Transactions of A Razmadze Mathematical Institute最新文献

We examine the third kind integral equations in Hölder class. The coefficients of the equations are piecewise strictly monotone functions having simple zeros. By singular integral equations theory, for solvability of considered equations, we give the necessary and sufficient conditions. Finding a solution is reduced to solving a regular integral equation of second kind.

The main research line of this paper is concerned with the existence and uniqueness of solutions for a certain class of coupled systems of Caputo type fractional $\Delta$-difference boundary value problems at resonance. To this aim, we use coincidence degree theory to obtain existence results and impose growth controlling conditions on nonlinearities, uniqueness results will be concluded. At the end by means of an illustrative example the obtained main results will be implemented.

The Morrey space $M_{\tilde{q}}^p$ is disproved dense in $M_q^p$ if $1<q<\tilde{q}<p$.

In this paper, a main theorem dealing with $|\bar{N},p_n{|}_k$ summability method has been generalized for $\phi -|\bar{N},p_n;\delta {|}_k$ summability by using different and general summability factors of Fourier series.

In the work by analysis of one-dimensional unsteady flows, based on the fundamental law of conservation with application of Fourier series is shown that in the presence of periodic, steady pulsations along the flow, the main frequency as well as all higher frequencies remain constant and only the amplitude of oscillations is changed that is in full agreement with the results of analysis of more complex three-dimensional flows. Thus, is confirmed the validity of the principle of conservation of frequencies or time scale along the flow. So, is obtained very interesting result for turbulence problem solution.

We formulate new identities involving new Green functions. Inequality of Popoviciu, which was improved by Vasić and Stanković (1976), is generalized by using newly introduced Green functions. We utilize Fink’s identity along with new Green’s function to generalize the known Popoviciu’s inequality from convex functions to higher order convex functions. Then we construct linear functionals from the generalized identities and formulate the monotonicity of these functionals utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are computed.

The paper deals with the linear theory of thermoelasticity for elastic isotropic microstretch materials with microtemperatures and microdilatations. For the differential equations of pseudo-oscillations the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

In this paper, using Sinc-Galerkin and Levenberg–Marquardt methods a stable numerical solution is obtained to a nonlinear inverse parabolic problem. Due to this, this problem is reduced to a parameter approximation problem. To approximate unknown parameters, we consider an optimization problem where objective function is minimized by Levenberg–Marquardt method. This objective function is obtained by using Sinc-Galerkin method and the overposed measured data. Finally, some numerical examples are given to demonstrate the accuracy and reliability of the proposed method.

We analyse some new aspects concerning application of the fundamental solution method to the basic three-dimensional boundary value problems, mixed transmission problems, and also interior and interfacial crack type problems for steady state oscillation equations of the elasticity theory. First we present existence and uniqueness theorems of weak solutions and derive the corresponding norm estimates in appropriate function spaces. Afterwards, by means of the columns of Kupradze’s fundamental solution matrix special systems of vector functions are constructed explicitly. The linear independence and completeness of these systems are proved in appropriate Sobolev–Slobodetskii and Besov function spaces. It is shown that the problem of construction of approximate solutions to the basic and mixed boundary value problems and to the interior and interfacial crack problems can be reduced to the problems of approximation of the given boundary vector functions by elements of the linear spans of the corresponding complete systems constructed by the fundamental solution vectors. By this approach the approximate solutions of the boundary value and transmission problems are represented in the form of linear combinations of the columns of the fundamental solution matrix with appropriately chosen poles distributed outside the domain under consideration. The unknown coefficients of the linear combinations are defined by the approximation conditions of the corresponding boundary and transmission data.

We prove that the group ${\mathrm{SO}}_3\left(Q\right)$ of rational rotations is the inverse limit of a family of finite solvable groups of order $2^{3k-2}\cdot 3$, whose $2$-Sylow subgroups have nilpotency class $2k-3$, exponent $2^{k-1}$, and Frattini subgroups coinciding with the commutator subgroups, and we give generators for these groups.