The Equadiff is a series of biannual conferences on mathematical analysis, numerical approximation and applications of differential equations. It is held in rotation in the Czech Republic, Slovakia and Western Europe. The last Equadiff (Equadiff 14 in the Czecho-Slovak series) was organized in Bratislava, Slovakia, July 24-28, 2017 by the Slovak University of Technology, in cooperation with Comenius University, Union of Slovak Mathematicians and Physicists, Slovak Mathematical Society and Algoritmy:SK s.r.o. The web page of the conference is: http://www.math.sk/equadiff/ During the last decades the Equadiff has clearly developed into the world platform for international exchange of ideas on all mathematical and numerical aspects of differential equations, ranging from fundamental concepts to applications. Download complete Proceedings of Equadiff 2017 Conference
Positive solutions for a system of fractional boundary value problems
We investigate the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with nonnegative nonlinearities which can be nonsingular or singular functions, subject to multi-point boundary conditions that contain fractional derivatives.
Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system
This paper is concerned with the two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics
On the Solution Set of a Nonconvex Nonclosed Second Order Inclusion
We consider a nonconvex and nonclosed second-order evolution inclusion and we prove the arcwise connectedness of the set of its mild solutions.
This paper discusses the existence theory to dynamical model of viscoplasticity and show possibility to obtain existence of solution without assuming weak safe-load condition.
Nonlinear diffusion equations with perturbation terms on unbounded domains
This paper considers the initial-boundary problem (P) for the nonlinear diffusion problem with the perturbation term
On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function
In this paper, we consider solutions to the following chemotaxis system with general sensitivity. We present a sufficient condition for the boundedness of solutions to the system and some related systems.
Viral infection model with diffusion and state-dependent delay: a case of logistic growth
We propose a virus dynamics model with reaction-diffusion and logistic growth terms, intracellular state-dependent delay and a general non-linear infection rate functional response. Classical solutions with Lipschitz in-time initial functions are investigated. This type of solutions is adequate to the discontinuous change of parameters due to, for example, drug administration. The Lyapunov functions approach is used to analyse stability of interior infection equilibria which describe the cases of a chronic disease.
Boundedness in a fully parabolic chemotaxis system with signal-dependent sensitivity and logistic term
This paper deals with the chemotaxis system with signal-dependent sensitivity and logistic term \begin{align*} &u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v) + \mu u(1-u), % \quad \\ &v_t=\Delta v + u - v \end{align*} in $\Omega\times (0,\infty)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $\mu > 0$ is a constant and $\chi$ is a function generalizing \begin{align*} \chi(s) = \frac{K}{(1+s)^2} \quad (K>0,\ s>0). \end{align*} In the case that $\mu=0$ global existence and boundedness were established under some conditions (\cite{Mizukami-Yokota_02}); however, conditions for global existence and boundedness in the above system have not been studied. The purpose of this paper is to construct conditions for global existence and boundedness in the above system.
Kolmogorov's epsilon-entropy of the attractor of the strongly damped wave equation in locally uniform spaces
We establish an upper bound on the Kolmogorov's entropy of the locally compact attractor for strongly damped wave equation posed in locally uniform spaces in subcritical case using the method of trajectories.
The Tree-Grid Method with Control-Independent Stencil
The Tree-Grid method is a novel explicit convergent scheme for solving stochastic control problems or Hamilton-Jacobi-Bellman equations with one space dimension. One of the characteristics of the scheme is that the stencil size is dependent on space, control and possibly also on time. Because of the dependence on the control variable, it is not trivial to solve the optimization problem inside the method. Recently, this optimization part was solved by brute-force testing of all permitted controls. In this paper, we present a simple modification of the Tree-Grid scheme leading to a control-independent stencil. Under such modification an optimal control can be found analytically or with the Fibonacci search algorithm.
Multiple positive solutions for a p-Laplace critical problem (p >1), via Morse theory
Propagation of errors in dynamic iterative schemes
We consider iterative schemes applied to systems of linear ordinary differential equations and investigate their convergence in terms of magnitudes of the coefficients given in the systems. We address the question of whether the reordering of equations in a given system improves the convergence of an iterative scheme.
We investigate the Lyapunov stability implying asymptotic behavior of a nonlinear ODE system describing stress paths for a particular hypoplastic constitutive model of the Kolymbas type under proportional, arbitrarily large monotonic coaxial deformations. The attractive stress path is found analytically, and the asymptotic convergence to the attractor depending on the direction of proportional strain paths and material parameters of the model is proved rigorously with the help of a Lyapunov function.
Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian
We consider a class of 1d Lagrangian systems with random forcing in the space-periodic setting. These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 15]. Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space $L_p$ for finite $p$, partially answering the conjecture formulated in \cite{GIKP05}. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin ans Zhang [13].
Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition
The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.
Numerical homogenization for indefinite H(curl)-problems
In this paper, we present a numerical homogenization scheme for indefinite, time-harmonic Maxwell's equations involving potentially rough (rapidly oscillating) coefficients. The method involves an $\mathbf{H}(\mathrm{curl})$-stable, quasi-local operator, which allows for a correction of coarse finite element functions such that order optimal (w.r.t.\ the mesh size) error estimates are obtained. To that end, we extend the procedure of [D.Gallistl, P.Henning, B.Verfurth, Numerical homogenization for H(curl)-problems, arXiv:1706.02966, 2017] to the case of indefinite problems. In particular, this requires a careful analysis of the well-posedness of the corrector problems as well as the numerical homogenization scheme.
Singularly perturbed set of periodic functional-differential equations arising in optimal control theory
We consider the singularly perturbed set of periodic functional-differential matrix Riccati equations, associated with a periodic linear-quadratic optimal control problem for a singularly perturbed delay system. The delay is small of order of a small positive multiplier for a part of the derivatives in the system. A zero-order asymptotic solution to this set of Riccati equations is constructed and justified.
Nonexistence of solutions of some inequalities with gradient nonlinearities and fractional Laplacian
We obtain necessary conditions of nonexistence of nontrivial solutions for some classes of nonlinear partial differential inequalities containing the fractional powers of the Laplace operator.
Semi-analytical approach to initial problems for systems of nonlinear partial differential equations with constant delay
This paper deals with the differential transform method for solving of an initial value problem for a system of two nonlinear functional partial differential equations of parabolic type. We consider non-delayed as well as delayed types of coupling and the different variety of initial functions are thought over. The convergence of solutions and the error estimation to the presented procedure is studied. Two numerical examples for non-delayed and delayed systems are included.
In this paper we deal with the mathematical modellng of rheological models with applications in various engineering disciplines and industry. We study the mechanical responsevisco-elasto-plastic materials. We describe the basic rheological elements and focus our attention to the specific model of concrete, for which we derive governing equations and discuss its solution. We provide an application of rheological model involving rigid-plastic element as well - mechanical and mathematical model of failure of one dimensional construction member, straight beam. Herein, the physical model is considered with a homogeneous isotropic material of the beam, quasi static regime is supposed.
Cross-Diffusion Systems with Entropy Structure
Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented.
Numerical modeling of heat exchange and unsaturated-saturated flow in porous media
We discuss the numerical modeling of heat exchange between the infiltrated water and porous media matrix. An unsaturated-saturated flow is considered with boundary conditions reflecting the external driven forces. The developed numerical method is efficient and can be used for solving the inverse problems concerning determination of transmission coefficients for heat energy exchange inside and also on the boundary of porous media. Numerical experiments support our method.
A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion
In this paper, we prove the existence and uniqueness of the solution of the initial boundary value problem for a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion together with a homogeneous Neumann boundary condition in an open bounded domain of $\mathbb{R}^n$ with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.
Vectorial quasilinear diffusion equation with dynamic boundary condition
In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)$_\varepsilon$ with a nonnegative constant $\varepsilon$, and for any $\varepsilon\ge0$, (S)$_\varepsilon$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma:=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method, which can enable us to handle the transmission systems of various vectorial mathematical models, such as the Bingham type flow equations, the Ginzburg--Landau type equations, and so on. On this basis, we set the goal of this paper to prove two Main Theorems, concerned with the well-posedness of (S)$_\varepsilon$ with the precise representation of solution, and $\varepsilon$-dependence of (S)$_\varepsilon$, for $\varepsilon \ge0$.
Remarks on the qualitative behavior of the undamped Klein-Gordon equation
We consider the undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions. For any real value of the initial energy, particularly for supercritical values of the energy, we give sufficient conditions to conclude blow up of weak solutions. The success of the analysis is based in a careful analysis of a differential inequality. Our results improve previous ones in the literature. Our result is extended to a general abstract non-dissipative wave equation.
Two approaches for the approximation of the nonlinear smoothing term in the image segmentation
Purpose of the paper is to study nonlinear smoothing term initiated in [3], [4],[6] and [7] for problems of image segmentation and missing boundaries completion. The generalization of approach presented in [1] is proposed and applied in the field of image segmentation. So called regularised Riemannian mean curvature flow equation is studied and the construction of the numerical scheme based on the finite volume method approach is explained. The principle of the level set, for the first time given in [2], is used. We mention two different approaches for the approximation of the nonlinear smoothing term in the equation and known theoretical results for both of them. We provide the numerical tests for both schemes. It the last section we discuss obtained results and propose possibilities for the future research.
Stability of ALE space-time discontinuous Galerkin method
We assume the heat equation in a time dependent domain, where the evolution of the domain is described by a given mapping. The problem is discretized by the discontinuous Galerkin (DG) method in space as well as in time with the aid of Arbitrary Lagrangian-Eulerian (ALE) method. The sketch of the proof of the stability of the method is shown.
Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds
We prove a generalization of the Douady-Oesterl\'{e} theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.
Gaussian curvature based tangential redistribution of points on evolving surfaces
There exist two main methods for computing a surface evolution, level-set method and Lagrangian method. Redistribution of points is a crucial element in a Lagrangian approach. In this paper we present a point redistribution that compress quads in the areas with a high Gaussian curvature. Numerical method is presented for a mean curvature flow of a surface approximated by quads.
Computational design optimization of low-energy buildings
European directives and related national technical standards force the substantial reduction of energy consumption of all types of buildings. This can be done thanks to the massive insulation and the improvement of quality of building enclosures, using the simple evaluation assuming the one-dimensional stationary heat conduction. However, recent applications of advanced materials, structures and technologies force the proper physical, mathematical and computational analysis coming from the thermodynamic principles. This paper shows the non-expensive evaluation of energy consumption of buildings with controlled indoor temperature, decomposing a building, considered as a thermal system, into particular subsystems and elements, coupled by interface thermal fluxes. We come to a rather large parabolic system of partial differential equations, containing the nonlinearities i) from the surface Stefan-Boltzmann radiation and ii) from the heating control; this can be handled using some properties of semilinear systems. The Fourier multiplicative decomposition, together with the finite element technique enables us to derive a sparse system of ordinary differential equations, appropriate for the input of climatic data (temperature, beam and diffuse solar radiation). For the approximate solutions the spectral analysis is helpful; all nonlinearities can be overcome thanks to quasi-Newton iterations. All above sketched simulations have been implemented in MATLAB. An example shows the validation of this approach, utilizing the time series of measured energy consumption from the real family house in Ostrov u Macochy(30 km northern from Brno). Additional procedures for the support of design of low-energy buildings come namely from the Nelder- Mead optimization algorithm.
A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint
We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than (8\pi)^2, whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known 8\pi​ problem in the Keller--Segel system to higher dimensions. We will compare mathematical structures of the Keller--Segel system and our system and discuss the difference.
A Note on the Uniqueness and Structure of Solutions to the Dirichlet Problem for Some Elliptic Systems
In this note, we consider some elliptic systems on a smooth domain or the whole space. By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.
Classical and generalized Jacobi polynomials orthogonal with different weight functions and differential equations satisfied by these polynomials
In this contribution we deal with classical Jacobi polynomials orthogonal with respect to different weight functions, their special cases - classical Legendre polynomials and generalized brothers of them. We derive expressions of generalized Legendre polynomials and generalized ultraspherical polynomials by means of classical Jacobi polynomials.
Stochastic Modulation Equations on Unbounded Domains
We study the impact of small additive space-time white noise on nonlinear SPDEs on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. One technical problem for establishing error estimates in the stochastic case rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time the error is always very large somewhere far out in space. Thus we have to work in weighted spaces that allow for growth at infinity. As a first example we study the stochastic one-dimensional Swift-Hohenberg equation on the whole real line. In this setting, because of the weak regularity of solutions, the standard methods for deterministic modulation equations fail, and we need to develop new tools to treat the approximation. Using energy estimates we are only able to show that solutions of the Ginzburg-Landau equation are Holder continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates.
An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems
This paper deals with nonlinear diffusion problems which include the Stefan problem, the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages, namely, it is very easy-to-implement, the ensuing linear algebraic systems are symmetric, it requires low computational cost, the accuracy is comparable to that of the well-studied nonlinear schemes, the computation is much faster than the nonlinear schemes to obtain the same level of accuracy. In this paper, numerical experiments are carried out to demonstrate efficiency of the proposed scheme.
Continuous dependence for BV-entropy solutions to strongly degenerate parabolic equations with variable coefficients
We consider the Cauchy problem for degenerate parabolic equations with variable coefficients. The equation has nonlinear convective term and degenerate diffusion term which depends on the spatial and time variables. In this paper, we prove the continuous dependence for entropy solutions in the space BV to the problem not only initial function but also all coefficients.
Numerical study on the blow-up rate to a quasilinear parabolic equation
In this paper, we consider the blow-up solutions for a quasilinear parabolic partial differential equations ut = u2 (uxx + u). We numerically investigate the blow-up rates of these solutions by using a numerical method which is recently proposed by the authors [3].
In this paper we describe two methods for optical flow estimation between two images. The both methods are based on the backward tracking of characteristics for advection equation and the difference is on the choice of advection vector field. We present numerical experiments on 2D data of cell nucleus.
Mathematically Modelling The Dissolution Of Solid Dispersions
A solid dispersion is a dosage form in which an active ingredient (a drug) is mixed with at least one inert solid component. The purpose of the inert component is usually to improve the bioavailability of the drug. In particular, the inert component is frequently chosen to improve the dissolution rate of a drug that is poorly soluble in water. The construction of reliable mathematical models that accurately describe the dissolution of solid dispersions would clearly assist with their rational design. However, the development of such models is challenging since a dissolving solid dispersion consitutes a non-ideal mixture, and the selection of appropriate forms for the activity coefficients that describe the interaction between the drug, the inert matrix, and the dissolution medium is delicate. In this paper, we present some preliminary ideas for modelling the dissolution of solid dispersions.
Toward a mathematical analysis for a model of suspension flowing down an inclined plane
We consider the Riemann problem of the dilute approximation equations with spatiotemporally dependent volume fractions from the full model of suspension, in which the particles settle to the solid substrate and the clear liquid film flows over the sediment [Murisic et al., J. Fluid. Mech. 717, 203–231 (2013)]. We present a method to find shock waves, rarefaction waves for the Riemann problem of this system. Our method is mainly based on [Smoller, Springer-Verlag, New York, second edition, (1994)].
Behaviour of the support of the solution appearing in some nonlinear diffusion equation with absorption
Numerical experiments suggest interesting properties in the several fields of fluid dynamics, plasma physics and population dynamics. Among such properties, we may observe the interesting phenomena; that is, the repeated appearance and disappearance phenomena of the region penetrated by the fluid in the flow through a porous media with absorption. The model equation in two dimensional space is written in the form of the initial-boundary value problem for a nonlinear diffusion equation with the effect of absorption. In this paper we show some numerical examples and prove such phenomena.
An elementary proof of asymptotic behavior of solutions
We provide an elementary proof of the asymptotic behavior of solutions of second order differential equations without successive approximation argument.
Nonlinear Tensor Diffusion in Image Processing
The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. First we briefly describe the diffusion model and provide its basic properties. Further we build a semi-implicit finite volume scheme for the above mentioned model with the help of a co-volume mesh. This method is well-known as diamond-cell method owing to the choice of co-volume as a diamond-shaped polygon. The scheme design is shown for $2$D as well as $3$D case, cf. [3,4]. Then the convergence and error estimate analysis for $2$D scheme is presented, cf. [3,2]. The convergence proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in tangential direction by means a gradient in normal direction. We proved that the error of the numerical solution in $L^2$-norm is of order $h$, where $h$ is a spatial step under the natural relation $k \approx h^2$ with a time step $k$. Last part is devoted to results of computational experiments. They confirm the usefulness this diffusion type especially as pre-processed algorithm. Let us note that this numerical technique was successfully applied within the framework of EU projects in such a way for the structure segmentation in zebrafish embryogenesis, cf [5].
New efficient numerical method for 3D point cloud surface reconstruction by using level set methods
In this article, we present a mathematical model and numerical method for surface reconstruction from 3D point cloud data, using the level-set method. The presented method solves surface reconstruction by the calculation of the distance function to the shape, represented by the point cloud, using the so called Fast Sweeping Method, and the solution of advection equation with curvature term, which creates the evolution of an initial condition to the final state. A crucial point for efficiency is a construction of initial condition by a simple tagging algorithm which allows us also to highly speed up the numerical scheme when solving PDEs. For the numerical discretization of the model we suggested a novel unconditionally stable method, in which the semi-implicit co-volume scheme is used in curvature part and implicit upwind scheme in advective part. The method was tested on representative examples and applied to real data representing the historical and cultural objects scanned by 3D laser scanners.
Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter
We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.