# $\mathbb{B}_0$-Valued Monogenic Functions to the theory of Plane Anisotropy

## TL;DR

This paper develops a hypercomplex function approach to solve a fourth-order elliptic PDE related to plane anisotropy in stress analysis, extending classical methods to non-isotropic materials.

## Contribution

It introduces a novel method using hypercomplex analytic functions in a semisimple algebra to address boundary value problems for anisotropic stress PDEs.

## Key findings

- Solution expressed via hypercomplex functions in bounded, simply-connected domains.
- Reduction of boundary value problems to hypercomplex function boundary conditions.
- Extension of classical isotropic PDE methods to anisotropic cases.

## Abstract

A solution of the elliptic type PDE of the 4th order, being a reduction of the Eqs. of stress function corresponding to any case of plane anisotropy which is not equal to isotropy (proved by S.\,G.~Mikhlin), is described in terms of hypercomplex `analytic' functions with values in two-dimensional semisimple algebra over the field of complex numbers in case when a domain under consideration is bounded and simply-connected. A boundary value problem on finding a function which satisfies this PDE in the considered domain (bounded and simply-connected) and permits continuations (to the boundary) of operators $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ acted to it, is reduced to certain BVP for these hypercomplex functions.