Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {<jats:italic>e</jats:italic><jats:sub>1</jats:sub>, <jats:italic>e</jats:italic><jats:sub>2</jats:sub>} satisfying the conditions
<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2017-0025_eq_001.png" />
<jats:tex-math> $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ </jats:tex-math></jats:alternatives></jats:inline-formula> Let <jats:italic>D</jats:italic> be a bounded simply-connected domain in ℝ<jats:sup>2</jats:sup>. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(<jats:italic>xe</jats:italic><jats:sub>1</jats:sub> + <jats:italic>ye</jats:italic><jats:sub>2</jats:sub>) = <jats:italic>U</jats:italic><jats:sub>1</jats:sub>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>)<jats:italic>e</jats:italic><jats:sub>1</jats:sub> + <jats:italic>U</jats:italic><jats:sub>2</jats:sub>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>)<jats:italic>i e</jats:italic><jats:sub>1</jats:sub> + <jats:italic>U</jats:italic><jats:sub>3</jats:sub>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>)<jats:italic>e</jats:italic><jats:sub>2</jats:sub> + <jats:italic>U</jats:italic><jats:sub>4</jats:sub>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>)<jats:italic>i e</jats:italic><jats:sub>2</jats:sub> having the classic derivative in the domain <jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub> = {<jats:italic>xe</jats:italic><jats:sub>1</jats:sub> + <jats:italic>ye</jats:italic><jats:sub>2</jats:sub> : (<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>) ∈ <jats:italic>D</jats:italic>}: to find a monogenic in <jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub> function Φ, which is continuously extended to the boundary ∂<jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub>, when values of two component-functions <jats:italic>U</jats:italic><jats:sub>1</jats:sub>, <jats:italic>U</jats:italic><jats:sub>4</jats:sub> are given on the boundary ∂<jats:italic>D</jats:italic>. 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.</jats:p>