The paper gives an explicit representation of the Green’s function of the Dirichlet boundary value problem for the polyharmonic equation in the unit ball. The solution of the homogeneous Dirichlet problem is found. An example of solving the hom...
The Green’s function of the Riquier–Neumann problem for the polyharmonic equation in the unit ball is constructed. Using the obtained Green’s function, an integral representation of the solution to the Riquier–Neumann problem...
In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the...
Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of...
In the last decade, the radial basis function-generated finite difference (RBF-FD) method has been widely used for numerical solutions of partial differential equations on smooth surfaces. In this paper, we present a comparison between two approaches...
In this article, the radial basis function method with polyharmonic polynomials for solving inverse problems of the stationary convection–diffusion equation is presented. We investigated the inverse problems in groundwater pollution problems fo...
This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the con...
This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a...
This paper proposes a localized Hermite method of approximate particular solutions (LHMAPS) for solving the 2D inhomogeneous Helmholtz-type equations. Building on the local scheme of the localized method of approximate particular solutions (LMAPS) fo...
This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball
To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously...
In this study, a novel space-time (ST) marching method is presented to solve linear and nonlinear transient flow problems in porous media. The method divides the ST domain into subdomains along the time axis. The solutions are approximated using ST p...
We develop a Pizzetti calculus for the rank-one Opdam–Cherednik setting based on the Weyl symmetrized generalized translation. Its spectral multiplier is the Jacobi function, and the Jacobi differential equation yields a Darboux transmutation b...
Radial basis function generated finite differences (RBF-FD) represent the latest discretization approach for solving partial differential equations. Their benefits include high geometric flexibility, simple implementation, and opportunity for large-s...
In mechanical engineering, the building industry, and many other branches of industry, vibration machines are widely used, in which circular and directed oscillations predominate in the form of movement of the working equipment. This article examines...