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Tuhtasin Ergashev

Publications and source records attributed to Tuhtasin Ergashev.

18 recordsLinked to original sources
arXiv2020

Double- and simple-layer potentials for generalized singular elliptic equations and their applications to the solving the Dirichlet problem

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with several singular coefficients are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using a decomposition formula and other identities for the Lauricella's hypergeometric function in many variables, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet problem for the generalized singular elliptic equation in the some part of the multidimensional ball.

math.AP↗
arXiv2020

Solving the Dirichlet and Holmgren problems for a three-dimensional elliptic equation by the potential method

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a three-dimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet and Holmgren problems for the three-dimensional singular elliptic equation in the half of the ball.

math.AP↗
arXiv2019

The generalized Holmgren problem for elliptic equation with several singular coefficients

Recently 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 generalized Holmgren 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.

math.AP↗
arXiv2019

Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables

In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the fundamental solutions of the generalized Helmholtz equation with several singular coefficients are written out through the newly introduced confluent hypergeometric function. Using the expansion formula established here for the confluent function, the order of the singularity of the fundamental solutions of the elliptic equation under this consideration is determined.

math.AP↗
arXiv2019

New decomposition formulas associated with the Lauricella multivariable hypergeometric functions

Decomposition formulas associated with the Lauricella multivariable hypergeometric functions were known, however, due to the recurrence of those formulas, additional difficulties may arise in the applications. Further study of the properties of the famous expansion formulas showed that it can be reduced to a more convenient form. In addition, this paper contains applications of new expansion formulas to the solving of boundary value problems for a multidimensional elliptic equation with several singular coefficients.

math.AP↗
arXiv2019

Some relations following from the decomposition formula for one multidimensional Lauricella hypergeometric function

Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, some relations following from the decomposition formula for one multidimensional Lauricell hypergeometric function are determined.

math.AP↗
arXiv2019

The Dirichlet problem for elliptic equation with several singular coefficients

Recently 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.

math.AP↗
arXiv2018

Decomposition formulas associated with the multivariable confluent hypergeometric functions

The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions of two and more variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities several decomposition formulas are found, which express the aforementioned hypergeometric functions in terms of such simpler functions as the products of the Gauss hypergeometric functions.

math.CA↗
arXiv2018

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.

math.AP↗
arXiv2018

Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation II

The double-layer potential plays an important role in solving boundary value problems for elliptic equations. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the next fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.

math.AP↗
arXiv2018

Third Double-layer Potential for generalized bi-axially symmetric Helmholtz equation

The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the third fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.

math.AP↗
arXiv2018

Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined. Key words: multidimensional elliptic equation with several singular coefficients; fundamental solutions; multiple hypergeometric functions; decomposition formula; order of the singularity.

math.AP↗
arXiv2018

Solutions of systems of the partial differential equations of Kampé de Fériet type functions

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this present investigation, we give the solutions of systems of partial differential equations for two Kampé de Fériet type functions of third and fourth orders and of two variables.

math.CA↗
arXiv2018

Generalized solutions of the degenerate hyperbolic equation of the second kind with a spectral parameter

For a degenerate hyperbolic equation of the second kind, and with a spectral parameter are studied the Cauchy problem, Cauchy-Goursat and Goursat in a new class of generalized solutions and is given an example that shows the importance of introducing the concept of such a class. Some operators with Bessel functions in the nucleus are introduced and their basic properties are studied. The important identities of these operators are helped to find an explicit representations of the stated problems.

math.AP↗
arXiv2018

The Fourth Double-Layer Potential for a Generalized Bi-Axially Symmetric Helmholtz Equation

The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholts equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the fourth fundamental solution.

math.AP↗
arXiv2018

The inversion formula for the Volterra integral equation with the Humbert function in the nuclear and its applications to the boundary value problems

Many problems of applied mathematics are reduced to the solution of integral equations with special functions in kernels, therefore the inversion formulas for such equations play an important role in solving boundary value problems for second-order partial differential equations. In this paper, we introduce one degenerate hypergeometric function of two variables through which the solution of the Volterra integral equation of the first kind studing here is expressed.

math.AP↗