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Search results for: FRACTIONAL DIFFERENTIAL EQUATIONS WITH ADVANCED ARGUMENTS
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Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: I
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Variable Order Differential Models of Bone Remodelling * *This work was supported by FCT, through IDMEC, under LAETA, projects UID/EMS/50022/2013, BoneSys, joint Polish-Portuguese project Modelling and controlling cancer evolution using fractional calculus, PERSEIDS (PTDC/EMS-SIS/0642/2014) and IF/00653/2012
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On Applications of Fractional Derivatives in Electromagnetic Theory
PublicationIn this paper, concepts of fractional-order (FO) derivatives are analysed from the point of view of applications in the electromagnetic theory. The mathematical problems related to the FO generalization of Maxwell's equations are investigated. The most popular formulations of the fractional derivatives, i.e., Riemann-Liouville, Caputo, Grünwald-Letnikov and Marchaud definitions, are considered. Properties of these derivatives are...
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Modelling and simulations in time-fractional electrodynamics based on control engineering methods
PublicationIn this paper, control engineering methods are presented with regard to modelling and simulations of signal propagation in time-fractional (TF) electrodynamics. That is, signal propagation is simulated in electromagnetic media described by Maxwell’s equations with fractional-order constitutive relations in the time domain. We demonstrate that such equations in TF electrodynamics can be considered as a continuous-time system of...
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The cohomological span of LS-Conley index
PublicationIn this paper we introduce a new homotopy invariant – the cohomological span of LS-Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form Lx+K(x), where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula...
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Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces
PublicationIn this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain...
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Integrate-and-fire models with an almost periodic input function
PublicationWe investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and μ-almost periodic functions. Special attention is paid to the properties of the firing map and its displacement, which give information about the spiking behavior of the considered system. We provide conditions under which such maps are well-defined and are uniformly continuous. We show that the LIF models with Stepanov almost periodic...
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Heteroclinic solutions for a class of the second order Hamiltonian systems
PublicationW pracy dowodzi się istnienia rozwiązań heteroklicznicznych dla pewnej klasy równań różniczkowych zwyczajnych drugiego rzędu typu hamiltonowskiego.
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Fixed point indices of iterated smooth maps in arbitrary dimension
PublicationWe give a complete description of possible sequences ofindices of iterations of f at an isolated fixed point, answering inaffirmative the Chow, Mallet-Paret and Yorke conjecture posed in[S.N. Chow, J. Mallet-Parret, J.A. Yorke, A periodic point index whichis a bifurcation invariant, in: Geometric Dynamics, Rio de Janeiro,1981, in: Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983,pp. 109-131].
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Straightened characteristics of McKendrick-von Foerster equation
PublicationWe study the McKendrick-von Foerster equation with renewal (that is the age-structured model, with total population dependent coefficient and nonlinearity). By using a change of variables, the model is then transformed to a standard age-structured model in which the total population dependent coefficient of the transport term reduces to a constant 1. We use this transformation to get existence, uniqueness of solutions of the problem...
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The saga of a fish: from a survival guide to closing lemmas
PublicationIn the paper by D. Burago, S. Ivanov and A. Novikov, “A survival guide for feeble fish”, it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe...
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Homoclinic solutions for a class of the second order Hamiltonian systems
PublicationW niniejszej pracy badamy istnienie orbit homoklinicznych dlaukładu Hamiltonowskiego drugiego rzędu: q^{..} + V_{q}(t,q) = f(t), gdzie V z iloczynu kartezjańskiego R x R^{n} do R jest postaciV(t,q) = -K(t,q) + W(t,q). Zakładamy, ze V jest T-okresowe ze względuna zmienną t, K spełnia tzw. ''pinching'' warunek, W jest superliniowew nieskończoności, a norma f w L^{2} jest wystarczająco mała.Orbitę homokliniczną takiego układu znajdujemy...
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Method of lines for physiologically structured models with diffusion
PublicationWe deal with a size-structured model with diffusion. Partial differential equations are approximated by a large system of ordinary differential equations. Due to a maximum principle for this approximation method its solutions preserve positivity and boundedness. We formulate theorems on stability of the method of lines and provide suitable numerical experiments.
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On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions
PublicationThe problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated...
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The modelling method of discrete-continuous systems
PublicationThe paper introduces a method of discrete-continuous systems modelling. In the proposed method a three-dimensional system is divided into finite elements in only two directions, with the third direction remaining continuous. The thus obtained discrete-continuous model is described by a set of partial differential equations. General difference equations of discrete system are obtained using the rigid finite element method. The limit...
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Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector
PublicationIn this paper, the formulation of time-fractional (TF) electrodynamics is derived based on the Riemann-Silberstein (RS) vector. With the use of this vector and fractional-order derivatives, one can write TF Maxwell’s equations in a compact form, which allows for modelling of energy dissipation and dynamics of electromagnetic systems with memory. Therefore, we formulate TF Maxwell’s equations using the RS vector and analyse their...
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A significance of multi slip condition for inclined MHD nano-fluid flow with non linear thermal radiations, Dufuor and Sorrot, and chemically reactive bio-convection effect
PublicationThe aim of this research is to discuss the significance of slip conditions for magnetized nanofluid flow with the impact of nonlinear thermal radiations, activation energy, inclined MHD, sorrot and dufour, and gyrotactic micro motile organisms over continuous stretching of a two-dimensional sheet. The governing equations emerge in the form of partial differential equations. Since the resultant governing differential equations...
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Parabolic Equations with Functional Dependence
PublicationWe consider the Cauchy problem for nonlinear parabolic equations with functional dependence and prove theorems on the existence of solutions to parabolic differential-functional equations.
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PROPERTIES OF ONE DIMENSIONAL OPEN-CHANNEL STEADY FLOW EQUATIONS
PublicationIn this paper properties of discrete forms of one dimensional steady gradually varied flow equations are discussed. Such forms of flow equations are obtained as a result of approximation of their differential forms, which is required to solve them numerically. For such purpose explicit or implicit numerical approximation schemes for ordinary differential equations can be applied. It turns out that dependently on the chosen approximation...
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Equations with Separated Variables on Time Scales
PublicationWe show that the well-known theory for classical ordinary differential equations with separated variables is not valid in case of equations on time scales. Namely, the uniqueness of solutions does not depend on the convergence of appropriate integrals.
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A DISCRETE-CONTINUOUS METHOD OF MECHANICAL SYSTEM MODELLING
PublicationThe paper describes a discrete-continuous method of dynamic system modelling. The presented approach is hybrid in its nature, as it combines the advantages of spatial discretization methods with those of continuous system modelling methods. In the proposed method, a three-dimensional system is discretised in two directions only, with the third direction remaining continuous. The thus obtained discrete-continuous model is described...
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Dyskretno-ciągła metoda modelowania układów dynamicznych
PublicationW artykule przedstawiono oryginalną metodę modelowania układów dyskretno-ciągłych. Metoda polega na dyskretyzowaniu układu trójwymiarowego jedynie w dwóch wybranych kierunkach. W trzecim z kierunków układ pozostaje ciągły. Otrzymany w ten sposób model jest modelem dyskretno-ciągłym. Opisany jest za pomocą równań różniczkowych cząstkowych. Ogólne równania różnicowe układu dyskretnego otrzymano, wykorzystując metodę sztywnych elementów...
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Dynamic modeling of non-cylindrical curved viscoelastic single-walled carbon nanotubes based on the second gradient theory
PublicationThis paper is devoted to the theoretical study of the dynamic response of non-cylindrical curved viscoelastic single-walled carbon nanotubes (SWCNTs). The curved nanotubes are largely used in many engineering applications, but it is challenging in understanding mechanically the dynamic response of these curved SWCNTs when considering the influences of the material viscosity. The viscoelastic damping effect on the dynamic response...
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Estimation of a Stochastic Burgers' Equation Using an Ensemble Kalman Filter
PublicationIn this work, we consider a difficult problem of state estimation of nonlinear stochastic partial differential equations (SPDE) based on uncertain measurements. The presented solution uses the method of lines (MoL), which allows us to discretize a stochastic partial differential equation in a spatial dimension and represent it as a system of coupled continuous-time ordinary stochastic differential equations (SDE). For such a system...
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Significant Production of Thermal Energy in Partially Ionized Hyperbolic Tangent Material Based on Ternary Hybrid Nanomaterials
PublicationNanoparticles are frequently used to enhance the thermal performance of numerous materials. This study has many practical applications for activities that have to minimize losses of energy due to several impacts. This study investigates the inclusion of ternary hybrid nanoparticles in a partially ionized hyperbolic tangent liquid passed over a stretched melting surface. The fluid motion equation is presented by considering the...
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FDTD Method for Electromagnetic Simulations in Media Described by Time-Fractional Constitutive Relations
PublicationIn this paper, the finite-difference time-domain (FDTD) method is derived for electromagnetic simulations in media described by the time-fractional (TF) constitutive relations. TF Maxwell’s equations are derived based on these constitutive relations and the Grünwald–Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced....
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Signal propagation in electromagnetic media described by fractional-order models
PublicationIn this paper, signal propagation is analysed in electromagnetic media described by fractional-order (FO) models (FOMs). Maxwell’s equations with FO constitutive relations are introduced in the time domain. Then, their phasor representation is derived for one-dimensional case of the plane wave propagation. With the use of the Fourier transformation, the algorithm for simulation of the non-monochromatic wave propagation is introduced....
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A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions
PublicationIn this paper we discuss a topological treatment for the planar system z' = f (t, z) + g(t, z) where f and g are T -periodic in time and g(t, z) is bounded. Namely, we study the effect of g(t, z) in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding...
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Local fixed point indices of iterations of planar maps
PublicationW artykule podana zostaje postać indeksów iteracji dla pewnej klasy odwzorowań planarnych. Podstawowymi narzędziami stosowanym w pracy są liczba Nielsena i indeks Conleya.
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On the Existence of Homoclinic Type Solutions of a Class of Inhomogenous Second Order Hamiltonian Systems
PublicationWe show the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems, where a C1-smooth potential satisfies a relaxed superquadratic growth condition, its gradient is bounded in the time variable, and a forcing term is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger...
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Sensitive Demonstration of the Twin-Core Couplers including Kerr Law Non-Linearity via Beta Derivative Evolution
PublicationTo obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr law is the sort of non-linearity addressed there. Because it offers solutions to problems with large tails or infinite fluctuations, the resulting...
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Diffusion equations with spatially dependent coefficients and fractal Cauer-type networks
PublicationIn this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0, can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization...
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Action-reaction based synthesis of acoustic wavefield equations
PublicationThe analysis of acoustic fields is usually based on the well-known mathematics of second order partial differential equations called wave equations. The author explores the duality and symmetry of linear fluid mechanics and develops two distinct equations of acoustics on the basis of a causal approach to local small-scale phenomena. Wavefields that are solutions of these equations have different composition, the spherical pressure...
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Numerical solution of threshold problems in epidemics and population dynamics
PublicationA new algorithm is proposed for the numerical solution of threshold problems in epidemics and population dynamics. These problems are modeled by the delay-differential equations, where the delay function is unknown and has to be determined from the threshold conditions. The new algorithm is based on embedded pair of continuous Runge–Kutta method of order p = 4 and discrete Runge–Kutta method of order q = 3 which is used for the...
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Numerical Analysis of Steady Gradually Varied Flow in Open Channel Networks with Hydraulic Structures
PublicationIn this paper, a method for numerical analysis of steady gradually varied fl ow in channel networks with hydraulic structures is considered. For this purpose, a boundary problem for the system of ordinary differential equations consisting of energy equation and mass conservation equations is formulated. The boundary problem is solved using fi nite difference technique which leads to the system of non-linear algebraic equations....
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Nonlinear Interaction of Modes in a Planar Flow of a Gas with Viscous and Thermal Attenuation
PublicationThe nonlinear interaction of wave and non-wave modes in a gas planar flow are considered. Attention is mainly paid to the case when one sound mode is dominant and excites the counter-propagating sound mode and the entropy mode. The modes are determined by links between perturbations of pressure, density, and fluid velocity. This definition follows from the linear conservation equations in the differential form and thermodynamic...
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On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory
PublicationIn this paper, concepts of fractional-order (FO) derivatives are reviewed and discussed with regard to element models applied in the circuit theory. The properties of FO derivatives required for the circuit-level modeling are formulated. Potential problems related to the generalization of transmission-line equations with the use of FO derivatives are presented. It is demonstrated that some formulations of FO derivatives have limited...
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Crank–Nicolson FDTD Method in Media Described by Time-Fractional Constitutive Relations
PublicationIn this contribution, we present the Crank-Nicolson finite-difference time-domain (CN-FDTD) method, implemented for simulations of wave propagation in media described by time-fractional (TF) constitutive relations. That is, the considered constitutive relations involve fractional-order (FO) derivatives based on the Grünwald-Letnikov definition, allowing for description of hereditary properties and memory effects of media and processes....
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Krzywa przejściowa z wygładzoną krzywizną dla dróg kolejowych
PublicationW pracy przedstawiono koncepcję nowej postaci krzywej przejściowej, o liniowym przebiegu krzywizny na długości i wygładzonymi rejonami skrajnymi. Może ona stanowić alternatywę dla tzw. gładkich krzywych przejściowych, o nieliniowym przebiegu krzywizny na całej długości. Została tutaj wykorzystana uniwersalna metoda identyfikacji krzywych przejściowych za pomocą równań różniczkowych. Wyznaczono ogólne równania krzywizny oraz odpowiednie...
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Inverse Flood Routing Using Simplified Flow Equations
PublicationThe paper considers the problem of inverse flood routing in reservoir operation strategy. The aim of the work is to investigate the possibility of determining the hydrograph at the upstream end based on the hydrograph required at the downstream end using simplified open channel flow models. To accomplish this, the linear kinematic wave equation, the diffusive wave equation and the linear Muskingum equation are considered. To achieve...
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Fractional Order Circuit Elements Derived from Electromagnetism
PublicationIn this paper, derivations of fractional-order (FO) circuit-element equations from electromagnetism are presented. Whilst many papers are devoted to FO modelling of electrical circuits, there are no strong foundations for such an approach. Therefore, we investigate relations between the FO electromagnetism and the FO circuit theory. Our derivations start from quasi-static (QS) approximations of Maxwell's equations in media with...
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Acoustic Heating Produced in the Thermoviscous Flow of a Shear-Thinning Fluid
PublicationThis study is devoted to the instantaneous acoustic heating of a shear-thinningfluid. Apparent viscosity of a shear-thinning fluid depends on the shear rate. Thatfeature distinguishes it from a viscous Newtonian fluid. The special linear combi-nation of conservation equations in the differential form makes it possible to derivedynamic equations governing both the sound and non-wave entropy mode inducedin the field of sound. These...
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Simulation of Wave Propagation in Media Described by Fractional-Order Models
PublicationIn this paper, algorithms for simulation of the wave propagation in electromagnetic media described by fractional-order (FO) models (FOMs) are presented. Initially, fractional calculus and FO Maxwell's equations are introduced. The problem of the wave propagation is formulated for media described by FOMs. Then, algorithms for simulation of the non-monochromatic wave propagation are presented which employ computations in the time...
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On Applications of Fractional Derivatives in Circuit Theory
PublicationIn this paper, concepts of fractional-order (FO) derivatives are discussed from the point of view of applications in the circuit theory. The properties of FO derivatives required for the circuit-level modelling are formulated. Potential problems related to the generalization of transmission line equations with the use of FO derivatives are presented. It is demonstrated that some of formulations of the FO derivatives have limited...
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Integrable zero-range potentials in a plane
PublicationWe examine general statements in the Wronskian representation of Darboux transformations for plane zero-range potentials. Such expressions naturally contain scattering problem solution. We also apply Abel theorem to Wronskians for differential equations and link it to chain equations for Darboux transforms to fix conditions for further development of the underlying distribution concept. Moutard transformations give a convenient...
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Analysis of a gene expression model
PublicationWe study a mathematical model of gene transcription and protein synthesis with negative feedback. We consider a system of equations taking into account the number of active binding sites, the way in which dimers bind to DNA and time delay in translation process. For a simplified model that consist of three ordinary differential equations with time delay we derive conditions for stability of the positive steady state and for the...
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Application of the distributed transfer function method and the rigid finite element method for modelling of 2-D and 3-D systems
PublicationIn the paper application of the Distributed Transfer Function Method and the Rigid Finite Element Method for modelling of 2-D and 3-D systems is presented. In this method an elastic body is divided into 1-D distributed parameter elements (strips or prisms). The whole body (divided into strips or prism) is described by a set of coupled partial differential equations. Solving this equations in the state space form it is possible...
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Thermal Buckling Analysis of Circular Bilayer Graphene sheets Resting on an Elastic Matrix Based on Nonlocal Continuum Mechanics
PublicationIn this article, the thermal buckling behavior of orthotropic circular bilayer graphene sheets embedded in the Winkler–Pasternak elastic medium is scrutinized. Using the nonlocal elasticity theory, the bilayer graphene sheets are modeled as a nonlocal double–layered plate that contains small scale effects and van der Waals (vdW) interaction forces. The vdW interaction forces between the layers are simulated as a set of linear springs...
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Application of muscle model to the musculoskeletal modeling
PublicationThe purpose of this paper is to investigate new fusiform muscle models. Each of these models treats a muscle as a system composedof parts characterized by different mechanical properties. These models explain the influence of differences in the stiffness of lateral parts and the degree of muscle model discretization. Each muscle model is described by a system of differential equations and a single integro-differential equation....
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The impact of methods the stochastic analysis on swimming safety of multihull floating units (Part1)
PublicationThe presented article concerns the application of the methods of the stochastic analysis to solve differential equations for multihull catamaran-type floating unit. There was described the continuous process of Markov and the method of equations of Focker-Planck-Kolmogorov. The analysis of dynamics of the multihull unit was carried out with the assumption that the system model is the linear model with six degrees of freedom, on...