S. A. Plaksa

Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.

Monogenic functions in the biharmonic boundary value problem

We consider a commutative algebra over the field of complex numbers with a basis {e1,e2} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and Dζ={xe1+ye2:(x,y)∈D}. Components of every monogenic function Φ(xe1+ye2) = U1(x,y)e1+U2(x,y)ie1+U3(x,y)e2+U4(x,y)ie2 having the classic derivative in Dζ are biharmonic functions in D, that is, Δ2Uj(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz-type boundary value problem for monogenic functions in a simply connected domain Dζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ∂V/∂x, ∂V/∂y are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy-type integral, we reduce the mentioned Schwarz-type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright

Dirichlet Problem for the Stokes Flow Function in a Simply-Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for the Stokes flow function in a simply-connected domain of the meridian plane to the Cauchy singular integral equation. For the case where the boundary of the domain is smooth and satisfies certain additional conditions, the regularization of the indicated singular integral equation is carried out.

Algebras of Hypercomplex Monogenic Functions and Axial-Symmetrical Potential Fields

System of equations

Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

y\frac{{\partial\varphi(x,y)}}{{{\partial_x}}}=\frac{{\partial\psi(x,y)}}{{{\partial_y}}},y\frac{{\partial\varphi (x,y)}}{{{\partial_y}}}=-\frac{{\partial\psi(x,y)}}{{{\partial_x}}}

MONOGENIC FUNCTIONS IN A BIHARMONIC ALGEBRA

(1) describes the spatial potential solenoid field, which is symmetrycal with respect to the axis Ox,in its meridianalplane xOy.Here φ,o is the axial-symmetrical potential and ψ is the Stokes flow function. The function ψ also should satisfy the condition

COMMUTATIVE ALGEBRAS OF HYPERCOMPLEX MONOGENIC FUNCTIONS AND SOLUTIONS OF ELLIPTIC TYPE EQUATIONS DEGENERATING ON AN AXIS

\psi (x,0) \equiv 0.