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Contents Preface Acknowledgements CHAPTER 1 Introduction 1.1 Historic Review 1.2 Basic Algorithms 1.3 Numerical Experiments 1.4 Characteristics of the MFS Part ¢ñ Laplace's Equation CHAPTER 2 Dirichlet Problems 2.1 Basic Algorithms of MFS 2.2 Preliminary Lemmas 2.3 Main Theorems 2.4 Stability Analysis for DiskDomains 2.5 Proof Methodology CHAPTER 3 Neumann Problems 3.1 Introduction 3.2 Method of Fundamental Solutions 3.2.1 Description of Algorithms 3.2.2 Main Results of Analysisand Their Applications 3.3 Stability Analysis of DiskDomains 3.4 Stability Analysis for BoundedSimply-Connected Domains 3.4.1 Trefftz Methods 3.4.2 Collocation Trefftz Methods 3.5 Error Estimates 3.6 Concluding Remarks CHAPTER 4 Other Boundary Problems 4.1 Mixed Boundary Condition Problems 4.2 Interior Boundary Conditions 4.3 Annular Domains CHAPTER 5 Combined Methods 5.1 Combined Methods 5.2 Variant Combinations of FS and PS 5.2.1 Simplified Hybrid Combination 5.2.2 Hybrid Plus Penalty Combination 5.2.3 Indirect Combination 5.3 Combinations of MFS with Other Domain Methods 5.3.1 Combined with FEM 5.3.2 Combined with FDM 5.3.3 Combined with Radial Basis Functions 5.4 Singularity Problems by Combination of MFS and MPS CHAPTER 6 Source Nodes on Elliptic Pseudo-Boundaries 6.1 Introduction 6.2 Algorithms of MFS 6.3 Error Analysis 6.3.1 Preliminary Lemmas 6.3.2 Error Bounds 6.4 Stability Analysis 6.5 Selection of Pseudo-Boundaries 6.6 Numerical Experiments 6.7 Concluding Remarks Part ¢ò.Helmholtz's Equations and Other Equations CHAPTER 7 Helmholtz Equationsin Simply-Connected Domains 7.1 Introduction 7.2 Algorithms 7.3 Error Analysis for Bessel Functions 7.3.1 Preliminary Lemmas 7.3.2 Error Bounds with Small k 7.3.3 Exploration of Bounded k 7.4 Stability Analysis for Disk Domains 7.5 Application to BKM CHAPTER 8 Exterior Problems of Helmholtz Equation 8.1 Introduction 8.2 Standard MFS 8.2.1 Basic Algorithms 8.2.2 Brief Error Analysis 8.3 Numerical Characteristics of Spurious Eigenvalues by MFS 8.4 Modified MFS 8.5 Error Analysis for Modified MFS 8.5.1 Preliminary Lemmas 8.5.2 Error Bounds 8.6 Stability Analysis for Modified MFS 8.7 Numerical Experiments 8.7.1 Circular Pseudo-Boundaries by Two MFS 8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS 8.8 Concluding Remarks CHAPTER 9 Helmholtz Equations in Bounded Multiply-Connected Domains 9.1 Introduction 9.2 Bounded Simply-Connected Domains 9.2.1 Algorithms 9.2.2 Brief Error Analysis 9.3 Bounded Multiply-Connected Domains 9.3.1 Algorithms 9.3.2 ErrorAnalysis 9.4 Stability Analysis for Ring Domains 9.5 Numerical Experiments 9.6 Concluding Remarks CHAPTER 10 Biharmonic Equations 10.1 Introduction 10.2 Preliminary Lemmas 10.3 Error Bounds 10.4 Stability Analysis for Circular Domains 10.4.1 Approaches for Seeking Eigenvalues 10.4.2 Eigenvalues ¦Ëk(¦µ) and ¦Ëk(D¦µ) 10.4.3 Bounds of Condition Number 10.5 Numerical Experiments CHAPTER 11 Elastic Problems 11.1 Introduction 11.2 Linear Elastostatics Problemsin2D 11.2.1 Basic Theory 11.2.2 Traction Boundary Conditions 11.2.3 Fundamental Solutions 11.2.4 Particular Solutions 11.3 HTM,MFS and MPS 11.3.1 Algorithms of HTM 11.3.2 Algorithms of MFS and MPS 11.4 Errors Between FS and PS 11.4.1 Preliminary Lemmas 11.4.2 Polynomials Pn Approximated by *and * 11.4.3 Other Proof for Theorem 11.4.1 11.4.4