(untitled) (Journal of Mathematics Mechanics and Computer Science, 2024)
The primary outcome of this study is the construction of partial solutions for a class of multidimensional partial differential equations with multiple singular coefficients of the second order. We consider the generalized multidimensional second-order Euler-Poisson-Darboux equation. Employing a well-known method, we reduce the generalized Euler-Poisson-Darboux equation to a second-order partial differential equation of the hypergeometric type. The solutions to this second order hypergeometric equation comprise 2n functions that contain the first Lauricella hypergeometric function. The Lauricella function, also known as an n-dimensional series, incorporates three distinct parameters- the Pohhammer polynomials. To study the properties of these particular solutions, we require a decomposition formula expressing the first Lauricell function as the product of simpler hypergeometric functions with fewer variables. Through this study of particular solutions and the determination of singularity order at the origin, we establish the uniqueness of these solutions. Thus, having proved the peculiarity of particular solutions at the origin, it can be argued that the constructed particular solutions are fundamental solutions of the generalized multidimensional second-order Euler-Poisson-Darboux equation.