On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer

Abstract

Abstract. We establish the necessary and sufficient conditions of existence of nontrivial quasi-polynomial solutions of the problem in a layer for homogeneous partial differential equation with s + 1 variables of second order in time variable and generally infinite order in other s (spatial) variables with Dirichlet boundary conditions in time. We apply the differential-symbol method for constructing such quasi-polynomial solutions. We also give examples of problems for which we construct other solutions besides of quasi-polynomial ones.

The differential-symbol method of constructing the quasi-polynomial solutions of two-point problem

“…where the functions̃︀ T 0 (t, ν),̃︀ T 1 (t, ν), ρ 0 (t, x, ν) are following:︀ Let us note that the found solution of problem (25), (26) is not unique in the class K C,C , since, for example, the functions U 1 (t, x) = c 1 e −x , U 2 (t, x) = c 2 (1 − t − x)e −x , c 1 , c 2 ∈ C, are solutions to the corresponding homogeneous problem and the sum of each of these solutions with the found solution U(t, x) is also a solution to problem (25), (26).…”

“…The present work continues the research of [24][25][26][27] and is devoted to the generalized problem (1), (7). The question of existence of solutions of the problem for the equation of the second order in time and generally infinite order in spatial variable with two-point conditions (7) in which A 1 , A 2 , B 1 , B 2 are operator coefficients in the variable x is being studied.…”

Analytical method to study a mathematical model of wave processes under two­point time conditions

“…Studying a set of non-trivial solutions to the homogeneous problem (6), (7) was addressed in paper [10] for s=1, in work [11] for s=1 and conditions (9), as well as in article [12] for s≥1.…”

“…Comment 1. If in equation 10quasi-polynomial f(t, x) has a more general form (11), then, according to the principle of linear superposition, a solution to problem (10), (7) takes the form of the following sum: Comment 2. Solutions to problems (10), (7) in the form of (14) and (16) are the quasi-polynomials.…”

Development of the quantitative method for automated text content authorship attribution based on the statistical analysis of N-grams distribution

“…Depending on the purpose of the study, a zone can be considered to be prose, poetry and drama in fiction; works by one author or a specific work; totality of words of a certain morpheme structure (for example, prefixed or monomorphemic), etc. [47].…”