# Mel'nichenko I. P.

### Algebras of Functionally Invariant Solutions of the Three-Dimensional Laplace Equation

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1284-1290

In commutative associative third-rank algebras with principal identity over a complex field, we select bases such that hypercomplex monogenic functions constructed in these bases have components satisfying the three-dimensional Laplace equation. The notion of monogeneity for these functions is similar to the notion of monogeneity in the complex plane.

### Potential fields with axial symmetry and algebras of monogenic functions of vector variables. III

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 228–243

We obtain new representations of the potential and flow function of three-dimensional potential solenoidal fields with axial symmetry, study principal algebraic analytic properties of monogenic functions of vector variables with values in an infinite-dimensional Banach algebra of even Fourier series, and establish the relationship between these functions and the axially symmetric potential or the Stokes flow function. The developed approach to the description of the indicated fields is an analog of the method of analytic functions in the complex plane used for the description of two-dimensional potential fields.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. II

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1695-1703

We obtain a new representation of potential and flow functions for spatial potential solenoidal fields with axial symmetry. We study principal algebraic-analytic properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. I

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1518-1529

We obtain a new representation of potential and flow functions for space potential solenoidal fields with axial symmetry. We study principal algebraic-analytical properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### Reduction of the principal biharmonic problem for a quadrant to nonsingular integral equations

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 775–784

The principal biharmonic problem for a quadrant with piecewise-continuous boundary conditions is reduced to a system of nonsingular integral equations.

### Biharmonic potentials and plane isotropic displacement fields

Kovalev V. F., Mel'nichenko I. P.

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 229-231

### Biharmonic bases in algebras of the second rank

Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 252–254