The Dirichlet problem for elliptic equation with several singular coefficients

AbstractRecently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables.

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Holmgren problem for Helmholtz equation with the three singular coefficients

e-Journal of Analysis and Applied Mathematics ◽

10.2478/ejaam-2019-0002 ◽

2019 ◽

Vol 2019 (1) ◽

pp. 15-30

Author(s):

Davlatjon R. Muydinjanov

Keyword(s):

Elliptic Equation ◽

Helmholtz Equation ◽

Hypergeometric Function ◽

Unique Solution ◽

Explicit Form ◽

Fundamental Solutions ◽

Confluent Hypergeometric Function ◽

Singular Coefficients

Abstract Fundamental solutions for a multidimensional Helmholtz equation with three singular coefficients have been constructed recently which are expressed in terms of the confluent hypergeometric function in four variables. In this paper, we study the Holmgren problem for a 3D elliptic equation with three singular coefficients. A unique solution of the problem is obtained in the explicit form.

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Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

Вестник КРАУНЦ. Физико-математические науки ◽

10.26117/2079-6641-2020-32-3-114-126 ◽

2020 ◽

pp. 114-126

Author(s):

T.G. Ergashev ◽

A. Hasanov

Keyword(s):

Elliptic Equation ◽

Fundamental Solution ◽

Hypergeometric Function ◽

Explicit Form ◽

Green's Function ◽

Explicit Solution ◽

Hypergeometric Functions ◽

Summation Formulae ◽

Singular Coefficients ◽

Abc Method

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem. В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

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ON AN ANALOGUE OF THE HOLMGREN'S PROBLEM FOR 3D SINGULAR ELLIPTIC EQUATION

Asian-European Journal of Mathematics ◽

10.1142/s1793557112500210 ◽

2012 ◽

Vol 05 (02) ◽

pp. 1250021 ◽

Cited By ~ 3

Author(s):

J. J. Nieto ◽

E. T. Karimov

Keyword(s):

Elliptic Equation ◽

Explicit Form ◽

Green's Function ◽

Unique Solvability ◽

Explicit Solution ◽

Hypergeometric Functions ◽

Three Dimensional ◽

Function Solution ◽

Singular Elliptic Equation ◽

Singular Coefficients

In the present paper an analogue of the Holmgren's problem for a three-dimensional elliptic equation with singular coefficients has been studied for the unique solvability. The uniqueness for the solution of considered problem is proved by an energy integral's method. Applying a method of Green's function, solution of the problem is found in an explicit form. Moreover, decomposition formulas, formulas of differentiation and some adjacent relations for Lauricella's hypergeometric functions were used to find explicit solution for aforementioned problem, which contains Appell's hypergeometric functions.

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