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

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Gryshchuk S. V., Plaksa S. A.

Open Mathematics

2017

tom 15

nr 1

374-381

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.

Ograniczanie wyników

1 Open Mathematics

1 Gryshchuk S. V.

1 Plaksa S. A.

1 2017

Wyniki wyszukiwania

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