2016/10/03 by S. V. Gryshchuk, Gryshchuk, S. V., С. А. Плакса +1
Mathematics · #30G35 #31A30 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1610.00436
openalex publication_date 2016/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A commutative algebra \mathbbB over the field of complex numbers with the bases \e1,e2\ satisfying the conditions (e12+e22)2=0, e12+e22≠ 0, is considered. The algebra \mathbbB is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem 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 values of two components U1, U4 are given on the boundary of a domain D lying in the Cartesian plane xOy. We develop a method of its solving which is based on expressions of monogenic functions via corresponding analytic functions of the complex variable. For a half-plane and for a disk, solutions are obtained in explicit forms by means of Schwartz-type integrals.