2020/03/19 by Tuhtasin Ergashev, Ergashev, Tuhtasin
Engineering · Mathematics · Physics and Astronomy · #33C65 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods in engineering #Primary 35J70 #Secondary 33C20
paper · pdf · doi:10.48550/arxiv.2003.08678
openalex publication_date 2020/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Potentials play an important role in solving boundary value problems for\nelliptic equations. In the middle of the last century, a potential theory was\nconstructed for a two-dimensional elliptic equation with one singular\ncoefficient. In the study of potentials, the properties of the fundamental\nsolutions of the given equation are essentially and fruitfully used. At the\npresent time, fundamental solutions of a three-dimensional elliptic equation\nwith one degeneration line are already known. In this paper, we investigate the\ndouble- and simple-layer potentials for this kind of elliptic equations.\nResults from potential theory allow us to represent the solution of the\nboundary value problems in integral equation form. By using some properties of\nGaussian hypergeometric function, we prove limiting theorems and derive\nintegral equations concerning a densities of the double- and simple-layer\npotentials. The obtained results are applied to find an explicit solution of\nthe Dirichlet and Holmgren problems for the three-dimensional singular elliptic\nequation in the half of the ball.\n