The Best Approximation Method in Computational Mechanics

1 Topics in Functional Analysis. - 1. 0 Introduction. - 1. 1 Set Theory. - 1. 2 Functions. - 1. 3 Matrices. - 1. 4 Solving Matrix Systems. - 1. 5 Metric Spaces. - 1. 6 Linear Spaces. - 1. 7 Normed Linear Spaces. - 1. 8 Approximations. - 2 Integration Theory. - 2. 0 Introduction. - 2. 1 Reimann and Lebesgue Integrals: Step and Simple Functions. - 2. 2 Lebesgue Measure. - 2. 3 Measurable Functions. - 2. 4 The Lebesgue Integral. - 2. 5 Key Theorems in Integration Theory. - 2. 6 Lp Spaces. - 2. 7 The Metric Space Lp. - 2. 8 Convergence of Sequences. - 2. 9 Capsulation. - 3 Hilbert Space and Generalized Fourier Series. - 3. 0 Introduction. - 3. 1 Inner Product and Hilbert Space. - 3. 2 Best Approximations in an Inner Product Space. - 3. 3 Approximations in L2(E). - 3. 4 Vector Representations and Best Approximations. - 3. 5 Computer Program. - 4 Linear Operators. - 4. 0 Introduction. - 4. 1 Linear Operator Theory. - 4. 2 Operator Norms. - 4. 3 Examples of Linear Operators in Engineering. - 4. 4 Superposition. - 5 The Best Approximation Method. - 5. 0 Introduction. - 5. 1 An Inner Product for the Solution of Linear Operator Equations. - 5. 2 Definition of Inner Product and Norm. - 5. 3 Generalized Fourier Series. - 5. 4 Approximation Error Evaluation. - 5. 5 The Weighted Inner Product. - 5. 6 Considerations in Choosing Basis Functions. - 6 The Best Approximation Method: Applications. - 6. 0 Introduction. - 6. 1 Sensitivity of Computational Results to Variation in the Inner Product Weighting Factor. - 6. 2 Solving Two-Dimensional Potential Problems. - 6. 3 Application to Other Linear Operators. - 6. 4 Computer Program: Two-Dimensional Potential Problems Using Real Variable Basis Functions. - 6. 5 Application of Computer Program. - 7 Solving Potential Problems using the Best Approximation Method. - 7. 0 Introduction. - 7. 1 The Complex Variable Boundary Element Method. -7. 2 Mathematical Development. - 7. 3 The CVBEM and W?. - 7. 4 The Space W?A. - 7. 5 Applications. - 7. 6 Computer Program: Two-Dimensional Potential Problems using Analytic Basis Functions (CVBEM). - 7. 7 Modelling Groundwater Contaminant Transport. - 7. 8 Three Dimensional Potential Problems. - 8 Applications to Linear Operator Equations. - 8. 0 Introduction. - 8. 1 Data Fit Analysis. - 8. 2 Ordinary Differential Equations. - 8. 3 Best Approximation of Function. - 8. 4 Matrix Systems. - 8. 5 Linear Partial Differential Equations. - 8. 6 Linear Integral Equations. - References. - Appendix A Derivation of CVBEM Approximation Function. - Appendix B Convergence of CVBEM Approximator. - Appendix C The Approximate Boundary for Error Analysis. Language: English