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Quaternionic boundary element method. (Russian) Zbl 0983.32003

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.

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.)