The Complex Variable Boundary Element Method

1: Flow Processes and Mathematical Models.- 1.0 Introduction.- 1.1 Ideal Fluid Flow.- 1.2 Steady State Heat Flow.- 1.3 Saturated Groundwater Flow.- 1.4 Steady State Fickian Diffusion.- 1.5 Use of the Laplace Equation.- 2: A Review of Complex Variable Theory.- 2.0 Introduction.- 2.1 Preliminary Definitions.- 2.2 Polar Forms of Complex Numbers.- 2.3 Limits and Continuity.- 2.4 Derivatives.- 2.5 The Cauchy-Riemann Equations and Harmonic Functions.- 2.6 Complex Line Integration.- 2.7 Cauchy's Integral Theorem.- 2.8 The Cauchy Integral Formula.- 2.9 Taylor Series.- 2.10 Program 1: A Complex Polynomial Approximation Method.- 2.11 Potential Theory and Analytic Functions.- 3: Mathematical Development of the Complex Variable Boundary Element Method.- 3.0 Introduction.- 3.1 Basic Definitions.- 3.2 Linear Global Trial Function Characteristics.- 3.3 The H1 Approximation Function.- 3.4 Higher Order Hk Approximation Functions.- 3.5 Engineering Applications.- 4: The Complex Variable Boundary Element Method.- 4.0 Introduction.- 4.1 A Complex Variable Boundary Element Approximation Model.- 4.2 The Analytic Function Defined by the Approximator $$rm hat{omega }$$(z).- 4.3 Program 2: A Linear Basis Function Approximator $$rm hat{omega }$$(z).- 4.4 A Constant Boundary Element Method.- 4.5 The Complex Variable Boundary Element Method (CVBEM).- 5: Reducing CVBEM Approximation Relative Error.- 5.0 Introduction.- 5.1 Application of the CVBEM to the Unit Circle.- 5.2 Approximation Error from the CVBEM.- 5.3 A CVBEM Modeling Strategy to Reduce Approximation Error.- 5.4 A Modified CVBEM Numerical Model.- 5.5 Program 3: A Modified CVBEM Numerical Model.- 5.6 Determining some Useful Relative Error Bounds for the CVBEM.- 6: Advanced Topics.- 6.0 Introduction.- 6.1 Expansion of the HkApproximation Function.- 6.2 Upper Half Plane Boundary Value Problems.- 6.3 Sources and Sinks.- 6.4 The Approximative Boundary for Error Analysis.- 6.5 Estimating Boundary Spatial Coordinates.