V. V. Karachik

Normalized system of functions with respect to the Laplace operator and its applications

A system of functions 0-normalized with respect to the operator ∆ in some domain Ω ⊂ R n is constructed. Application of this system to boundary value problems for the polyharmonic equation is considered. Connection between harmonic functions and solutions of the Helmholtz equation is investigated.

Construction of polynomial solutions to some boundary value problems for Poisson’s equation

A polynomial solution of the inhomogeneous Dirichlet problem for Poisson’s equation with a polynomial right-hand side is found. An explicit representation of the harmonic functions in the Almansi formula is used. The solvability of a generalized third boundary value problem for Poisson’s equation is studied in the case when the value of a polynomial in normal derivatives is given on the boundary. A polynomial solution of the third boundary value problem for Poisson’s equation with polynomial data is found.

ON ONE SET OF ORTHOGONAL HARMONIC POLYNOMIALS

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others. Introduction For the investigation of harmonic polynomials a scalar product for homogeneous polynomials of degree m in the form 〈Pm(x), Qm(x)〉 = Pm(D)Qm(x) was introduced in [1]—one of the basic works on harmonic analysis— where the operator Pm(D) is obtained from the polynomial Pm(x) by replacing each variable xi on the differential operator ∂/∂xi. If we denote a set of all polynomials over C by P , then that scalar product can be extended on P in the following way: 〈P (x), Q(x)〉 = P (D)Q(x)|x=0. This idea proved to be very successful for the investigation of polynomial solutions to systems of PDE with constant coefficients (see for instance [2], [3]). In the present work we shall consider a full set of harmonic polynomials which are furthermore orthogonal both in L2(∂Sn) (Sn is a unit ball in R) and in P , and we shall give one interesting property of theirs (Corollary 2). 1. System of harmonic polynomials G(ν)(x) Let k, s ∈ N (N ≡ N ∪ {0}) and n ∈ N but n > 1. Consider the polynomials

On some integro-differential operators in the class of harmonic functions and their applications

We study properties of integro-differential operators generalizing the operators of the Riemann-Liouville and Caputo fractional differentiation in the class of harmonic functions. The properties obtained are applied to examine some local and nonlocal boundary value problems for the Laplace equation in the unit ball.

On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball

Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous Neumann problem for a polyharmonic equation in the unit ball.

Polynomial Solutions of the Dirichlet Problem for the Biharmonic Equation in the Ball

We find a polynomial solution of the Dirichlet problem for the inhomogeneous biharmonic equation with polynomial right-hand side and polynomial boundary data in the unit ball. To this end, we use the closed-form representation of harmonic functions in the Almansi formula.

Solvability conditions for the Neumann problem for the homogeneous polyharmonic equation

We obtain solvability conditions for the Neumann problem for the homogeneous polyharmonic equation in the unit ball. We study the arithmetic triangle that arises in these conditions. For the elements of that triangle, we obtain recursion relations similar to the relations for the Pascal, Euler, and Stirling triangles. We establish a relationship between the Neumann problem and the Dirichlet problem.

On the solution of the inhomogeneous polyharmonic equation and the inhomogeneous helmholtz equation

We present formulas that simplify finding the solutions of the Poisson equation, the inhomogeneous polyharmonic equation, and the inhomogeneous Helmholtz equation in the case of a polynomial right-hand side. They are based on the representation of an analytic function by harmonic functions. The resulting formulas remain valid for some analytic right-hand sides for which the corresponding operator series converge.

On some special polynomials

New special functions called G-functions are introduced. Connections of G-functions with the known Legendre, Chebyshev and Gegenbauer polynomials are given. For G-functions the Rodrigues formula is obtained.

Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.