Milena Dimova

A new conservative finite difference scheme for Boussinesq paradigm equation

A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the proposed family of schemes is about two times more accurate than the family of schemes studied in [Kolkovska N., Two families of finite difference schemes for multidimensional Boussinesq paradigm equation, In: Application of Mathematics in Technical and Natural Sciences, Sozopol, June 21–26, 2010, AIP Conf. Proc., 1301, American Institute of Physics, Melville, 2010, 395–403].

Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein–Gordon equation with combined power-type nonlinearity is studied. Finite time blow-up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data. A new functional, invariant under the flow of the equation, is introduced and investigated.

Comparison of some finite difference schemes for boussinesq paradigm equation

The aim of the paper is to propose and study families of finite difference schemes for solving the Boussinesq Paradigm Equation. The nonlinear term of the equation is approximated in three different ways. We obtained a pair of implicit (with respect to the nonlinearity) families of schemes and an explicit one. All schemes have second rate of convergence in space and time. Numerical tests performed confirm our theoretical results regarding accuracy and convergence of all three schemes.

Comparison of Two Numerical Approaches to Boussinesq Paradigm Equation

In order to study the time behavior and structural stability of the solutions of Boussinesq Paradigm Equation, two different numerical approaches are designed. The first one A1 is based on splitting the fourth order equation to a system of a hyperbolic and an elliptic equation. The corresponding implicit difference scheme is solved with an iterative solver. The second approach A2 consists in devising of a finite difference factorization scheme. This scheme is split into a sequence of three simpler ones that lead to five-diagonal systems of linear algebraic equations. The schemes, corresponding to both approaches A1 and A2, have second order truncation error in space and time. The results obtained by both approaches are in good agreement with each other.

Finite time blow up of the solutions to boussinesq equation with linear restoring force and arbitrary positive energy

Abstract Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems — multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.

Structures and Waves in a Nonlinear Heat-Conducting Medium

This paper is an overview of the main contributions of a Bulgarian team of researchers to the problem of finding the possible structures and waves in the open nonlinear heat-conducting medium, described by a reaction–diffusion equation. Being posed and actively worked out by the Russian school of A.A. Samarskii and S.P. Kurdyumov since the seventies of the last century, this problem still contains open and challenging questions.

Numerical investigation of spiral structure solutions of a nonlinear elliptic problem

The nonlinear elliptic problem considered arises when investigating a class of self-similar solutions of a reaction-diffusion equation. We focus our study on the solutions of spiral structure. The proposed approach is based on the continuous analog of the Newtons method and on the Galerkin finite element method. To reveal solutions of spiral structure appropriate initial approximations are used. The last ones are expressed by the confluent hypergeometric function 1F1(a, b; z). Algorithms for accurate, fast and reliable computation of its values for broad ranges of the parameters a and b and of the variable z are worked out. A detailed numerical analysis of the evolution of the spiral structure solutions with respect to the medium parameters, including critical values, is carried out.

Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.