Schwartz-type boundary-value problems for canonical domains in a biharmonic plane.

A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]