Построение Полиномиальных Решений Задачи Дирихле Для Полигармонического Уравнения В Шаре

Карачик, В. В.

doi:10.7868/s0044466914070072

Cited by 4 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…В них получен явный вид функции Грина. Явное представление функции Грина третьей краевой задачи для уравнения Пуассона получено в статье [8], а в [9] для 3-гармонического уравнения в шаре представлен оператор Грина, действующий на полиномиальные данные.…”

unclassified

See 1 more Smart Citation

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

Karachik

2023

MMPh

View full text Add to dashboard Cite

In this paper, an elementary solution for polyharmonic equations is determined and its properties are given. This elementary solution coincides with previously known elementary solutions of biharmonic and triharmonic equations. Using the elementary solution, an integral representation of the solutions of a non-homogeneous polyharmonic equation in a bounded domain with a smooth boundary is found. Based on the integral representation, the solvability of the Riquier–Neumann problem is investigated. First, the concept of the Green's function of the Riquier–Neumann problem is defined, and then the Green's function is proved. Using the integral representation of the solutions of the polyharmonic equation and the Green's function of the Riquier–Neumann problem, the integral representation of the solution of the Riquier–Neumann problem in a unit ball is found. An example of the solution of the Neumann problem for the Poisson equation with the simplest right-hand side is given, which is necessary in what follows. On the basis of the Green's function of the Riquier–Neumann problem, a theorem on the integral representation of the solution of the Riquier–Neumann boundary value problem with boundary data, the integral of which over the unit sphere vanishes, is proved. In conclusion, on the basis of the theorem, an example of calculating the solution of the Riquier–Neumann problem with boundary functions coinciding with the traces of homogeneous harmonic polynomials on a unit sphere is given.

show abstract

unclassified

“…Ex  -были введены в работах [9,17,18]. Кроме того, в этих работах были найдены функции Грина соответствующих задач Ди-…”

unclassified

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

Karachik

2023

MMPh

View full text Add to dashboard Cite

show abstract

Exact Solutions to the Boundary-Value Problems for the Helmholtz Equation in a Layer With Polynomials in the Right-Hand Sides of the Equation and of the Boundary Conditions

Algazin

2020

Bulletin of the MSRU (Physics and Mathematics)

View full text Add to dashboard Cite

In a multidimensional infinite layer bounded by two hyperplanes, the inhomogeneous Helmholtz equation with a polynomial right-hand side is considered. It is shown that the Dirichlet and Dirichlet-Neumann boundaryvalue problems with polynomials in the right-hand sides of the boundary conditions have a solution that is a quasipolynomial that contains, in addition to the power functions, also hyperbolic or trigonometric functions. This solution is unique in the class of slow growth functions if the parameter of the equation is not an eigenvalue. An algorithm for constructing this solution is given and examples are considered.

show abstract

О Задаче Дирихле-Рикье Для Бигармонического Уравнения

Карачик¹,

Карачик²,

Torebek³

2017

Matematicheskie Zametki

View full text Add to dashboard Cite

Исследуется существование решения задачи Дирихле-Рикье для однородного бигармонического уравнения в единичном шаре с граничными операторами до третьего порядка включительно, содержащие нормальные производные и лапласиан. Доказаны теоремы существования решения задачи. Библиография: 30 названий.

show abstract

Построение Полиномиальных Решений Задачи Дирихле Для Полигармонического Уравнения В Шаре

Cited by 4 publications

References 0 publications

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

Exact Solutions to the Boundary-Value Problems for the Helmholtz Equation in a Layer With Polynomials in the Right-Hand Sides of the Equation and of the Boundary Conditions

О Задаче Дирихле-Рикье Для Бигармонического Уравнения

Contact Info

Product

Resources

About