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This paper deals with radial and non-radial solutions for local and nonlocal Liouville type equations. At first non-degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. Non-radial solution is constructed in the case of Blaschke type nonlinearity. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinearities is considered and C^2 smooth monotonically decreasing radial solution u ( r ) is found. Moreover, u ( r ) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R^4 and their Gaussian, respectively normal curvatures are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are given.

In this chapter we present mathematical study of memristor systems. More precisely, we apply local activity theory in order to determine the edge of chaos regime in reaction-diffusion memristor cellular nanoscale networks (RD-MCNN) and in memristor hysteresis CNN (M-HCNN). First we give an overview of mathematical models of memristors, CNN and complexity. Then we consider the above mentioned two models and we develop constructive algorithm for determination of edge of chaos in them. Based on these algorithms numerical simulations are provided. Two applications of M-HCNN model in image processing are presented.

[Slavova Angela; Славова Анжела]; [Zafirova Zoya; Зафирова Зоя] In this paper we study a class of integro-differential equations. First we present Cellular Nonlinear Networks (CNN) modeling of integro-differential equations. We map the equation under consideration into CNN architecture and study its dynamics. Applying harmonic balance method we obtain periodic solutions of the model. We provide numerical simulations which illustrate the obtained theoretical results. 2010 Mathematics Subject Classification: 92B20, 35B10, 35Q68, 35Q92.

[Slavova Angela; Славова Анжела]; [Bobeva Galina; Бобева Галина] In this paper we study the dynamics of piezoelectrical material (PEM) with nano-inhomogeneities. The boundary value model defined by the system of two partial differential equations and the boundary conditions for the generalized stress is mapped into Cellular Nanoscale Networks architecture. We study the dynamics of the obtained CNN model via harmonic balance technique. Validation and simulations are provided which illustrate the theoretical results. 2010 Mathematics Subject Classification: 92B20,74F15, 35K57, 35Q68.

[Slavova Angela; Славова Анжела]; [Bobeva Galina; Бобева Галина] In this paper different reaction-diffusion Cellular Nonlinear Networks (RDCNN) models will be presented. Dynamics of Oregonator RD-CNN model will be studied. 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. 2010 Mathematics Subject Classification: 92B20, 34C28, 35K57.

[Slavova Angela; Славова Анжела]; [Markova Maya; Маркова Мая] In this paper a memristor-based chaotic system model will be studied. 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. 2010 Mathematics Subject Classification: 92B20, 34C55, 34C28.

[Slavova Angela; Славова Анжела] Piezoelectrical material with heterogeneities of nano-holes or nano-inclusions is considered in the case when it is subjected to time harmonic electromechanical 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. 2010 Mathematics Subject Classification: 92B20,74F15, 35K57, 35Q68.

[Slavova Angela; Славова Анжела] 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. 2010 Mathematics Subject Classification: 92B20, 34C55, 34C28.

[Slavova Angela; Славова Анжела]; [Markova Maya; Маркова Мая] 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. 2010 Mathematics Subject Classification: 92B20, 74F15, 35K57, 35Q68.

[Slavova Angela; Славова Анжела]; [Kyurkchiev Nikolay; Кюркчиев Николай] 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 . Then we propose -subordinated Brownian motion and option pricing with and without transaction costs modules. 2010 Mathematics Subject Classification: 91B25, 91B24, 91B02, 34K50, 65M12, 65Y20.

[Slavova Angela; Славова Анжела]; [Kyurkchiev Nikolay; Кюркчиев Николай] In this paper we propose new modules in programming environment 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. 2000 Mathematics Subject Classification: 65M12, 65Y20.

[Slavova Angela; Славова Анжела] 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. 2000 Mathematics Subject Classification: 76B15, 35Q80, 92B20.

This book deals with equations of mathematical physics as the different modifications of the KdV equation, the Camassa-Holm type equations, several modifications of Burger's equation, the Hunter-Saxton equation, conservation laws equations and others. The equations originate from physics but are proposed here for their investigation via purely mathematical methods in the frames of university courses. More precisely, we propose classification theorems for the traveling wave solutions for a sufficiently large class of third order nonlinear PDE when the corresponding profiles develop different kind of singularities (cusps, peaks), existence and uniqueness results, etc. The orbital stability of the periodic solutions of traveling type for mKdV equations are also studied. Of great interest too is the interaction of peakon type solutions of the Camassa-Holm equation and the solvability of the classical and generalized Cauchy problem for the Hunter-Saxton equation. The Riemann problem for special systems of conservation laws and the corresponding -shocks are also considered. As it concerns numerical methods we apply the CNN approach. The book is addressed to a broader audience including graduate students, Ph.D. students, mathematicians, physicist, engineers and specialists in the domain of PDE.