Methods of quaternionic analysis for the treatment of nonlinear boundary value problems. (Q2711296)

The authors continue their study of quaternionic analysis for solving partial differential equations. In this rich paper analytical methods as well as numerical methods are applied for several types of nonlinear equations. As an example the following equation may serve: Let \(G \subset \mathbb R\) be a bounded domain with a piecewise Lyapunov boundary. For NEWLINE\[NEWLINE\underline{a} =\sum_{i=1}^{3}a_ie_i, \quad \text{grad}_{\underline{a}}u: = \sum_{i=1}^{3} a_i \partial_i u e_i, \quad \text{div}_{\underline{a}}\underline{u}: = \sum_{i=1}^{3}a_i \partial_i u_i, \quad \text{rot}_{\underline{a}}\underline{u}: = \sum_{i=1}^{3} a_i[\text{rot }\underline{u}]_iNEWLINE\]NEWLINE the following equation is studied NEWLINE\[NEWLINE \text{grad }b\text{ div}_{\underline{a}} \underline{u}-\text{div\,rot}_{\underline{a}}\underline{u} +\text{rot\,rot}_{\underline{a}} \underline{u}+c\,\text{grad }(\underline{\lambda}(\underline{u})\cdot\text{rot }\underline{u}) = f(\underline{u})NEWLINE\]NEWLINE in \(G\) and \(\underline{u} = \underline{g}\) on \(\partial G\). \(\lambda\) should be a \(C^{\infty}\)-mapping and \(f\) a mapping which satisfies a Lipschitz condition, the components \(a_i\) od \(a\) are real functions which have to be strictly positive, \(b\) also has to be a \(C^{\infty}\) function, \(c\) is constant.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00011].