zbMATH — the first resource for mathematics

Quaternionic analysis and elliptic boundary value problems. (English) Zbl 0699.35007
Mathematical Research, 56. Berlin: Akademie-Verlag. 253 p. (1989).
Hamilton constructed the quaternions - his “Elements of quaternions” appeared in 1866 - but in contrast with the theory of complex numbers, the impact of quaternion theory in analysis and applications has been very small. The main aim of the book at hand “consists in the statement of a new strategy for solving linear and nonlinear boundary value problems of partial differential equations of mathematical physics by the help of hypercomplex analysis”. In the first chapters the authors introduce quaternionic analysis, formulate and study properties of operators in this context, which also includes the orthogonal decomposition of a suitable space. The boundary value problems tackled are Dirichlet problems for Laplace and Helmholtz’ equations, the time- dependent Maxwell equations, Stokes and Navier-Stokes equations. A chapter on collocation methods introduces a new collocation procedure. In the last chapter a discrete quaternionic function theory is developed which is demonstrated for the discrete Navier-Stokes problem. An appendix summarizes some new developments and related ideas, there is a useful list of references.
The text contains a healthy mix of pure and applied-numeric analysis; the ideas are interesting and they are presented in a clear manner. It will be interesting to see what the impact will be of this detailed proposal to use quaternions.
Reviewer: F.Verhulst

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q30 Navier-Stokes equations
35Q99 Partial differential equations of mathematical physics and other areas of application
30G35 Functions of hypercomplex variables and generalized variables
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs