Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients

Abstract

We consider an equationHere α, β, γ are constants, moreover 0 < 2α, 2β, 2γ < 1. Main result of this paper is a construction of eight fundamental solutions for above-given equation in an explicit form. They are expressed by Lauricella's hypergeometric functions of three variables. Using expansion of Lauricella's hypergeometric function by products of Gauss's hypergeometric functions, it is proved that the found solutions have a singularity of the order 1/r at r → 0. Furthermore, some properties of these solutions, which will be used at solving boundary-value problems for afore-mentioned equation are shown.

Fundamental Solutions for a Class of Multidimensional Elliptic Equations with Several Singular Coefficients

“…For example, in the works of Barros-Neto and Gelfand [1][2][3] fundamental solutions for Tricomi operator, relative to an arbitrary point in the plane were explicitly calculated. In this direction we would like to note the works [4,5], where three-dimensional fundamental solutions for elliptic equations were found. In the works [6][7][8] , fundamental solutions for a class of multidimensional degenerate elliptic equations with spectral parameter were constructed.…”

“…Various modifications of the equation (1) in the two-and three-dimensional cases were considered in many papers [4,[18][19][20][21][22][23][24][25][26][27].…”

“…Since the aforementioned properties of hypergeometric functions of Gauss, Appell, Kummer were known [28], results on investigations of elliptic equations with one or two singular coefficients were successful. In the paper [4] when finding and studying the fundamental solutions of equation (1) for m = 3, an important role was played the decomposition formula of Hasanov and Srivastava [29,30], however, the recurrence of this formula did not allow further advancement in the direction of increasing the number of singular coefficients.…”

LINEARLY INDEPENDENT SOLUTIONS FOR THE HYPERGEOMETRIC EXTON FUNCTIONS X1AND X2

“…For instance, in [18], a comprehensive list of hypergeometric functions of three variables as many as 205 is recorded, together with their regions of convergence. It is noted that Riemann's functions and the fundamental solutions of the degenerate second-order partial differential equations are expressible by means of hypergeometric functions of several variables (see [7,8,9,11,12,13,14,15,16,17,19,20]). Therefore, in investigation of boundary-value problems for these partial differential equations, we need to study the solution of the system of hypergeometric functions and find explicit linearly independent solutions (see [12,13,14,15,16]).…”

FUNCTIONAL RELATIONS INVOLVING SARAN'S HYPERGEOMETRIC FUNCTIONS FEAND F(3)

“…In fact, in Srivastava and Karlsson's work [35], there is an extensive list of as many as 205 hypergeometric functions of three variables together with their region of convergence. It is noted that Riemann's functions and the fundamental solutions of the degenerate second-order partial differential equations are expressible by using hypergeometric functions of several variables (see [2,4,5,6,14,15,16,17,18,19,28,31,38,39,40]). For solutions of the boundary-value problems for the involved partial differential…”