An Introduction to the Method of Fundamental Solutions

Contents
Preface
About the Authors
Part I: Fundamentals
	1. Introduction
		1.1 What is a Fundamental Solution
		1.2 Its Heritage
		1.3 Use of Fundamental Solutions in Integral Equations
			1.3.1 Green’s theorem
			1.3.2 Fredholm integral equations
			1.3.3 Null-field integral equations
		1.4 Origin of the MFS
		1.5 Link Between the MFS and Integral Equations
		1.6 Simple MFS Codes
		1.7 A Literature Review
	2. Fundamental Solutions
		2.1 Dirac Delta Function
		2.2 Existence of Fundamental Solutions
		2.3 Laplace Equation
		2.4 Helmholtz-Type Equations
			2.4.1 Helmholtz equation
			2.4.2 Modified Helmholtz equation
			2.4.3 Complex variable form
		2.5 Diffusion Equation
		2.6 Wave Equation
		2.7 Advection–Diffusion Equation
		2.8 Elastostatics (Cauchy–Navier Equations)
			2.8.1 Point force solution
			2.8.2 Double force, quadrupole, and hexapole
			2.8.3 Center of dilatation
			2.8.4 Displacement discontinuity
		2.9 Stokes Flow
		2.10 Elastodynamics
			2.10.1 Force fundamental solution
			2.10.2 Spherical wave fundamental solution
		2.11 Biharmonic and Polyharmonic Equations
		2.12 Polymetaharmonic Equations
			2.12.1 Product of Helmholtz operators
			2.12.2 Product of modified Helmholtz operators
			2.12.3 Product of mixed metaharmonic operators
			2.12.4 Product of mixed harmonic and metaharmonic operators
			2.12.5 Plate equations
		2.13 Anisotropic Elliptic Equations
			2.13.1 Quasi-harmonic equation
			2.13.2 Helmholtz-type equations
			2.13.3 Advection–diffusion type equations
		2.14 Anisotropic Elasticity
			2.14.1 3D general anisotropy
			2.14.2 2D general anisotropy
			2.14.3 Transverse isotropy
			2.14.4 Plane strain orthotropy
			2.14.5 Elastodynamics
		2.15 Variable Coefficients
			2.15.1 Quasi-harmonic equation
			2.15.2 Helmholtz-type equations
		2.16 Variable Anisotropic Coefficients
			2.16.1 Quasi-harmonic equation
			2.16.2 Helmholtz-type equations
			2.16.3 Cauchy–Navier equations
		2.17 Half Plane and Half Space Solutions
			2.17.1 Laplace equation
			2.17.2 Helmholtz-type equations
			2.17.3 Cauchy–Navier equations
			2.17.4 Layered geometry
		2.18 Coupled Equations
		2.19 Nonlinear PDEs
	3. Basis Functions
		3.1 T-Complete General Solutions
			3.1.1 Laplacian operator
			3.1.2 Helmholtz-type operators
			3.1.3 Biharmonic operator
			3.1.4 Cauchy–Navier operator
			3.1.5 Stokes flow
			3.1.6 Variable coefficients
		3.2 Polynomials
			3.2.1 General polynomials
			3.2.2 Chebyshev polynomials
		3.3 Radial Basis Functions
			3.3.1 Interpolation of functions
			3.3.2 Solving PDEs
	4. Particular Solutions
		4.1 Method of Particular Solutions
		4.2 Monomials
			4.2.1 Laplacian operator
			4.2.2 Helmholtz-type operators
			4.2.3 General second-order PDEs
			4.2.4 Polyharmonic operators
			4.2.5 Polymetaharmonic operators
			4.2.6 Cauchy–Navier operator
			4.2.7 Other operators
		4.3 Homogeneous Polynomials
		4.4 Chebyshev Polynomials
			4.4.1 Laplacian operator
			4.4.2 Helmholtz-type and other operators
		4.5 Polyharmonic Splines
			4.5.1 Laplacian operator
			4.5.2 Helmholtz-type operators
			4.5.3 Polyharmonic operators
			4.5.4 Polymetaharmonic operators
			4.5.5 Cauchy–Navier operator
		4.6 Multiquadric RBF
			4.6.1 Laplacian operator
			4.6.2 Biharmonic operator
			4.6.3 Cauchy–Navier operator
		4.7 Gaussian RBF
			4.7.1 Laplacian operator
			4.7.2 Biharmonic operator
	5. Solving Partial Differential Equations
		5.1 Mesh versusMeshlessMethods
		5.2 Interpolation of Scattered Data
			5.2.1 Interpolation using basis functions
			5.2.2 Interpolation using nodal values
			5.2.3 Partition of unity
			5.2.4 Least squares approximation
			5.2.5 Moving least squares approximation
		5.3 Local versus Global Interpolation
			5.3.1 Smoothness
			5.3.2 Accuracy and efficiency
		5.4 Condition Number and Stability
			5.4.1 Condition number
			5.4.2 Effective condition number
			5.4.3 Schaback’s uncertainty principle
			5.4.4 High precision computation
		5.5 Solving PDEs—Strong versusWeak Form
			5.5.1 Strong form—Collocation method
			5.5.2 Weak form—Method of weighted residuals
		5.6 Summary
	6. The Method of Fundamental Solutions
		6.1 Singular and Nonsingular Basis Functions
			6.1.1 Potential problems
			6.1.2 Elastostatics
			6.1.3 Elastodynamics
			6.1.4 Non-singular bases
			6.1.5 Distributed fundamental solutions
		6.2 Trefftz Collocation Method
		6.3 Invertibility of MFS Matrices
		6.4 Degenerate Scale
		6.5 Spurious Frequency ofWave Equation
		6.6 BVPs with Harmonic and Metaharmonic BCs
			6.6.1 Error estimate and condition number
			6.6.2 Limit as R→∞
			6.6.3 2D interior potential problem
			6.6.4 Trefftz collocation method
			6.6.5 3D interior potential problem
			6.6.6 Helmholtz-type equations
			6.6.7 Summary
		6.7 BVPs with Non-harmonic and Non-metaharmonic BCs
			6.7.1 Maximum principle
			6.7.2 OverdeterminedMFS by least squares
			6.7.3 Error estimate and condition number
			6.7.4 2D interior potential problem
			6.7.5 3D interior potential problem
			6.7.6 Exterior domain Helmholtz problem
			6.7.7 Summary
		6.8 Localized Method of Fundamental Solutions
			6.8.1 The LMFS methodology
			6.8.2 Locally variable coefficient
Part II: Advanced Topics
	7. Solution of Elliptic BVPs
		7.1 LOOCV
		7.2 Laplace Equation
			7.2.1 Numerical implementation in 2D
			7.2.2 Numerical implementation in 3D
		7.3 Helmholtz Equation
			7.3.1 Numerical implementation in 2D
			7.3.2 Numerical implementation in 3D
			7.3.3 Implementation for exterior problems
		7.4 Biharmonic Equation
			7.4.1 Numerical implementation in 2D
			7.4.2 Numerical implementation in 3D
			7.4.3 Alternative biharmonic formulations
		7.5 Cauchy–Navier Equations
			7.5.1 Numerical implementation in 2D
			7.5.2 Numerical implementation in 3D
	8. The Method of Particular Solutions
		8.1 Two-Step MPS-MFS
		8.2 MPS-MFS by Multiquadric Basis Functions
			8.2.1 Polymetaharmonic operators
			8.2.2 Helmholtz-type operators
			8.2.3 Ghost point method
			8.2.4 Numerical implementation
		8.3 MPS-MFS by Polynomial Basis Functions
			8.3.1 Polymetaharmonic operator
			8.3.2 Numerical implementation
		8.4 One-Step MPS-MFS
		8.5 MPS-MFS by Chebyshev Polynomials
		8.6 Particular Solutions via Fundamental Solutions
			8.6.1 Poisson problems
			8.6.2 Inhomogeneous Helmholtz-type problems
	9. Matrix Decomposition Algorithms for Axisymmetric BVPs
		9.1 Preliminaries
		9.2 Laplace Equation
			9.2.1 2D implementation
			9.2.2 3D implementation
		9.3 Helmholtz Equation
			9.3.1 2D implementation
			9.3.2 3D implementation
		9.4 Biharmonic Equation
			9.4.1 2D implementation
			9.4.2 3D implementation
	10. The Localized Method of Fundamental Solutions
		10.1 Laplacian BVPs
			10.1.1 Numerical implementation in 2D
			10.1.2 Numerical implementation in 3D
		10.2 Biharmonic BVPs
	11. Inverse Problems
		11.1 Cauchy Problems
			11.1.1 Laplace equation
			11.1.2 Helmholtz equation
			11.1.3 Biharmonic equation
			11.1.4 Cauchy–Navier equations
		11.2 Inverse Geometric Problems
			11.2.1 Laplace equation
			11.2.2 Helmholtz equation
			11.2.3 Biharmonic equation
			11.2.4 Cauchy–Navier equations
		11.3 Other Inverse Problems
			11.3.1 Inverse source problems
			11.3.2 Inverse boundary coefficient problems
			11.3.3 Inverse initial/boundary value problems
		11.4 Related Methods and Problems
			11.4.1 Free boundary problems
			11.4.2 Related methods
	12. Geometric Modeling Using the MFS
		12.1 PF/MFS for Surface Construction
		12.2 Design and Construction of Times Roman Font
		12.3 3D Surface Reconstruction
Appendix A: List of Acronyms
Appendix B: List of Fundamental Solutions
Appendix C: Additional MATLAB® Codes
References
Index