The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.

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Several passive energy dissipation devices have been implemented in practice as the seismic protective systems to mitigate structural damage caused by earthquakes. The solid viscoelastic dampers are among such passive energy dissipation systems. To examine the response reducing effectiveness of these dampers, it is necessary that engineers are able to conduct response analysis of structures installed with added dampers accurately and efficiently. The main objective of this work, therefore, is to develop formulations that can be effectively used with various models of the viscoelastic dampers to calculate the seismic response of a structure-damper system. To incorporate the mechanical effect from VE dampers in the structural dynamic design, it is important to use a proper force-deformation model to correctly describe the frequency dependence of the damper. The fractional derivative model and the general linear model are capable of capturing the frequency dependence of viscoelastic materials accurately. In our research, therefore, we have focused on the development of systematic procedures for calculating the seismic response for these models. For the fractional derivative model, we use the G1 and L1 algorithms to derive various numerical schemes for solving the fractional differential equations for earthquake motions described by acceleration time histories at discrete time points. For linear systems, we also develop a modal superposition method for this model of the damper. This superposition approach can be implemented to obtain the response time history for seismic input defined by the ground acceleration time history. For random ground motion that is described stochastically by the spectral density function, we derive an expression based on random vibration analysis to compute the mean square response of the system. It is noted that the numerical computations involved with the fractional derivative model can be complicated and cumbersome. To alleviate computation difficulty, we explore the use of a general linear model with Kelvin chain analog as a physical representation of the damper properties. The parameters in the model are determined through a curve fitting optimization process. To simplify the analytical work, a self-adjoint system of state equations are formulated by introducing auxiliary displacements for the internal elements in the Kelvin chain. This self-adjoint system can then be solved by using the modal superposition method, which can be extended to develop a response spectrum approach to calculate the seismic design response for the structural system for seismic inputs defined by design ground response spectra. Numerical studies are carried out to demonstrate the applicability of these formulations. Results show that all the proposed approaches provide accurate response values, and the response reduction effects of the viscoelastic dampers can be evaluated to assess their performance using these models and methods. However, the use of a general linear model of the damper is the most efficient. It can capture frequency dependence of the storage and loss moduli as well as the fractional derivative model. The calculation of the response by direct numerical integration of the equations of motion or through the use of the modal superposition approach is significantly simplified, and response spectrum formulation for the calculation of seismic response of design interest can be conveniently formulated. / Ph. D.

Abstract in English / In this dissertation we apply linear evolution equations to the Newtonian derivative, Caputo time fractional derivative and $-time fractional derivative. It is notable that the most utilized fractional order derivatives for modelling true life challenges are Riemann- Liouville and Caputo fractional derivatives, however these fractional derivatives have the same weakness of not satisfying the chain rule, which is one of the most important elements of the match asymptotic method [2, 3, 16]. Furthermore the classical bounded perturbation theorem associated with Riemann-Liouville and Caputo fractional derivatives has con rmed not to be in general truthful for these models, particularly for solution operators of evolution systems of a derivative with fractional parameter ' that is less than one (0 < ' < 1) [29]. To solve this problem, we introduce the derivative with new parameter, which is de ned as a local derivative but has a fractional order called $-derivative and apply this derivative to linear evolution equation and to support what we have done in the theory, we utilize application to population dynamics and we provide the numerical simulations for particular cases. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this work, we establish the Lyapunov-type inequalities for the fractional boundary value problems with Hilfer derivative for different boundary conditions. We apply this inequality to fractional eigenvalue problems and prove one of the important results of real zeros of certain Mittag-Leffler functions and improve the bound of the eigenvalue using the Cauchy-Schwarz inequality and Semi-maximum norm. We extend it for higher order cases.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19121号 / 情博第567号 / 新制||情||100(附属図書館) / 32072 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 木上 淳, 教授 磯 祐介, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

In this dissertation, a new numerical method for solving fractional calculus problems is presented. The method is based upon the fractional Taylor vector approximations. The operational matrix of the fractional integration for the fractional Taylor vector is introduced. This matrix is then utilized to reduce the solution of the fractional calculus problems to the solution of a system of algebraic equations. This method is used to solve fractional differential equations, Bagley-Torvik equations, fractional integro-differential equations, and fractional duffing problems. Illustrative examples are included to demonstrate the validity and applicability of this technique.

In this thesis, we provide a comprehensive overview of classical fractional derivatives and collect results on mapping properties. In particular, we discuss mapping properties e.g. we prove that the 𝛼 order fractional derivative maps the Sobolev space W_0^(p,s) to the fractional Sobolev-Slobodeckij space W^(p,s-α) for all 𝛼 < 𝑠 < 1. Further, we present several definitions of “Bernstein fractional derivatives” using the Bernstein function and in particular, we study the Bernstein censored fractional derivative by using the Picard method to get its inverse Bernstein censored fractional integral. Moreover, we use analytic tools to get the existence and uniqueness of the solution of the corresponding resolvent equation. Finally, we construct a stochastic process through Ikeda–Nagasawa–Watanabe (INW) piecing together procedure such that its generator is the Bernstein censored fractional derivative. Additionally, we show that this process gives a Feller semigroup.:Introduction 1 Basics 1.1 Some results in functional analysis 1.2 Fourier, Laplace and Mellin transforms 1.3 Regularly varying functions 1.4 Markov processes 1.5 Lévy processes and subordinators 2 Fractional derivatives and integrals 2.1 Classical fractional integrals and derivatives 2.2 Mapping properties of fractional integrals and derivatives 2.3 Bernstein functions 2.4 Fractional derivatives based on Bernstein Functions 2.5 Probabilistic interpretation of fractional derivatives 2.6 Fractional Laplace operator 3 Censored Bernstein fractional derivative and integral 3.1 Sonine pairs 3.2 Examples of Sonine pairs 3.3 Mapping properties of general fractional derivatives 3.4 Censored Bernstein fractional derivative and integral 3.5 Linear censored initial value problem 4 Censored process 4.1 Construction 4.2 Probabilistic representation 5 Application 5.1 Censored subordinator for a regularly varying kernel 5.2 Linear censored initial value problem for regularly varying kernels Bibliography

In this dissertation, we use the Natural transform decomposition method to obtain approximate analytical solution of fractional differential equations. This technique is a combination of decomposition methods and natural transform method. We use the Adomian decomposition, the homotopy perturbation and the Daftardar-Jafari methods as our decomposition methods. The fractional derivatives are considered in the Caputo and Caputo- Fabrizio sense. / Mathematical Sciences / M. Sc. (Applied Mathematics)

This work aims at enhancing the understanding and to provideimproved models of the dynamic behavior of rubber vibrationisolators which are widely used in mechanical systems.Initially, a time domainmodel relating compressions tocomponent forces accounting for preload effects, frequency anddynamic amplitude dependence is presented. The problem ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, where theelastic force response is modelled either by a fully linear ora nonlinear shape factor based approach, displaying resultsthat agree with those of a neo-Hookean hyperelastic isolatorunder a long term precompression. The viscoelastic force ismodelled by a fractional derivative element, while the frictionforce governs from a generalized friction element displaying asmoothed Coulomb force. This is a versatile one-dimensionalcomponent model effectively using a small number of parameterswhile exhibiting a good resemblance to measured isolatorcharacteristics. Additionally, the nonlinear excitationeffects on dynamic stiffness and damping of a filled rubberisolator are investigated through measurements. It is shownthat the well-known Payne effect - where stiffness is high forsmall excitation amplitudes and low for large amplitudes whiledamping displays a maximum at intermediate amplitudes -evaluated at a certain frequency, is to a large extentinfluenced by the existence of additional frequency componentsin the signal. Finally, a frequency, temperature and preloaddependent dynamic stiffness model is presented covering theranges from 20 to 20 000 Hz, -50 to +50 °C at 0 to 20 %precompression. A nearly incompressible, thermo-rheologicallysimple material model is adopted displaying viscoelasticitythrough a time - strain separable relaxation tensor with asingle Mittag-Leffler function embodying its time dependence.This fractional derivative based function successfully fitsmaterial properties throughout the whole audible frequencyrange. An extended neo-Hookean strain energy function, beingdirectly proportional to the temperature and density, isapplied for the finite deformation response with componentproperties solved by a nonlinear finite element procedure. Thepresented work is thus believed to enlighten workingconditions’impact on the dynamic properties of rubbervibration isolators, while additionally taking some of thesemost important features into account in the presentedmodels.