Class Of Dynamic Equations Research Articles

Numerical Computations for One Class of Dynamical Mathematical Models in Quasi-Sobolev Space

The article studies some mathematical models that represent one class of dynamical equations in quasi-Sobolev space. The analytical investigation of solvability of the Cauchy problem in the quasi-Sobolev space and theoretical results used to enhance and develop an algorithm structure of the numerical procedures to find approximate solutions for models, the steps of algorithm based on the theoretical investigation of models, new algorithm of numerical method allowing to find approximate solutions of mathematical models under study in quasi-Sobolev space. Construction a program implements an algorithm of numerical method that allow finding approximate solutions for models. To construct the theory of degenerate holomorphic semigroups of operators in quasi-Banach spaces of sequences, we used the classical methods of functional analysis, theory of linear bounded operators, spectral theory. To construct the operators of resolving semigroups we used the Laplace transform of operator-valued functions in quasi-Banach spaces of sequences. The numerical investigation for models generate some approximate solutions which are normally based on the modified projection method. The convergence of the approximate solution to the exact one theoretically is justified by the convergence of the corresponding series, the agreement of approximate computations with the theoretical solution is established.

SOME GRONWALL{BELLMAN TYPE INEQUALITIES ON TIME SCALES FOR VOLTERRA-FREDHOLM DYNAMIC INTEGRAL EQUATIONS

In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalitiesof Gronwall{Bellman{Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussedin the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given herecan be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Someexamples are presented to demonstrate the applications of our results.

Some dynamic integral inequalities with mixed nonlinearities on time scales

The objective of this paper is to study some dynamic integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results include many known ones in the literature and can be used as tools in the study of qualitative theory of certain classes of dynamic equations with mixed nonlinearities on time scales.

Solvability of Initial Problems for One Class of Dynamical Equations in Quasi-Sobolev Spaces

The equations, which are not solved with respect to the highest derivative, are now actively studied. Such equations are also called the Sobolev type equations. Note that these equations in Banach spaces are studied quite well. Quasi-Sobolev spaces are quasi normalized complete spaces of sequences. Recently these spaces began to be studied. The interest to such spaces and its equations is connected with a desire to fill up the theory more than with practical applications. The paper is devoted to the study of solvability of the Cauchy problem and the Showalter -- Sidorov problem for a class of equations considered in the quasi-Sobolev spases. To this end we use properties of the equation operators, namely the relative boundedness of the operators. To illustrate abstract results we consider an analogue of the Hoff equation in the quasi-Sobolev spaces.

Some nonlinear dynamic integral inequalities on time scales

In this paper, we study some dynamic integral inequalities with mixed nonlinearities on time scales, which provide explicit bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations with mixed nonlinearities on time scales.

Some new integral inequalities on time scales

By introducing two adjusting parameters, we investigate some new nonlinear integral inequalities on time scales, which provide explicit bounds on unknown functions, and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales. Mathematics subject classification (2010): 26D15, 26E70.

Some Sublinear Dynamic Integral Inequalities on Time Scales

We study some nonlinear dynamic integral inequalities on time scales by introducing two adjusting parameters, which provide improved bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales.

Dynamic double integral inequalities in two independent variables on time scales

First, we establish some new nonlinear dynamic inequalities in two independent vari- ables ofPachpatte type, that might beuseful tools inthe study ofqualitative properties ofsolutions of certain classes of dynamic equations on time scales. These results extend recent inequalities fordifference equations tothe general time-scale setting. Then, after establishing anabla Jensen's inequality, we relate several inequalities of Hilbert-Pachpatte type that extend and unify recent continuous and discrete inequalities of this type.