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We consider a commutative algebra  over the field of complex numbers with a basis { e  1 ,  e  2 } satisfying the conditions(   e   1    2    +   e   2    2)   2    =  0  ,   e   1    2    +   e   2    2    ≠  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 ℝ 2 . We consider ( 1 - 4 )-problem for monogenic -valued functions Φ( xe  1  +  ye  2 ) =  U  1 ( x ,  y ) e  1  +  U  2 ( x ,  y ) i e  1  +  U  3 ( x ,  y ) e  2  +  U  4 ( x ,  y ) i e  2  having the classic derivative in the domain  D   ζ   = { xe  1  +  ye  2  : ( x ,  y ) ∈  D }: to find a monogenic in  D   ζ   function Φ, which is continuously extended to the boundary ∂ D   ζ  , when values of two component-functions  U  1 ,  U  4  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.

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