On One Evolution Equation of Parabolic Type with Fractional Differentiation Operator in S Spaces

In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operator A=I−Δω/2, Δ=d2/dx2, and ω∈1;−2 is a fixed parameter. The operator A is treated as a pseudodifferential operator in a certain space of type S. The solvability of this problem is proved. The representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution are investigated. The behavior of the solution at t⟶+∞ (solution stabilization) in the spaces of generalized functions of type S′ and the uniform stabilization of the solution to zero on ℝ are studied.

A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

Correctness conditions for boundary value problems

The article deals with the issues of mathematical modeling of technological systems that contain physical fields’ sources. It is believed that in the case of a simple spatial form of the object under study, the boundary value problems will be correct. The interest lies in mathematical models for nonlinear, multilayer objects under the influence of load sources, for which, using the traditional theory of existence and unity, it is impossible to guarantee the correctness of boundary value problems. The author considers boundary value problems for systems of differential and pseudo differential equations in a multilayer medium which describe the state of the studied systems under the action of discrete load sources. The correctness of such problems is proven using the theory of distributions over the space of generalized functions. The object of research is boundary value problems for systems of differential and pseudo differential equations in a multilayer medium. The aim of the research is to build correct boundary value problems, which underlie the calculated mathematical models of the process of action of physical fields on multilayer objects. The necessary and sufficient conditions for the correctness of the parabolic boundary value problem in the space of generalized functions are obtained in the article. It is shown that its solution is infinitely differentiated by a spatial variable. The results of the research can be used to obtain the conditions for the correctness of the boundary value problem for differential equations with variable coefficients. Note that, in some cases, the correctness of the calculated mathematical models determines the correctness of applied optimization mathematical models. The application of the author's research is possible when proving the correctness of boundary value problems for a number of technological processes. The universality of the research allows to widely usage of the results obtained in this work to improve the quality of technological processes.

ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

Approximation of fracture energies with p-growth via piecewise affine finite elements

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.

KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST: A CORRIGENDUM

The paper “Kernel estimation when density may not exist” (Zinde-Walsh, 2008) considered density as a generalized function given by a functional on a space of smooth functions; this made it possible to establish the limit properties of the kernel estimator without assuming the existence of the density function. This note corrects an error in that paper in the derivation of the variance of the kernel estimator. The corrected result is that in the space of generalized functions the parametric rate of convergence of the kernel density estimator to the limit Gaussian process is achievable.

Some extensions of a certain integral transform to a quotient space of generalized functions

In this paper, we establish certain spaces of generalized functions for a class of ɛ