moduli and describe the laminate dissipative linear behavior with the classical laminate theory. Results obtained on a quasi-isotropic laminate show that this model adequately…

Materials sciencebusiness.industry[SPI] Engineering Sciences [physics]Loss factorComposite numberStiffnessGeneral MedicineStructural engineering[SPI.MAT] Engineering Sciences [physics]/MaterialsViscoelasticityFractional calculusVibrationDissipative systemmedicineStandard linear solid modelComposite materialmedicine.symptombusinessComputingMilieux_MISCELLANEOUS
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