Quaternionic Analysis and Elliptic Boundary Value Problems

1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU’s Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of the T-Operator.- 2.4. VEKUA’s Theorems.- 2.5. Some Integral Operators on the Manifold.- 3. Orthogonal Decomposition of the Space L2,H(G).- 4. Some Boundary Value Problems of DIRICHLET’s Type.- 4.1. LAPLACE Equation.- 4.2. HELMHOLTZ Equation.- 4.3. Equations of Linear Elasticity.- 4.4. Time-independent MAXWELL Equations.- 4.5. STOKES Equations.- 4.6. NAVIER-STOKES Equations.- 4.7. Stream Problems with Free Convection.- 4.8. Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity.- 5. H-regular Boundary Collocation Methods.- 5.1. Complete Systems of H-regular Functions.- 5.2. Numerical Properties of H-complete Systems of H-regular Functions.- 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems.- 5.4. Numerical Examples.- 6. Discrete Quaternionic Function Theory.- 6.1. Fundamental Solutions of the Discrete Laplacian.- 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator.- 6.3. Elements of a Discrete Quaternionic Function Theory.- 6.4. Main Properties of Discrete Operators.- 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations.- 6.6. Concluding Remarks.- References.- Notations.