﻿ Pliska Studia Mathematica Volume 26, 2016

### Pliska Studia Mathematica

Volume 26, 2016

Proceedings of the Third International conference on New Trends of the Applications of Differential Equations in Sciences (NTADES 2016) in Cooperation of Society of Industrial and Applied Mathematics (SIAN)

GUEST EDITOR: A. Slavova

Sofia, 2016

#### A B S T R A C T S

THE SIS-MODEL ON TIME SCALES
Martin Bohner bohner@mst.edu,
Sabrina H. Streipert shsbrf@mst.edu

2010 Mathematics Subject Classification: 34N05, 92D25.
Key words: dynamic equations, time scales, epidemic models, stability.

In this paper, we introduce the epidemic model following the hypothesis of the disease flow SusceptibleInfectedSusceptible, short SIS, on time scales. After a brief introduction of time scales, we present dynamic systems representing the SIS-model on time scales and derive its solution sets. We are discussing the stability of the steady states before investigating a modified SIS-model including a birth and death rate. Throughout, examples are used to illustrate the results.

ON WAVE OPERATOR IMAGE OF PERTURBED HAMILTONIANS ON THE LINE
Anna Rita Giammetta anna_rita_giammetta@hotmail.it

2010 Mathematics Subject Classification: 35Q40, 35R15, 42B37.
Key words: Homogeneous Sobolev norms, Paley Littlewood decomposition. Elliptic estimates, Laplace operator with potential.

We consider the following perturbed Hamiltonian H= −∂x2 + V(x) on the real line. The potential V(x), satisfies a short range assumption of type
(1+|x|)γ V(x) ∈ L1(R),     γ > 1.
We study the wave operator images of classical homogeneous Sobolev type spaces [\dot H]sp(R), p ∈ (1, ∞). It is shown that the assumption zero is not a resonance guarantees that the corresponding wave operators leave classical homogeneous Sobolev spaces of order s ∈ [0, 1/p) invariant.

NEW TYPES OF TWO COMPONENT NLS-TYPE EQUATIONS
V. S. Gerdjikov vgerdjikov@math.bas.bg,
A. A. Stefanov aleksander.a.stefanov@gmail.com

2010 Mathematics Subject Classification: 35Q15, 31A25, 37K10, 35Q58.
Key words: inverse scattering transform, multi-component NLS equations, Lax representation, the group of reductions.

We study MNLS related to the D.III-type symmetric spaces. Applying to them Mikhailov reduction groups of the type Zr×Z2 we derive new types of 2-component NLS equations. These are not counterexamples to the Zakharov-Schulman theorem because the corresponding interaction Hamiltonians depend not only on |uk|2, but also on u1u2* + u1*u2.

BIFURCATIONS OF PERIODIC SOLUTIONS TO DIFFERENTIAL EQUATIONS AND MULTIVALENT GUIDING FUNCTIONS METHOD
Valeri Obukhovskii valerio-ob2000@mail.ru,
Pietro Zecca kornev_vrn@rambler.ru,
Nguyen Van Loi zecca@unifi.it,
Sergei Kornev nguyenvanloi1@tdt.edu.vn

2010 Mathematics Subject Classification: 34C23, 34C25.
Key words: multivalent guiding function, global bifurcation, periodic solution.

In this paper, we define a new class of multivalent guiding functions called local multivalent guiding functions and use it to study the global bifurcation problem of periodic soluions to a parameterized differential equation.

BOUNDARY-VALUE PROBLEMS FOR PDES ARISING IN~THE VALUATION OF STRUCTURED FINANCIAL PRODUCTS
Rossella Agliardi rossella.agliardi@unibo.it

2010 Mathematics Subject Classification: 35K20, 91G20, 91G80.
Key words: Boundary value problems, Black-Scholes PDE, structured bonds.

We explicitly solve some mixed initial/boundary value problems for generalized Black-Scholes PDEs with financially relevant boundary conditions. As an illustration, new pricing formulas are obtained for convertible and reverse convertible bonds under credit risk.

MODELS OF INTERNAL WAVES IN THE PRESENCE OF CURRENTS
Alan Compelli alan.compelli@dit.ie,
Rossen Ivanov rossen.ivanov@dit.ie

2010 Mathematics Subject Classification: 35Q35, 37K05, 74J30.
Key words: Internal waves, linear equations, wave-current, Hamiltonian system, KdV equation.

A fluid system consisting of two domains is examined. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. An internal wave propagating in one direction, driven by gravity, acts as a free common interface between the fluids. Various current profiles are considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are formulated. The presented models provide potential applications to modelling of internal geophysical waves.

NUMERICAL SIMULATION OF THE FLOW AROUND TWO BLUFF BODIES SEPARATED BY A GAP
Sonia Tabakova stabakova@gmail.com

2010 Mathematics Subject Classification: 76D05, 76M12.
Key words: numerical solution, hydrodynamic interaction, drag coefficient.

The aim of the present work is to study numerically the shielding effects of a semi-infinite cylinder and a protruding disk placed in an incompressible viscous flow. The continuity and momentum equations are solved numerically for different flow regimes: laminar and turbulent using the CFD software ANSYS/FLUENT. Different geometrical parameters (diameters and gap aspect ratios) and a wide spectrum of the Reynolds numbers: 5 ≤ Re ≤ 5 × 105 are considered. The results for the drag force coefficient and the axial velocity patterns are compared with the classical experimental results of Koening and Roshko [An experimental study of geometrical effects on the drag and flow field of two bluff bodies separated by a gap. J. Fluid Mech. 156 (1985), 167−204] at Re = 5 × 105.

GLOBAL SOLVABILITY TO DOUBLE DISPERSION EQUATION WITH BERNOULLI TYPE NONLINEARITY VIA ONE PARAMETRIC FAMILY OF POTENTIAL WELLS
N. Kutev kutev@math.bas.bg,
M. Dimova mdimova@unwe.bg,
N. Kolkovska natali@math.bas.bg,

2010 Mathematics Subject Classification: 35L30, 35L75.
Key words: double dispersion equation, global existence, potential well method.

One parametric family of potential wells for double dispersion equation with Bernoulli type nonlinearity is introduced. Sign preserving properties of the Nehari functionals are obtained. Global existence of the weak solution to the Cauchy problem is proved for wider class of initial data than the corresponding ones in the classical potential well method.

ON THE NUMERICAL SOLUTION OF THE GENERAL "LIGAND--GATED NEURORECEPTORS MODEL" VIA CAS MATHEMATICA
Nikolay Kyurkchiev nkyurk@math.bas.bg

2010 Mathematics Subject Classification: 92B05, 92E10, 92C40, 92C45.
Key words: Neuroreceptor model, neurotransmitter concentration, bipartite graph of the kinetic "ligand--gated neureceptors" model, open state concentration solution.

We present a software module for analysis of the general "ligand--gated neuroreceptors model" (GLGNM) within the programming environment of CAS Mathematica. Numerical examples which demonstrate scientific applications and visualization properties of the module are presented.

CONTROL OF CHAOTIC BEHAVIOR OF~INTEGRO-DIFFERENTIAL CNN MODEL ARISING IN PIEZOELECTRIC MATERIAL WITH NANO-HETEROGENEITIES
Elena Litsyn elena@cs.bgu.ac.il,
Angela Slavova slavova@math.bas.bg

2010 Mathematics Subject Classification: 92B20,74F15, 35K57, 35Q68.
Key words: piezoelectric solid, Cellular Nonlinear/Nanoscale Networks (CNN), dynamic behaviour, edge of chaos, feedback control.

Piezoelectrical material with heterogeneities of nano-holes or nano-inclusions is considered in the case when it is subjected to time harmonic electro-mechanical load. The model is reduced to a system of integro-differential equations (IDE). We construct Cellular Nonlinear/Nanoscale Network (CNN) architecture for the boundary value IDE problem under consideration. For such IDE CNN model we shall determine "edge of chaos" region of the parameter set.Validation will be provided as well. Feedback control will be applied in order to stabilize the model. The computer simulations will illustrate the obtained theoretical results.

AERODYNAMIC CHARACTERISTICS OF JOUKOWSKY LIKE WINGS
Yoshihiro Mochimaru ymochima-1947@cx.117.cx

2010 Mathematics Subject Classification: 31A35, 33E05, 65N06, 76D05.
Key words: Jpukowsky airfoil, wing, numerical simulation.

Aerodynamic characteristics of Joukowsky like airfoils are analyzed, under the assumption that the flow near the airfoil is governed by two-dimensional incompressible isothermal Newtonian fluid flow, beyond the viscous boundary-layer of which is approximated by a potential flow with a new parameter. Vorticity transport equation is solved numerically, using a spectral finite difference scheme to give steady-state various characteristics.

SOLVABILITY IN CLASSICAL SENSE OF QUASI-LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS AND APPLICATION
P. Popivanov popinano@math.bas.bg,
Y. Markov ymarkov@gmail.com

2010 Mathematics Subject Classification: 35J47, 35J57.
Key words: Elliptic systems, non-cooperative system, existence, sub- and super-solution.

In this article is studied the solvability in classical C2(Ω) ∩ C(Ω)\$ sense of quasi-linear non-cooperative weakly coupled systems of elliptic second-order PDE. The main tool for the research is the method of sub- and super-solutions. The result is applied to a model example describing two dimensional non-super-conformal minimal surface M2 in R4.

DYNAMIC BEHAVIOUR OF A CRACK AT MACRO- AND NANO- SCALE IN ANISOTROPIC PLANE BY BIEM
Tsviatko Rangelov rangelov@math.bas.bg,
Petia Dineva petia@imbm.bas.bg

2010 Mathematics Subject Classification: 35Q74, 74S15, 74H35.
Key words: Elastodynamics, Macro/Nano-crack, Plain strain state, General anisotropy, Frequency domain, BIEM, SCF, SIF.

The two-dimensional ‘in-plane’ time-harmonic elastodynamic problem for anisotropic infinite plane with a crack at macro and nano scale subjected to incident plane longitudinal P or shear SV wave is studied. The continuum mechanics model of surface elasticity proposed by Gurtin and Murdoch [A continuum theory of elastic material surfaces. Arch. Ration. Mach. Anal. 57 (1975), 291−323] is applied to account for the effects of surface elasticity for a crack at nano-level. The non-hypersingular traction boundary integral equation method (BIEM) is used in conjunction with closed form frequency dependent fundamental solution, obtained by Radon transform. In addition a parametric study for the dynamic stress intensity factor (SIF) and stress concentration field (SCF) sensitivity to the frequency, crack-size, surface effects and material anisotropy is presented.

A GENERALIZED MOUNTAIN PASS THEOREM
Hans-Jörg Ruppen hans-joerg.ruppen@epfl.ch

2010 Mathematics Subject Classification: 58E05, 58E30.
Key words: Variational principles, critical points.

We present a new variational characterization of multiple critical points for even energy functionals functionals corresponding to non-linear Schrödinger equations of the following type:
−Δ u + V(x) u - q(x) |u|σ u = λ u,   (x ∈ RN)

u ∈ H1(RN) \ {0}.
We assume N ≥ 3, q(x) ∈ L(RN), q(x)> 0 a.e. with lim|x|→∞ q(x) = 0 and 0 < σ < 4/(N−2). Our results cover the following three cases in a uniform way:
1. V(x) ≡ 0;
2. V(x) is a Coulomb potential and
3. V(x) ∈ LRN with V(x+k) ≡ V(x) for all k ∈ ZN.
The eigenvalue λ &isin: R \ σ(−Δ + V) thereby may or may not lie inside a spectral gap.
Our variational characterization is "simple" and well suited for discussing multiple bifurcation of solutions.
The detailed presentation of all the results can be found in [A generalized min-max theorem for functionals of strongly indefinite sign. Calc. Var. Partial Differential Equations 50, 1−2 (2014), 231--255], [Odd linking and bifurcation in gaps: the weakly indefinite case. Proc. Roy. Soc. Edinburgh Sect. A 143, 5 (2013), 1061−1088] and [A generalized mountain-pass theorem: the existence of multiple critical points. Calc. Var. Partial Differential Equations 55, 5 (2016), 55--89].

DYNAMICS OF HYSTERESIS CNN WITH MEMRISTOR SYNAPSES
Angela Slavova slavova@math.bas.bg,
Ronald Tetzlaff Ronald.Tetzlaff@tu-dresden.de

2010 Mathematics Subject Classification: 92B20, 34C55, 34C28.
Key words: hysteresis Cellular Nonlinear Networks (CNN), dynamic behaviour, memristor.

In this paper a new hysteresis Cellular Nonlinear Networks (CNN) model will be studied in which we shall introduce memristor in the synapses. Dynamics of such model will be investigated. Local activity theory will be applied in order to determine the edge of chaos domain of the parameter set in which the model under consideration can exhibit complexity. Simulations and applications will be provided.

A NEW APPROACH TO IMPROVE THE NUMERICAL PROCEDURE FOR 2D ANTI-PLANE CRACK PROBLEMS IN FUNCTIONALLY GRADED MAGNETO-ELECTRO-ELASTIC MATERIALS
Yonko Stoynov ids@tu-sofia.bg

2010 Mathematics Subject Classification: 65M38, 65N80, 65Z05.
Key words: functionally graded magneto-electro-elastic materials, Radon transform, Fourier transform, Kelvin functions, SIF, BIEM.

Magneto-electro-elastic composite materials have extensive applications in modern smart structures, because they possess good coupling between mechanical, electrical and magnetic fields. This new effect was reported for the first time by Van Suchtelen [Product properties: a new application of composite materials. Phillips Research Reports 27 (1972), 28−37] in 1972. Due to their ceramic structure cracks inevitably exist in these materials. If these cracks extend the material may lose its structural integrity and/or functional properties.
In this study we consider functionally graded magneto-electro-elastic materials (MEEM) subjected to anti-plane time-harmonic load. Our purpose is to evaluate the dependence of the stress concentration near the crack tips on the frequency of the applied external load. We use boundary integral equation method (BIEM) for the numerical solution.
For materials with complex geometry of cracks the numerical procedure becomes too cumbersome. To increase the speed of the computations we derive new fundamental solutions by the Fourier transform. The more simple form of these fundamental solutions leads to decreasing of the number of the numerical computations. Asymptotic for small arguments of the new solutions will be presented. The results can be used to improve the numerical procedure based on the BIEM for complex crack configurations in MEEM.

DISCRETE DIRAC−KÄHLER EQUATION AND ITS FORMULATION IN ALGEBRAIC FORM
Volodymyr Sushch volodymyr.sushch@tu.koszalin.pl

2010 Mathematics Subject Classification: 39A12, 39A70, 81Q05.
Key words: Dirac−Kähler equation, Hestenes equation, discrete models, Clifford algebra.

A relationship between the discrete Dirac−Kähler equation and discrete analogues of some Dirac type equations in the geometric spacetime algebra is discussed. We show that a solution of the discrete Dirac−Kähler equation can be represented as the sum and difference of solutions of discrete Dirac type equations with the corresponding sign of the mass term.

SCATTERING FOR SYSTEMS OF N WEAKLY COUPLED NLS EQUATIONS ON Rd×M2 IN THE ENERGY SPACE
Mirko Tarulli mta@tu-sofia.bg,
George Venkov gvenkov@tu-sofia.bg

2010 Mathematics Subject Classification: 35J10, 35Q55, 35G50, 35P25.
Key words: Nonlinear Schrödinger systems, scattering theory, weakly coupled equations.

We study scattering properties in the energy space of the solution to the following system of N nonlinear Schrödinger equations (NLS), with N ≥ 2, posed on product spaces Rn × M2, for d ≥ 1 and M2 any 2-dimensional compact Riemaniann manifold:
i∂t uμ + (Δx + Δy) uμ + ΣNμ,ν = 1 Gμν(uμ, uν)uμ = 0,       μ = 1, ..., N,

(uμ(0,·, ·))Nμ = 1 = (uμ,0)Nμ = 1 ∈ H1(Rd×M2)N.

Here, for all μ, ν = 1,...,N, uμ = uμ(t,x,y) : R × Rd × M2C, (uμ)Nμ = 1 = (u1,..., uN), moreover we require that each function Gμν = Gμν(·, ·) : C × CC is measurable and such that
| Gμν(x,x)| ≤ βμν|x|4/d,
for any x ∈C and with βμν ≥0 being coupling parameters, for any μ, ν = 1,...,N.

ON SOLITON EQUATIONS IN CLASSICAL DIFFERENTIAL GEOMETRY
Tihomir Valchev tiv@math.bas.bg

2010 Mathematics Subject Classification: 37K25, 53A04, 53A05.
Key words: soliton equations, curve motion, isothermic surfaces.

In this survey report we shall briefly sketch certain problems from the classical differential geometry of curves and surfaces that lead to nonlinear partial differential equations known from soliton theory. Thus an alternative viewpoint on these completely integrable equations will be presented.