Pliska Studia Mathematica

Volume 25, 2015

Proceedings of International conference on New Trends of the Applications of Differential Equations in Sciences (NTADES 2015)


GUEST EDITOR: A. Slavova

Sofia, 2015

C O N T E N T S

• Slavova, A. Preface (pp. 3−4)
  
  Plenary Talks
  
  
• Compelli, A., R. Ivanov. Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid (pp. 7−18)
• Georgiev, V., G. Venkov. Optimal Interpolation Constant for the Generalized Schrödinger–Newton System (pp. 19−28)
• Guo, Q., M. W. Wong. Heat Kernels and Green Functions of Some Pseudo-Differential Operators in Relativistic Quantum Mechanics (pp. 29−40)
• Tabakova, S. A Study on the Stability of some Free Films (pp. 41−54)
  
  Contributed Talks
  
  
• Akça, H., V. Covachev, Z. Covacheva. Existence of a Mild Solution to a Second-Order Impulsive Functional-Differential Equation with a Nonlocal Condition (pp. 57−66)
• Fabricant, A., N. Kutev, T. Rangelov. Improved Hardy Inequality and Applications (pp. 67−80)
• Kulikov, A., D. Kulikov. Bifurcations in Kuramoto-Syvashinsky equation (pp. 81−90)
• Kutev, N., M. Dimova, N. Kolkovska. Application of the Improved Concavity Method to Sixth Order Boussinesq Equations with Arbitrary High Initial Energy (pp. 91−104)
• Kyurkchiev, V. On a Modification of the Model of Phillips for Stabilization Control and Adequate Intervention of the Authorized Body Overseeing the Implementation of the Projects of the Operational Programme (pp. 105−110)
• Mochimaru, Y., D. Akita. Heat Transfer Analysis from an Elliptic Cylinder at Moderately High Reynolds Number Flows (pp. 111−118)
• Petrova, T., E. Kirilova, W. Becker, J. Ivanova. Monitoring of Adhesive Joint Used in Lightweight Devices (pp. 119−128)
• Popivanov, P. On a Nonstandard Boundary Value Problem for the Laplace Operator in the Plane (pp. 129−136)
• Popivanov, P., G. Boyadzhiev, Y. Markov. Existence of Classical Solutions of Quasi-linear Non-cooperative Elliptic Systems (pp. 137−146)
• Raeva, E., V. Pavlov. Modeling Issues of the Claim Process and Insurance Risk (pp. 147−154)
• Rangelov, T., S. Parvanova, P. Dineva. 2D Elastodynamic Problems for Anisotropic Solids with Defects at Macro- and Nano- Scale by Integro-Differential Equations (pp. 155−166)
• Shayganmanesh, A., A. Saeedi. Stability and Accuracy of RBF Direct Method for Solving a Dynamic Investment Model (pp. 167−174)
• Slavova, A., N. Kyrkchiev. On an Implementation of α-Subordinated Brownian Motion and Option Pricing with and without Transaction Costs via CAS MATHEMATICA (pp. 175−182)
• Slavova, A., M. Markova. CNN Modelling of Nano-Inclusions (pp. 183−192)
• Stoynov, Y. The Influence of Inhomogeneity on the Dynamic Behavior of Functionally Graded Magneto-electro-elastic Materials with Cracks (pp. 193−202)
• Valchev, T. On Solutions of the Rational Type to Multicomponent Nonlinear Equations (pp. 203−212)
• Vasileva, D., I. Bazhlekov, E. Ayryan, E. Bazhlekova. A Compact Alternating Direction Implicit Scheme for Two-dimensional Fractional Oldroyd-B Fluids (pp. 213−224)

A B S T R A C T S

HAMILTONIAN APPROACH TO INTERNAL WAVE-CURRENT INTERACTIONS IN A TWO-MEDIA FLUID WITH A RIGID LID
Alan Compelli,
Rossen Ivanov rossen.ivanov@univie.ac.at

2010 Mathematics Subject Classification: 35Q35, 37K05, 74J30.
Key words: Internal waves, vorticity, current, shear flow, Hamiltonian system.

We examine a two-media 2-dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface with wind generated surface waves but considered bounded above by a lid by an assumption that surface waves have negligible amplitude. An internal wave driven by gravity which propagates in the positive x-direction acts as a free common interface between the media. The current is such that it is zero at the flatbed but a negative constant, due to an assumption that surface winds blow in the negative x-direction, at the lid. We are concerned with the layers adjacent to the internal wave in which there exists a depth dependent current for which there is a greater underlying than overlying current. Both media are considered incompressible and having non-zero constant vorticities. The governing equations are written in canonical Hamiltonian form in terms of the variables, associated to the wave (in a presence of a constant current). The resultant equations of motion show that wave-current interaction is influenced only by the current profile in the 'strip' adjacent to the internal wave.

OPTIMAL INTERPOLATION CONSTANT FOR THE GENERALIZED SCHRÖDINGER−NEWTON SYSTEM
Vladimir Georgiev georgiev@dm.unipi.it,
George Venkov gvenkov@tu-sofia.bg

2010 Mathematics Subject Classification: 35A05, 35A15, 35Q51, 35Q55.
Key words: generalized Choquard equation, Schrödinger−Newton system, radially symmetric solutions, energy minimizers, shooting method.

In the present article we prove non-existence of radial solutions to the generalized Choquard equation of the form
Δ u(x) + ω u(x) = (∫_R³(|u(y)|^pdy)/(4π|y|) − ∫_R³(|u(y)|^p dy)/(4π|x-y|)) |u(x)|^p−2 u(x)
for 2 < p < 7/3 and ω > 0. The solutions can be associated with solutions to the Schrödinger−Newton system in R³
Δ u(x) + ω u(x) = A(x)|u(x)|^p−2 u(x)
Δ A(x) = |u(x)|^p,
with a prescribed asymptotic behavior
lim_{|x| → ∞} A(x) = ∫_R³ (|u(y)|^p dy)/(4π|y|)
at infinity. Using the Kato result for the absence of embedded eigenvalues for short-range potential perturbations of the Laplace operator we show that any H¹ radial solution to the generalized Choquard equation is identically zero. Further, we propose a variational problem that will lead to generalized Choquard equation of the form
Δ u(x) + ω u(x) = (δ ∫_R³(|u(y)|^p dy)/(4π|y|) − ∫_R³(|u(y)|^p dy)/(4π |x-y|)) |u(x)|^p−2 u(x)
for 2 < p < 7/3, δ ∈ [0, 1/2) and ω > 0. The variational setting will give a radial decreasing H¹_rad solution to this equation.

HEAT KERNELS AND GREEN FUNCTIONS OF SOME PSEUDO-DIFFERENTIAL OPERATORS IN RELATIVISTIC QUANTUM MECHANICS
Qiang Guo pangpangguo@gmail.com,
M. W. Wong mwwong@mathstat.yorku.ca

2010 Mathematics Subject Classification: Primary: 47G30, 81Q10.
Key words: relativistic Hamiltonian, pseudo-differential operator, Poisson kernel, Bessel potential, Bessel−Poisson kernel, Laplace transform, heat semigroup, L^p-Lr estimates, upper incomplete gamma function, photon, Schrödinger kernel, free propagator, powers of the Hermite operator, traces.

A new formula for the Green function of the pseudo-differential operator (− Δ + m²)^½ on Rⁿ, where m > 0, is derived using its heat kernel, which can be used to obtain L^p-Lr estimates of the heat semigroup generated by (− Δ + m²)^½ on Rⁿ. The Bessel potential of order 1 is then shown to be equal to the Laplace transform of an upper incomplete gamma function. An explicit formula is given for the kernel of the Schrödinger semigroup generated by the Hamiltonian of a free photon moving in Rⁿ. In the case when m = 1, the trace of the heat semigroup and the zeta function regularized trace of the inverse of the operator (− Δ + 1 + |x|²)^s, s > 0, are also given.

A STUDY ON THE STABILITY OF SOME FREE FILMS
Sonia Tabakova tabakova@gmail.com

2010 Mathematics Subject Classification: 76E30, 76E17, 76A05, 76A20.
Key words: variational inequalities, semi-coercive bilinear forms, recession cone.

In this work the special case of free films of liquid (bounded by two interfaces between liquid and gas or liquid and two other liquids) is considered. The films are assumed to be viscous (Newtonian or non-Newtonian), with fully mobile interfaces (with unknown velocity, but with given shear stresses on the interfaces), laterally bounded (with fixed film thickness or fixed thickness gradient on the lateral boundaries), with planar located mid-surface (symmetric interfaces with respect to the middle plane). Additional action of different forces on the interfaces is applied, such as capillary forces (through a constant or temperature/surfactant dependent surface tension), van der Waals forces, etc.
Typical solutions of the nonlinear evolution equations are discussed. In the cases when the film static shapes exist (which depends on the combination of the different parameters), a linear and nonlinear stability analysis is also presented for them, when squeezing perturbations are applied on the shape itself, on the velocity or on the temperature, etc.

EXISTENCE OF A MILD SOLUTION TO A SECOND-ORDER IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATION WITH A NONLOCAL CONDITION
Haydar Akça haydar.akca@adu.ac.ae,
Valéry Covachev vcovachev@hotmail.com,
Zlatinka Covacheva zkovacheva@hotmail.com

2010 Mathematics Subject Classification: 34A37, 34G20.
Key words: impulse effect, nonlocal condition, cosine family.

An abstract second-order semilinear functional-differential equation such that the linear part of its right-hand side is given by the infinitesimal generator of a strongly continuous cosine family of bounded linear operators, and provided with impulse and nonlocal conditions is studied. Under not too restrictive conditions the existence of a mild solution is proved using Schauder's fixed point theorem.

IMPROVED HARDY INEQUALITY AND APPLICATIONS
Alexander Fabricant,
Nikolai Kutev,
Tsviatko Rangelov rangelov@math.bas.bg

2010 Mathematics Subject Classification: 26D10.
Key words: Hardy inequality, first eigenvalue of p−Laplacian.

New Hardy type inequality with double singular kernel in a bounded domain Ω ∈ Rⁿ is proved. When Ω is an annulus or a ball, a generalization of the well known Hardy inequalities with kernels singular only on the boundary or at the origin is given. The inequality is is with optimal constant and has an additional positive term depending on |∇ u|.
As an application of the improved Hardy inequality a new analytical lower bound for the first eigenvalue of the p−Laplacian is obtained.

BIFURCATIONS IN KURAMOTO−SYVASHINSKY EQUATION
Anatoli Kulikov,
Dmitri Kulikov kulikov_d_a@mail.ru

2010 Mathematics Subject Classification: 35K35, 35B32.
Key words: Kuramoto−Sivashinsky equation, boundary-value problem, stability, bifurcation, invariant manifolds.

Kuramoto−Sivashinsky equation with periodic boundary-value conditions is considered. The stability of the homogeneous equilibrium is investigated as well as the local bifurcation of the spatially nonhomogeneous t-periodic solutions. It is shown that the two-dimensional invariant manifolds are composed of these solutions. These manifolds can be stable or unstable, but all solutions belonging to these manifolds are always unstable.
The bifurcation problem can be reduced to investigate certain system of ordinary differential equations (normal form). This normal form was constructed by a modified Krylov−Bogolubov algorithm. These normal forms can be used to explain a ripple topography induced by ion bombardment.

APPLICATION OF THE IMPROVED CONCAVITY METHOD TO SIXTH ORDER BOUSSINESQ EQUATIONS WITH ARBITRARY HIGH INITIAL ENERGY
N. Kutev kutev@math.bas.bg,
M. Dimova mkoleva@math.bas.bg,
N. Kolkovska natali@math.bas.bg,

2010 Mathematics Subject Classification: 35L30, 35L75, 35B44.
Key words: Sixth order Boussinesq equation, finite time blow up, arbitrary initial energy.

Finite time blow up of the solutions to sixth order Boussinesq equation with arbitrary positive initial energy is proved. An improved variant of the concavity method of Levine is applied. This new method allows us to derive nonexistence of global solutions under conditions on the initial data which are more general than the assumptions used in the literature.

ON A MODIFICATION OF THE MODEL OF PHILLIPS FOR STABILIZATION CONTROL AND ADEQUATE INTERVENTION OF THE AUTHORIZED BODY OVERSEEING THE IMPLEMENTATION OF THE PROJECTS OF THE OPERATIONAL PROGRAMME
Vesselin Kyurkchiev vesko123@gmail.com

2010 Mathematics Subject Classification: 97B10, 97M10, 97U70.
Key words: model of Philips, stabilization control, e-learning, learning administration, adaptive learning, DisPeL.

The hereby article has been viewed the possibility of adapting the classical model of Phillips for stabilization control and adequate intervention of authorized bodies − Observation Committees of Operational Programmes (OPs). Proposals are: software tools for adaptive e-learning, as well as specialized module in CAS MATHEMATICA.

HEAT TRANSFER ANALYSIS FROM AN ELLIPTIC CYLINDER AT MODERATELY HIGH REYNOLDS NUMBER FLOWS
Yoshihiro Mochimaru ymochima-1947@cx.117.cx,
Daisuke Akita akita@ide.titech.ac.jp

2010 Mathematics Subject Classification: 35C10, 35Q30, 65N06, 76D05, 76R05, 80M22.
Key words: heat transfer, high Reynolds number, spectral analysis, finite difference.

Heat transfer from an elliptic cylinder placed normal to a uniform flow at moderately high Reynolds numbers is analyzed, using a spectral finite difference scheme. The subcritical field near the cylinder is assumed to be governed by a laminar potential flow with parametric variables. Not only a uniform surface temperature condition (Dirichlet type) but uniform heat flux condition (Neumann type) can be supported to show good agreement with traditional experimental data.

MONITORING OF ADHESIVE JOINT USED IN LIGHTWEIGHT DEVICES
T. Petrova,
E. Kirilova,
W. Becker,
J. Ivanova ivanova@imbm.bas.bg

2010 Mathematics Subject Classification: 74B15, 34A33.
Key words: smart structure, shear lag model, interface delamination, static and electric load, environmental conditions.

The excellent performance of the shear lag method for modeling smart pre-damaged bi-material structures under static and dynamic loading lies on the obtained important analytical formulae. The authors developed this method and applied it to investigate the piezoelectric response of a smart structure consisting in a piezoelectric patch over a host layer under static load and affected by electrical load at environment conditions. The interface delamination is investigated and the analytically calculated debond length is found, which is not considered in the typical local techniques. The numerical examples are oriented to the real materials used in the solar cells and other devices. The results are presented in figures and discussed in detail.

ON A NONSTANDARD BOUNDARY VALUE PROBLEM FOR THE LAPLACE OPERATOR IN THE PLANE
Petar Popivanov popinano@math.bas.bg

2010 Mathematics Subject Classification: 35J05, 35J65, 35J67, 30D55.
Key words: Laplace operator in R², nonlinear boundary value problem, classical solution, generalized solution, Hardy H_p space.

This paper deals with a nonstandard boundary value problem(BVP) for the Laplace operator in the plane. On the boundary of a bounded simply connected domain, say the unit disk, |∇ u| = m is prescribed and it is shown that in general the corresponding bvp possesses infinitely many solutions which can be classical or generalized depending on the function m > 0, m ∈ C⁰, respectively m > 0 a.e., log m ∈ L₁, m ∈ L_p, p ≥ 1. We shortly discuss the same problem in doubly connected domain.

EXISTENCE OF CLASSICAL SOLUTIONS OF QUASI-LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
P. Popivanov popinano@math.bas.bg,
G. Boyadzhiev jds@tu-sofia.bg,
Y. Markov popinano@math.bas.bg

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

Method of sub- and super-solutions is applied in investigation of solvability in classical C²(Ω) ∩ C(Ω⁻) sense of quasi-linear non-cooperative weakly coupled systems of elliptic second-order PDE.

MODELING ISSUES OF THE CLAIM PROCESS AND INSURANCE RISK
Elitsa Raeva eraeva@uni-ruse.bg,
Velizar Pavlov vpavlov@uni-ruse.bg

2010 Mathematics Subject Classification: 62C42, 62G22.
Key words: statistical distribution, models, insurance, claim, risk.

This work presents a brief overview of some proper statistical distributions for modeling of claims and risk in general insurance. Applications of normal approximation, normal power transformation and power transformation for modeling the number and the size of claims have been considered. Some problems in terms of modern risk theory are described and their reformulation in the actuarial practice are made. There are numerical examples to determine the amount of insurance premiums in order to limit to one percent the probability of insolvency. Some comparisons were made.

2D ELASTODYNAMIC PROBLEMS FOR ANISOTROPIC SOLIDS WITH DEFECTS AT MACRO- AND NANO- SCALE BY INTEGRO-DIFFERENTIAL EQUATIONS
Tsviatko Rangelov rangelov@math.bas.bg,
Sonia Parvanova petia@imbm.bas.bg,
Petia Dineva slp_fce@uacg.bg

2010 Mathematics Subject Classification: 74J20, 74S15, 74G70.
Key words: Elastodynamics; Macro/nano-holes, inclusions, cracks; General anisotropy; Boundary integral equations; Stress concentration factor; Stress intensity factor.

The aim of the study is to propose, develop and validate an accurate and efficient boundary integral equation method (BIEM) and apply it for solution of plane dynamic problems for anisotropic composite solids with cracks, inclusions and holes at macro and nano level. The modeling approach is based on the frame of continuum mechanics, linear wave propagation theory, linear fracture mechanics and surface elasticity theory. The computational tool is displacement and non-hypersingular traction BIEM based on frequency dependent fundamental solution. The obtained results reveal the sensitivity of the dynamic stress concentrations fields to the: (a) type of the defect-crack, hole or inclusion; (b) type of the boundary condition at macro or nano scale; (c) characteristics of the dynamic load; (d) material anisotropy; (e) wave-defect, defect-defect interaction. The non-classical boundary conditions and a localized constitutive equation for the matrix-inclusion interfaces within the framework of the Gurtin-Murdoch surface elasticity theory are developed, applied, and reported for the case of isotropic media. The relevant solid matrix could be an infinite or finite-sized medium containing multiple nano-cavities and/or elastic nano-inclusions of arbitrary shape and configuration.
The application of the near-field results is in computational fracture mechanics, while the information for the scattered wave field can be used for development of new efficient non-destructive test methods for monitoring of the integrity and reliability of the composite materials and the engineering structures based on them.

STABILITY AND ACCURACY OF RBF DIRECT METHOD FOR SOLVING A DYNAMIC INVESTMENT MODEL
Ahmad Shayganmanesh golbabai@iust.ac.ir,
Ahmad Saeedi saeidi@iust.ac.ir

2010 Mathematics Subject Classification: Primary 49Mxx; Secondary 37Mxx.
Key words: Radial Basis Functions, accuracy, stability, variational problems, Dynamic Investment problem.

In this paper we consider a Dynamic investment model. In the model, firm's objective is maximizaing discounted sum of profits over an interval of time. The model assumes that firm's capital in time t increases with investment and decreases with depreciation rate that can be expressed by means of differential equation.
We propose a direct method for solving the problem based on Radial Basis Functions (RBFs). The authors describe operational matrices of RBFs and use them to reduce the variational problem to a static optimization problem which can be solved via some optimization techniques. Next, we describe some economic interpretation of the solution. Finally, the accuracy and stability of the Multiquadric (MQ), and Gaussian (GA) RBFs are illustrated by conducting some numerical experiments.

ON AN IMPLEMENTATION OF α-SUBORDINATED BROWNIAN MOTION AND OPTION PRICING WITH AND WITHOUT TRANSACTION COSTS VIA CAS MATHEMATICA
Angela Slavova slavova@math.bas.bg,
Nikolay Kyrkchiev nkyurk@math.bas.bg

2010 Mathematics Subject Classification: 91B25, 91B24, 91B02, 34K50, 65M12, 65Y20.
Key words: α-subordinated Brownian motion, self-financing delta-hedging strategy, CAS MATHEMATICA, option pricing with and without transaction costs modules

In this we suppose that the underlying of the option contract is driven by a subordinated geometric Brownian motion. Firstly, we investigate the case when there is no transaction cost during trading. We derive the pricing formula for a European option in this case. Then, we study the case with transaction costs. We apply the mean self-financing delta-hedging strategy. We develop α-subordinated Brownian motion and option pricing without transaction costs module via CAS MATHEMATICA. We obtain bounds for call and put options for various values of α. Then we propose α-subordinated Brownian motion and option pricing with and without transaction costs modules.

CNN MODELLING OF NANO-INCLUSIONS
Angela Slavova slavova@math.bas.bg,
Maya Markova maya.markova@gmail.com

2010 Mathematics Subject Classification: 92B20,74F15, 35K57, 35Q68.
Key words: Piezoelectric solid, Cellular Nonlinear Nanoscale Networks (CNN), dynamic behaviour, harmonic balance method.

Piezoelectrical material with heterogeneities of nano-inclusions is considered in the case when it is subjected to time harmonic electro-mechanical load. The model is defined by the system of two partial differential equations and the boundary conditions for the generalized stress. On the exterior boundary, boundary conditions prescribe traction on the part of the boundary and prescribe displacement on the complemented part. We construct Cellular Nonlinear/Nanoscale Network (CNN) architecture for the boundary value problem. The dynamics of the obtained CNN model is studied via harmonic balance method. Traveling wave solutions are obtained numerically. The simulations are provided which illustrate the theoretical results. The obtained results are applicable in the field of non-destructive testing and fracture mechanics of multi-functional materials and structural elements based on them.

THE INFLUENCE OF INHOMOGENEITY ON THE DYNAMIC BEHAVIOR OF FUNCTIONALLY GRADED MAGNETO-ELECTRO-ELASTIC MATERIALS WITH CRACKS
Yonko Stoynov ids@tu-sofia.bg

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

Functionally graded materials are extensively used in modern industry. They are composite materials with continuously varying properties in one or more spacial dimensions, according to the specific purpose. In view of their applications, stress analysis of such materials is important for their structural integrity. In this study we will consider cracked functionally graded magneto-electro-elastic materials subjected to SH waves. We assume that the material properties vary in one and the same way, described by an inhomogeneity function. The boundary value problem is reduced to a system of integro-differential equations based on the existence of fundamental solutions. Different inhomogeneity classes are used to obtain a wave equation with constant coefficients. Radon transform is applied to derive the fundamental solution in a closed form. Program code in FORTRAN 77 is developed and validated using available examples from literature. Simulations show the dependence of stress on the frequency of the applied time-harmonic load for different types of material inhomogeneity.

ON SOLUTIONS OF THE RATIONAL TYPE TO MULTICOMPONENT NONLINEAR EQUATIONS
Tihomir Valchev tiv@math.bas.bg

2010 Mathematics Subject Classification: 35C05, 35Q55, 37K15, 74J30.
Key words: rational solutions, multicomponent nls equations, generalized Heisenberg model.

In this report we shall propose an algorithm to construct rational type solutions to multicomponent nonlinear evolution equations solvable through inverse scattering transform. The algorithm to be demonstrated is based on Zakharov-Shabat's dressing technique. As an illustration of our approach we shall consider in more detail the derivation of rational solutions to a generalized Heisenberg ferromagnet equation.