Pliska Studia Mathematica Bulgarica
Volume 23, 2014
Proceedings of International conference on
New Trends of the Applications of Differential Equations in Sciences (NTADES'2014)
This volume is dedicated to the 145 anniversary
of the Bulgarian Academy of Sciences
GUEST EDITOR: A. Slavova
Sofia, 2014
C O N T E N T S
- Slavova, A. Preface (pp. 3−4)
- Georgiev, V., A. R. Giammetta. Bernstein Inequality for 1 − d Hamiltonians without Resonances (pp. 5−24)
- Popivanov, P. Explicit Formulas to the Solutions of Several Equations of Mathematical Physics (pp. 25−38)
- Boyadzhiev, G. 3D Modelling of Wave Propagation in Solid Media and Applications in Geophysics (pp. 39−48)
- Chobanov, G. Existence Results for Some Variational Inequalities Involving Non-negative, Non-coercitive Bilinear Forms (pp. 49−56)
- Donchev, T., D. Kolev, A. Nosheen, M. Rapaqat, A. Zeinev. Numerical Methods for Delayed Differential Equations with Discontinuites (pp. 57−66)
- Fabricant, A., N. Kutev, Ts. Rangelov. Sharp Hardy Inequalities in a Ball (pp. 67−80)
- Kutev, N., N. Kolkovska, M. Dimova. Global Behavior of the Solutions to Sixth Order Boussinesq Equation with Linear Restoring Force (pp. 81−94)
- Mauro, J. A.On the Regularity Properties of the Pressure Field Associated to a Hopf Weak Solution to the Navier-Stokes Equations (pp. 95−118)
- Mochimaru, Y. Spectral Finite Difference Analysis of Natural Convection in a Two-dimensional Enclosure of a Three-fins Type (pp. 119−126)
- Rangelov, Ts., P. Dineva. Anti-plane Scattering by Heterogeneities in Piezoelectric Plane by BIEM (pp. 127−140)
- Slavova, A., N. Kyurkchiev. Programme Packages for Implementation of Modifications of Black-Scholes Model and web Applications (pp. 141−158)
- Slavova, A., P. Zecca. On Supra-Bayesian Weighted Combination of Available Data Determined by Kerridge Inaccuracy and Entropy (pp. 159−174)
- Stoynov, Y. 2D Fracture Problems in Magneto-electro-elastic Composite Materials under Anti-plane Waves by BIEM (pp. 175−188)
- Yordanov, B., R. Yordanova. On the Existence of
Drifting Orbits for Non-convex Hamiltonian Systems (pp. 189−209)
A B S T R A C T S
BERNSTEIN INEQUALITY FOR 1 − D HAMILTONIANS
WITHOUT RESONANCES
Vladimir Georgiev
georgiev@dm.unipi.it,
Anna Rita Giammetta
giammetta@mail.dm.unipi.it
2010 Mathematics Subject Classification: 35P05, 42B25, 46E25, 42B15.
Key words: Bernstein inequality, Hamiltonians with potential, Sobolev spaces, Besov spaces.
We consider 1-D Laplace operator with short range potential
W(x) and prove the Bernstein inequality for this perturbed Laplacian. It is shown that non resonance assumption at zero and sufficiently fast decay of
the potential at infinity guarantee that the Hamiltonian obeys the Bernstein
inequality.
EXPLICIT FORMULAS TO THE SOLUTIONS OF SEVERAL
EQUATIONS OF MATHEMATICAL PHYSICS
Petar Popivanov
popivano@math.bas.bg
2010 Mathematics Subject Classification: 35C05, 35L05, 35Q53, 35S10.
Key words: Klein-Gordon equation, wave equation, semilinear hyperbolic equation,
Kadomtsev-Petviashvili equation, solution into closed form.
Explicit formulas to the solutions of several equations of math-
ematical physics including semilinear multidimensional Klein-Gordon equa-
tion, the wave equation, Kadomtsev-Petviashvili equation and cubic first
order hyperbolic pseudodifferential equation are proposed.
3D MODELLING OF WAVE PROPAGATION IN SOLID
MEDIA AND APPLICATIONS IN GEOPHYSICS
Georgi Boyadzhiev
gpb@math.bas.bg
2010 Mathematics Subject Classification: 35L53, 35Q86, 86A15.
Key words: key words Strongly coupled linear hyperbolic systems, modelling of multi-layered
solid body, applications in Geophysics.
In this paper the geometrical properties of the bi-characteristic
curves are employed in developing a new approach to 3D modelling of elastic piecewise homogeneous media, in particular Earth crust and upper Mantle.
The method is based on tomography and the refraction, respectively, reflection, of the bi-characteristic curves at the layer boundaries of multi − layered
media.
EXISTENCE RESULTS FOR SOME VARIATIONAL
INEQUALITIES INVOLVING NON-NEGATIVE,
NON-COERCITIVE BILINEAR FORMS
Georgi Chobanov
chobanov@math.bas.bg
2010 Mathematics Subject Classification: 35J85.
Key words: variational inequalities, semi-coercive bilinear forms, recession cone.
In the present paper the existence of solutions to variational
inequalities for semi-coercive bilinear forms is studied. The result generalizes
a result by Lions-Stampacchia and is close to an abstract result by Fichera.
NUMERICAL METHODS FOR DELAYED DIFFERENTIAL
EQUATIONS WITH DISCONTINUITES
T. Donchev
tzankodd@gmail.com,
D. Kolev
kolev@mmu.uctm.edu,
kolev@uctm.edu
A. Nosheen
hafiza@gmail.com,
M. Rapaqat
m.rafaqat50@gmail.com,
A. Zeinev
a_zejnev@uctm.edu
2010 Mathematics Subject Classification: 34A37.
Key words: Impulsive differential equations, Runge-Kutta methods, delay.
The numerical approximation of higher order are used to solve
differential equations with discontinuous solutions and fixed time delay. The
accuracy of these methods is investigated.
SHARP HARDY INEQUALITIES IN A BALL
Alexander Fabricant,
Nikolai Kutev,
Tsviatko Rangelov
rangelov@math.bas.bg
2010 Mathematics Subject Classification: 26D10.
Key words: Hardy inequality, sharp estimates.
Several Hardy−type inequalities in a sectorial area and in a
ball are obtained. Sharpness of the inequalities is shown. An application
to the lower bound of the first eigenvalue for the p−Laplacian in bounded
domains is given.
GLOBAL BEHAVIOR OF THE SOLUTIONS TO SIXTH
ORDER BOUSSINESQ EQUATION WITH LINEAR
RESTORING FORCE
N. Kutev,
N. Kolkovska
natali@math.bas.bg,
M. Dimova
mkoleva@math.bas.bg
2010 Mathematics Subject Classification: 35L30, 76B15, 65M06.
Key words: Sixth order Boussinesq equation, potential well method, finite time blow up, arbitrary high positive initial energy.
Potential well method is established to sixth order Boussinesq equation with linear restoring force and subcritical initial energy. For supercritical initial energy finite time blow up of the solutions is proved under general structural conditions on the initial data. Numerical experiments, illustrating the theoretical results, are presented.
ON THE REGULARITY PROPERTIES OF THE PRESSURE
FIELD ASSOCIATED TO A HOPF WEAK SOLUTION TO
THE NAVIER-STOKES EQUATIONS
Jmmy Alfonso Mauro
jmmyamauro@math.bas.bg
2010 Mathematics Subject Classification: 76D05, 35Q30, 76D03.
Key words: Navier-Stokes equations, Leray-Hopf weak solutions, regularity of the pressure field.
We give some new a priori estimates for the pressure field associated to a Hopf weak solution, under the minimal assumption that the initial data v_{0} is in L^{2}(Ω). Then, such estimates are applied to obtain
an existence theorem of suitable weak solutions on a bounded or exterior
domain Ω⊂R^{3}, with the minimal assumption v_{0}∈L^{2}(Ω).
SPECTRAL FINITE DIFFERENCE ANALYSIS OF
NATURAL CONVECTION IN A TWO-DIMENSIONAL
ENCLOSURE OF A THREE-FINS TYPE
Yoshihiro Mochimaru
2010 Mathematics Subject Classification: 30C20, 65N06, 76R10.
Key words: spectral analysis, natural convection, finite difference.
Spectral finite difference scheme is applied to two-dimensional
natural laminar convection around a three-fins type enclosure, including a shape control parameter supplemented with a condition of doubly-connectedness. Streamlines, characteristics for the maximum stream function and the mean Nusselt number against a Grashof number are presented.
ANTI-PLANE SCATTERING BY HETEROGENEITIES IN
PIEZOELECTRIC PLANE BY BIEM
Tsviatko Rangelov
rangelov@math.bas.bg,
Petia Dineva
petia@imbm.bas.bg
2010 Mathematics Subject Classification: 74J20, 74S15, 74G70.
Key words: MDL, multifunctional nano-structured materials, electro-mechanical dynamic load,
integro-differential equations, boundary integral equation method.
Considered is homogeneous or functional graded piezoelectric
material with heterogeneities of different type (hole, crack, inclusion, nano-hole, nano-inclusion) subjected to time-harmonic wave. With respect to the boundary conditions along the interface between the heterogeneity and
the infinite matrix different boundary value problems are formulated and
solved. Boundary Integral Equation Method is applied to evaluate both:
(a) the wave far-field due to the wave scattering and diffraction; (b) the
stress concentration field near heterogeneities.
The obtained results are applicable in the field of non-destructive testing,
material science and fracture mechanics of multi-functional materials and
structural elements based on them.
PROGRAMME PACKAGES FOR IMPLEMENTATION OF
MODIFICATIONS OF BLACK-SCHOLES MODEL AND WEB
APPLICATIONS
Angela Slavova
slavova@math.bas.bg,
Nikolay Kyurkchiev
nkyurk@math.bas.bg
2010 Mathematics Subject Classification: 65M12, 65Y20.
Key words: Black-Scholes model, market price, coefficient of variation, Garman-Kohlhagen
model, programming environment MATHEMATICA.
In this paper we propose new modules in programming en-
vironment MATHEMATICA for the generalizations of Black-Scholes (BS)
model taking into account the market price and coefficient of variation. First we derive generalization of Black-Scholes PDE and present its explicit solution. Then we derive the Garman-Kohlhagen's model as generalization of BS model. The proposed modules gives the possibility for visualization and hypersensitive analysis.
APPLICATIONS OF EQUATIONS OF MATHEMATICAL
PHYSICS IN STUDYING TSUNAMI WAVES
Angela Slavova
slavova@math.bas.bg,
Pietro Zecca
zecca@unifi.it
2010 Mathematics Subject Classification: 76B15, 35Q80, 92B20.
Key words: Tsunami waves, equations of mathematical physics, viscoelastic Burgers equation, Cellular Nonlinear Networks, traveling wave solution, generalizations.
In this paper we present equations of mathematical physics
which have applications in studying tsunami waves. First we investigate
an interesting system of non-linear PDE − the viscoelastic generalization of the Burger's equation. In the above mentioned system we are looking for travelling wave solutions and we are studying their profiles. Then we derive travelling wave solutions of the viscoelastic Burgers' equation. Cellular Nonlinear Networks (CNN) model of this equation is constructed. Using its computer realization new wave profiles of the travelling wave solutions are obtained.
2D FRACTURE PROBLEMS IN
MAGNETO-ELECTRO-ELASTIC COMPOSITE MATERIALS
UNDER ANTI-PLANE WAVES BY BIEM
Yonko Stoynov
jds@tu-sofia.bg
Magneto-electro-elastic material with two cracks is studied.
The material is subjected to harmonic anti-plane mechanical and in-plane
electric and magnetic load. The boundary value problem for the coupled
system of partial differential equations is solved numerically by the Boundary integral equation method (BIEM). Program code in FORTRAN 77 based on the BIEM is developed. The numerical solution is compared with the results obtained by the dual integral equations method. Numerical simulations show the dependence of the stress, electric and magnetic field concentration near the crack tips on the normalized frequency of the applied dynamic load for different locations and dispositions of the cracks.
ON THE INSTABILITY OF ACTION VARIABLES IN
NON-CONVEX HAMILTONIAN SYSTEMS
Borislav Yordanov
byordanov@math.bas.bg,
Roumyana Yordanova
ryordanova@math.bas.bg
2010 Mathematics Subject Classification: Primary 37J25, 37J40; Secondary 37C10, 37C40.
Key words: Hamiltonian systems, instability, resonances, Arnold diffusion.
We show the instability of action variables on resonant orbits
of certain non-convex Hamiltonian systems with several degrees of freedom.
Such orbits remain in the vicinity of resonant surfaces where the action
variables can undergo changes O(1) infinitely often although the size of perturbations O(ε) can be arbitrarily small.
We also perform numerical simulations to compare the effects of two
condition for instability in two four-dimensional examples with random parameters.