Search
Department of Differential Equations and Mathematical Applications
Publications
Filters
total: 270
Catalog Publications
Year 2015
-
Efficient quadrature for fast oscillating integralof paraxial optics
PublicationThe study concerns the determination of quadrature for the integral solutionof the paraxial wave equation. The difficulty in computation of the integral isassociated with the rapid change of the integrand phase. The developed quadraturetakes into account the fast oscillating character of the integrand. The presentedmethod is an alternative to the commonly used methods based on the use of theFourier transform. The determination...
-
Entropic Measures of Complexity of Short-Term Dynamics of Nocturnal Heartbeats in an Aging Population
PublicationTwo entropy-based approaches are investigated to study patterns describing differences in time intervals between consecutive heartbeats. The first method explores matrices arising from networks of transitions constructed following events represented by a time series. The second method considers distributions of ordinal patterns of length three, whereby patterns with repeated values are counted as different patterns. Both methods provide...
-
Entropy Measures in the Assessment of Heart Rate Variability in Patients with Cardiodepressive Vasovagal Syncope
PublicationSample entropy (SampEn) was reported to be useful in the assessment of the complexity of heart rate dynamics. Permutation entropy (PermEn) is a new measure based on the concept of order and was previously shown to be accurate for short, non-stationary datasets. The aim of the present study is to assess if SampEn and PermEn obtained from baseline recordings might differentiate patients with various outcomes of the head-up tilt test...
-
Existence and uniqueness of solutions for single-population McKendrick-von Foerster models with renewal
PublicationWe study a McKendrick-von Foerster type equation with renewal. This model is represented by a single equation which describes one species which produces young individuals. The renewal condition is linear but takes into account some history of the population. This model addresses nonlocal interactions between individuals structured by age. The vast majority of size-structured models are also treatable. Our model generalizes a number...
-
First-order differential equations with nonlocal boundary conditions
PublicationWe study a first-order boundary value problem subject to some boundary conditions given by Riemann-Stieltjes integrals. Using a monotone iterative method, we formulate sufficient conditions which guarantee the existence of extremal or quasi-solutions in the corresponding region bounded by upper and lower solutions of our problems. The case when a unique solution exists is also investigated. Some examples are given to illustrate...
-
Fractional differential equations with causal operators
PublicationWe study fractional differential equations with causal operators. The existence of solutions is obtained by applying the successive approximate method. Some applications are discussed including also the case when causal operator Q is a linear operator. Examples illustrate some results.
-
Generalised heart rate statistics reveal neurally mediated homeostasis transients
PublicationDistributions of accelerations and decelerations, obtained from increments of heart rate recorded during a head-up tilt table (HUTT) test provide short-term characterization of the complex cardiovascular response to a rapid controlled dysregulation of homeostasis. A generalised statistic is proposed for evaluating the neural reflexes responsible for restoring the homeostatic dynamics. An evaluation of the effects on heart rate...
-
Highly porous nanoberyllium for X-ray beam speckle suppression
PublicationThe speckle suppressor device containing highly porous nanoberyllium is proposed for manipulating with the spatial coherence length and removing undesirable speckle structure during the imaging experiments. We report a special device called the speckle suppressor, which contains the highly porous nanoberyllium plate, compacted from both sides by two beryllium windows. By insertion the speckle suppressor in the X-ray beam it allows...
-
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.
-
On the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation
PublicationIn this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers-Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19, 1907{1920 (2014)]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some...
-
On the regularity of the displacement sequence of an orientation preserving circle homeomorphism
PublicationWe investigate the regularity properties of the displacemnet sequence of an orientation preserving circle homeomorphism. is rational, then ηn(z) is asymptotically periodic with semi-period q. This
-
Positive solutions to advanced fractional differential equations with nonlocal boundary conditions
PublicationWe study the existence of positive solutions for a class of higher order fractional differential equations with advanced arguments and boundary value problems involving Stieltjes integral conditions. The fixed point theorem due to Avery-Peterson is used to obtain sufficient conditions for the existence of multiple positive solutions. Certain of our results improve on recent work in the literature.
-
Systems of boundary value problems of advanced differential equations
PublicationThis paper considers the existence of extremal solutions to systems of advanced differential equations with corresponding nonlinear boundary conditions. The monotone iterative method is applied to obtain the existence results. An example is provided for illustration.
-
Systems of Nonlinear Fractional Differential Equations
PublicationUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives D(T)(q)x and D(T)(q)y. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.
-
The Hopf theorem for gradient local vector fields on manifolds
PublicationWe prove the Hopf theorem for gradient local vector fields on manifolds, i.e., we show that there is a natural bijection between the set of gradient otopy classes of gradient local vector fields and the integers if the manifold is connected Riemannian without boundary.
-
Thermal ablation modeling via the bioheat equation and its numerical treatment
PublicationThe phenomenon of thermal ablation is described by Pennes’ bioheat equation. This model is based on Newton’s law of cooling. Many approximate methods have been considered because of the importance of this issue. We propose an implicit numerical scheme which has better stability properties than other approaches.
-
Weighted difference schemes for systems of quasilinear first order partial functional differential equations
PublicationThe paper deals with initial boundary value problems of the Dirichlet type for system of quasilinear functional differential equations. We investigate weighted difference methods for these problems. A complete convergence analysis of the considered difference methods is given. Nonlinear estimates of the Perron type with respect to functional variables for given functions are assumed. The proof of the stability of difference problems...
-
Wspomnienie o prof. dr. hab. inż. Zbigniewie Sikorze (1954-2015)
PublicationWspomnienie.
Year 2014
-
Acoustic Streaming Induced by Periodic and Aperiodic Sound in a Bubbly Liquid
PublicationThe vortex ow which follows intense sound propagating in a bubbly liquid, is considered. The reasons for acoustic streaming are both nonlinearity and dispersion. That makes streaming especial as compared with that in a Newtonian uid. Conclusions concern the vortex ow induced in a half-space by initially harmonic or impulse Gaussian beam. The vortex ow recalls a turbulent ow with increasing in time number of small-scale vortices...
-
Boundary problems for fractional differential equations
PublicationIn this paper, the existence of solutions of fractional differential equations with nonlinear boundary conditions is investigated. The monotone iterative method combined with lower and upper solutions is applied. Fractional differential inequalities are also discussed. Two examples are added to illustrate the results.
-
Difference functional inequalities and applications.
PublicationThe paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.
-
Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization a` la Mickens of the generalized Burgers–Huxley equation.
PublicationDeparting from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity,...
-
Existence of solutions with an exponential growth for nonlinear differential-functional parabolic equations
PublicationWe consider the Cauchy problem for nonlinear parabolic equations with functional dependence.We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equations.
-
Explicit and implicit difefrence methods for quasilinear first order partial functional differential equations.
PublicationInitial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Suffcient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes...
-
Fractional problems with advanced arguments
PublicationThis paper concerns boundary fractional differential problems with advanced arguments. We investigate the existence of initial value problems when the initial point is given at the end point of an interval. Nonhomogeneous linear fractional differential equations are also studied. The existence of solutions for fractional differential equations with advanced arguments and with boundary value problems has been investigated by using...
-
Modelling gene expression of a self-regulating protein
PublicationWe analyze a model of gene transcription and protein synthesis. We take into account the number of sites on the protein’s promoter at which the protein’s dimers can bind blocking transcription of protein mRNA.
-
Monotone iterative method for first-order differential equations at resonance
PublicationThis paper concerns the application of the monotone iterative technique for first-order differential equations involving Stieltjes integrals conditions. We discuss such problems at resonance when the measure in the Stieltjes integral is positive and also when this measure changes the sign. Sufficient conditions which guarantee the existence of extremal, unique and quasi-solutions are given. Three examples illustrate the results.
-
Network Approach to Increments of RR-intervals for Visualization of Dynamics of Cardiac Regulation
PublicationThe transition network for RR -increments is pre- sented in a directed and weighted graph, with vertices represent- ing RR -increments and edges corresponding to the order in a sequence of increments. The adjacency matrix and the transition matrix of this network provide a graphical tool which could be useful in the assessment of cardiac regulation. As an example, the method is applied in detecting differences between diurnal activity...
-
On the convergence of a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation
PublicationIn this note, we establish the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The numerical method was previously investigated in the literature and, amongst other features of interest, it is a fast and nonlinear technique that is capable of preserving positivity, boundedness and monotonicity....
-
Ordinal Pattern Statistics for RR Intervals during Head-Up Tilt Test in Patients with the History of Vasovagal Syncope
PublicationWe apply ordinal pattern analysis to quantify differences in distribution of patterns of length 3 and 4 in basal state and during head-up tilt test (HUTT) in patients with the history of syncope and positive (HUTT(+)) or negative (HUTT(-)) responses to the test. We identify the patterns related to prevalence of sympathetic or parasympathetic cardiac modulation as well as describe the relations between the response to the test and...
-
Positive solutions to fractional differential equations involving Stieltjes integral conditions
PublicationIn this paper, we investigate nonlocal boundary value problems for fractional differential equations with dependence on the first-order derivatives and deviating arguments. Sufficient conditions which guarantee the existence of at least three positive solutions are new and obtained by using the Avery–Peterson theorem. We discuss problems (1) and (2) when argument b can change the character on [0, 1], so in some subinterval I of...
-
Positive solutions to Sturm–Liouville problems with non-local boundary conditions
PublicationIn this paper, the existence of at least three non-negative solutions to non-local boundary-value problems for second-order differential equations with deviating arguments α and ζ is investigated. Sufficient conditions, which guarantee the existence of positive solutions, are obtained using the Avery–Peterson theorem. We discuss our problem for both advanced and delayed arguments. An example is added to illustrate the results.
-
Temporal Changes in Complexity of Cardiovascular Regulation during Head-Up Tilt Test by Entropic Measures of Fluctuations of Heart Period Intervals and Systolic Blood Pressure
PublicationTemporal changes in complexity of cardiovascular regulation during head-up tilt test by entropic measures of fluctuations of heart period intervals and systolic blood pressure
-
The finite-difference simulation of x-rays propagation through a system of lenses
PublicationThe propagation of X-ray waves through an optical system consisting of 33 aluminum X-ray refractive lenses is considered. For solving the problem, a finite-difference method is suggested and investigated. It is shown that very small steps of the difference grid are necessary for reliable computation of propagation of X-ray waves through the system of lenses. It is shown that the wave phase is a function very quickly increasing...
Year 2013
-
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...
-
Centrum Zastosowań Matematyki
PublicationProjekt Centrum Zastosowan Matematyki realizowany jest przy Wydziale Fizyki Technicznej i Matematyki Stosowanej Politechniki Gdanskiej od wrzesnia 2013 do sierpnia 2015 roku. W drodze konkursu organizowanego przez Narodowe Centrum Badan i Rozwoju uzyskał on dofinansowanie w wysokosci niemal dwa miliony złotych ze srodków Europejskiego Funduszu Społecznego. Zadania projektu obejmuja: stworzenie platformy internetowej, organizacje cyklu...
-
Existence of solutions for a coupled system of difference equations with cousal operators
PublicationPraca dotyczy układów równań różnicowych. Podano warunki dostateczne na istnienie rozwiązań takich problemów. Badano również nierówności różnicowe.
-
Existence results to delay fractional differential equations with nonlinear boundary conditions
PublicationPraca dotyczy problemów brzegowych dla ułamkowych równań różniczkowych z opóźnionym argumentem. Podano warunki dostateczne na istnienie rozwiązań ekstremalnych takich zagadnień.
-
Fractional equations of Volterra type involving a Riemann Liouville derivative
PublicationIn this paper, we discuss the existence of solutions of fractional equations of Volterra type with the Riemann Liouville derivative. Existence results are obtained by using a Banach fixed point theorem with weighted norms and by a monotone iterative method too. An example illustrates the results.
-
Generalized quazilinearization for systems of degenerate singular perturbation problem
PublicationPraca dotyczy ogólnej metody kwazilinearyzacji dla układów równań różniczkowych z parametrami.
-
Green function diagonal for a class of heat equations
PublicationA construction of the heat kernel diagonal is considered as element of generalized zeta function theory, which gradient at the origin defines determinant of a differential operator in a technique for regularizing quadratic path integral. Some classes of explicit expressions of the Green function in the case of finite-gap potential coefficient of the heat equation are constructed. An algorithm and program for Mathematica are presented...
-
Initial value problems for neutral fractional differential equations involving a Riemann-Liouville derivative
PublicationBadano równania neutralne typu ułamkowego z odchylonym argumentem. Podano warunki dostateczne na istnienie jednego rozwiązania.
-
Method of lines for Hamilton-Jacobi functional differential equations.
PublicationInitial boundary value problems for nonlinear first order partial functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A method of quasi linearization is adopted. Suffcient conditions for the convergence of the method of lines and error estimates for approximate solutions are presented. The proof of the stability of the diffrential difference...
-
Method of lines for nonlinear first order partial functional differential equations.
PublicationClassical solutions of initial problems for nonlinear functional differential equations of Hamilton--Jacobi type are approximated by solutions of associated differential difference systems. A method of quasilinearization is adopted. Sufficient conditions for the convergence of the method of lines and error estimates for approximate solutions are given. Nonlinear estimates of the Perron type with respect to functional variables...
-
Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives
PublicationStosując tw. Avery-Petersona o punkcie stałym, podano warunki dostateczne na istnienie nieujemnych rozwiązań dla układów równań różniczkowych rzędu drugiego z argumentami opóźnionymi i wyprzedzonymi oraz warunkami brzegowymi zawierającymi całki Stieltjesa. Praca zawiera wiele przykładów.
-
O Centrum Zastosowań Matematyki
PublicationCentrum Zastosowań Matematyki to projekt realizowany w ramach Programu Operacyjnego Kapitał Ludzki wyłoniony w drodze konkursu zorganizowanego przez Narodowe Centrum Badań i Rozwoju.
-
Ordinal pattern statistics for the assessment of heart rate variability
PublicationThe recognition of all main features of a healthy heart rhythm (the so-called sinus rhythm) is still one of the biggest challenges in contemporary cardiology. Recently the interesting physiological phenomenon of heart rate asymmetry has been observed. This phenomenon is related to unbalanced contributions of heart rate decelerations and accelerations to heart rate variability. In this paper we apply methods based on the concept...
-
Path components of the space of gradient vector fields on the two dimensional disc
PublicationWe present a short proof that if two gradient maps on the twodimensional disc have the same degree, then they are gradient homotopic.
-
Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions
PublicationIn this paper, we consider the existence of positive solutions for second-order differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundary value problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments α and also in the case when α(t)=t, t∈[0,1]....
-
Pseudopotentials via Moutard Transformations and Differential Geometry
Publication..