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

• Autorzy: Serhii V. Gryshchuk , Sergiy A. Plaksa
• Języki publikacji: EN

Abstrakty

EN

We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions [...] (e12+e22)2=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.

Słowa kluczowe

EN

Biharmonic equation; Biharmonic algebra; Monogenic function; Schwarz-type boundary value problem; Displacements-type boundary value problem

Kategorie tematyczne

• 74B05: Classical linear elasticity
• 31A30: Biharmonic, polyharmonic functions and equations, Poisson's equation
• 30G35: Functions of hypercomplex variables and generalized variables

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2017
• Tom: 15
• Numer: 1
• Strony: 374-381

Daty

wydano

2017-01-01

otrzymano

2016-10-22

zaakceptowano

2017-02-12

online

2017-04-01

Twórcy

autor

Serhii V. Gryshchuk

• Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str. 3, 01004, Kiev-4,

autor

Sergiy A. Plaksa

• Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str. 3, 01004, Kiev-4,

Identyfikatory

DOI

10.1515/math-2017-0025