Alekhin, V. V.; Grigor’ev, Yu. V. Quaternionic boundary element method. (Russian) Zbl 0983.32003 Sib. Zh. Ind. Mat. 2, No. 1, 47-52 (1999). Let \(\mathbf i\), \(\mathbf j\), \(\mathbf k\) be the basis quaternionic unity elements and let \(\mathbf f(\mathbf r) = f_0(x,y,z) + \mathbf if_x(x,y,z) + \mathbf jf_y(x,y,z) + \mathbf kf_z(x,y,z)\) be a quaternionic function, where \(\mathbf r\) is a radius vector. Given the scalar part \(\mathbf f_0\in C^{\alpha}\) and normal component \(\mathbf f^n\in C^{\alpha}\) of a function \(\mathbf f\) on \(S=\partial\Omega\subset\mathbb R^3\), where \(\Omega\) is a bounded domain with piecewise smooth boundary \(S\), the problem reads as follows: Reconstruct in \(\Omega\) a regular function \(\mathbf f\) satisfying the additional conditions \[ \nabla\cdot\mathbf f = 0,\qquad \nabla \mathbf f_0 + \nabla\times\mathbf f = 0. \] The method for solving the problem consists in finding a sequence of approximate solutions based on using the Cauchy integral formula. Unknown boundary components of \(\mathbf f\) are determined from the singular integral equation \[ \mathbf f(\mathbf r) -(K_S\mathbf f)(\mathbf r) = 0,\qquad \mathbf r\in S, \] where \(K_S\) is the spatial analog of the Cauchy integral operator.The rest of the article is devoted to a numerical example which illustrates the method presented. Reviewer: V.Grebenev (Novosibirsk) Cited in 2 Documents MSC: 32A30 Other generalizations of function theory of one complex variable 32A40 Boundary behavior of holomorphic functions of several complex variables 45G05 Singular nonlinear integral equations 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) Keywords:quaternionic boundary element method; singular integral equation; Cauchy formula; approximate solution × Cite Format Result Cite Review PDF