Monogenic functions in two dimensional commutative algebras to equations of plane orthotropy

Abstract

Among all two-dimensional commutative and assosiative algebras of the second rank with the unity \(e\) over the field of complex numbers \(\mathbb{C}\) we find a semi-simple algebra \(\mathbb{B}_{0} := \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\}\), \(\omega^2=e\), containing a basis \((e_1,e_2)\), such that \( e_1^4 + 2p e_1^2 e_2^2 + e_2^4 = 0 \) for any fixed \( p \) such that \(-1 \lt p \gt 1 \). A domain \(\mathcal{B}_{1}=\{(e_1,e_2)\}\), \(e_1=e\), is discribed in an explicit form. We consider an approach of \(\mathbb{B}_{0}\)-valued ''analytic'' functions \(\Phi(xe_1+ye_2) = U_{1}(x,y)e_1 + U_{2}(x,y)ie_1+ U_{3}(x,y)e_2 + U_{4}(x,y)ie_2\) (\((e_1,e_2)\in \mathcal{B}\), \(x\) and \(y\) are real variables) such that their real-valued components \(U_{k}\), \(k=\overline{1,4}\), satisfy the equation on finding the stress function \(u\) in the case of orthotropic plane deformations (with absence of body forses): \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4 }{\partial x^2\partial y^2}+ \frac{\partial^4 }{\partial y^4} \right) u(x,y)=0\) for every \((x,y)\in D\), where \(D\) is a domain of the Cartesian plane \(xOy\). A characterization of solutions \(u\) for this equation in a bounded simply-connected domain via real components \(U_{k}\), \(k=\overline{1,4}\), of the function \(\Phi\) is done in the following sense: let \(D\) be a bounded and simply-connected domain, a solution \(u\) is fixed, then \(u\) is a first component of monogenic function \(\Phi_{u}\). The variety of such \(\Phi_{u}\) is found in a complete form. We consider a particular case of \((e,e_2)\in \mathcal{B}_{1}\) for which \(\Phi_{u}\) can be found in an explicit form. For this case a function \(\Phi_{u}\) is obtained in an explicit form. Note, that in case of orthotropic plane deformations, when Eqs. of the stress function is of the form: \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4}{\partial x^2\partial y^2}+\frac{\partial^4 } {\partial y^4} \right) u(x,y)=0\), here \(p\) is a fixed number such that \(p>1\), a similar research is done in [Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I. Ukr. Mat. Zh. 2018. 70, No. 8. pp. 1058-1071 (Ukrainian); Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. II. Ukr. Mat. Zh. 2018. 70, No. 10. pp. 1382-1389 (Ukrainian)].