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

Serhii V. Gryshchuk, and Sergiy A. Plaksa. "Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations." Open Mathematics 15.1 (2017): 374-381. <http://eudml.org/doc/287985>.

@article{SerhiiV2017,
abstract = {We consider a commutative algebra 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\}\ne 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic -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.},
author = {Serhii V. Gryshchuk, Sergiy A. Plaksa},
journal = {Open Mathematics},
keywords = {Biharmonic equation; Biharmonic algebra; Monogenic function; Schwarz-type boundary value problem; Displacements-type boundary value problem; biharmonic algebra; biharmonic monogenic functions; boundary value problems},
language = {eng},
number = {1},
pages = {374-381},
title = {Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations},
url = {http://eudml.org/doc/287985},
volume = {15},
year = {2017},
}

TY - JOUR
AU - Serhii V. Gryshchuk
AU - Sergiy A. Plaksa
TI - Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 374
EP - 381
AB - We consider a commutative algebra 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}\ne 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic -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.
LA - eng
KW - Biharmonic equation; Biharmonic algebra; Monogenic function; Schwarz-type boundary value problem; Displacements-type boundary value problem; biharmonic algebra; biharmonic monogenic functions; boundary value problems
UR - http://eudml.org/doc/287985
ER -