Power Series.Orthogonal Operational Expansions

Formulation of the classic Taylor series as an orthogonal concept based on identifying the expansions coefficients as differential transformation applied to a unique function; definition of operational orthogonality by analogy with the Hilbert space and identification of the nth derivative at a point based on the improper integral on the positive semiaxis ; deduction of inversion integrals for Laplace transforms for analytical functions

Power Series.Orthogonal Operational Expansions

Formulation of the classic Taylor series as an orthogonal concept based on identifying the expansions coefficients as differential transformation applied to a unique function; definition of operational orthogonality by analogy with the Hilbert space and identification of the nth derivative at a point based on the improper integral on the positive semiaxis ; deduction of inversion integrals for Laplacen transforms for analytical functions.

ESTIMATES OF POSITIVE NONTRIVIAL SOLUTIONS OF A DIFFERENTIAL EQUATION WITH POWER NONLINEARITY

Differential equations in paper with power nonlinearity are considered. Solutions which are defined in some neighborhood of plus infinity are called proper solutions. It is proved that propersolution to the equation is kneser solution, which means that solution and it’s quasiderivatives change their signs and tend to zero. The integral representation for proper solutions is proved. Upper estimates for solution and it’s quasiderivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with quasiderivative are proved. Upper and lowerestimates of solution and it’s derivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with derivative are provedDifferential equationsy[n] = rn(x)ddx(rn

The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

We introduce the Szász and Chlodowsky operators based on Gould-Hopper polynomials and study the statistical convergence of these operators in a weighted space of functions on a positive semiaxis. Further, a Voronovskaja type result is obtained for the operators containing Gould-Hopper polynomials. Finally, some graphical examples for the convergence of this type of operator are given.

On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

We consider the time-fractional derivative in the Caputo sense of orderα∈(0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function inR+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit whenα↗1of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems whenα= 1, and the fractional diffusion equation becomes the heat equation.

Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at+∞and−∞inℝN(N=2,3). Namely, it is 4-linear at−∞and 4-superlinear at+∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.

Long-Time Behaviour of Solutions for Autonomous Evolution Hemivariational Inequality with Multidimensional “Reaction-Displacement” Law

We consider autonomous evolution inclusions and hemivariational inequalities with nonsmooth dependence between determinative parameters of a problem. The dynamics of all weak solutions defined on the positive semiaxis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. New properties of complete trajectories are justified. We study classes of mathematical models for geophysical processes and fields containing the multidimensional “reaction-displacement” law as one of possible application. The pointwise behavior of such problem solutions on attractor is described.