Applied Mathematical Sciences

I: Presentation of Numerical Methods.- 1. Finite-Difference Methods for Boundary-Value Problems.- 1.1. Sample Problems.- 1.2. Finite-Difference Methods for Linear, Second Order Ordinary Differential Equations.- 1.3. A Finite-Difference Approximation for the Cantilevered Beam Problem.- 1.4. Finite-Difference Methods for a Nonlinear Boundary-Value Problem.- 1.5. Finite-Difference Approximations for Two-Dimensional Elliptic Equations.- References.- 2. Projection Methods for Variational Equations.- 2.1. Basic Properties of Variational Equations and Sample Problems.- 2.2. Sample Problems (Revised).- 2.3. The Ritz Method.- 2.4. Galerkin Methods and the Method of Least Squares.- 2.5. Projection Methods for Nonlinear Problems.- References.- 3. Approximation Methods for Integral Equations of the Second Kind.- 3.1. Linear Integral Equations of the Second Kind.- 3.2. Quadrature Methods.- 3.3. Projection Methods.- 3.4. Approximations of Nonlinear Integral Equations.- References.- 4. Approximation Methods for Initial Value Problems in Partial Differential Equations 74.- 4.1. Difference Methods for the Heat Equation.- 4.2. Galerkin Methods for the Heat Equation.- 4.3. Numerical Methods for the Wave Equation.- 4.4. The Numerical Solution of Nonlinear Initial Value Problems.- 4.5. Pure Initial Value Problems and a General Representation of Approximation Methods.- References.- II: Convergence Theory.- 5. The Concepts of Discrete Convergence and Discrete Approximations.- 5.1. Definitions, Basic Properties, and First Examples.- 5.2. Restriction and Embedding Operators.- 5.3. Discrete Uniform Convergence of Continuous Functions.- 5.4. Discrete Approximations of Lp-Spaces and Weak Convergence of Measures.- References.- 6. Discrete Convergence of Mappings and Solutions of Equations.- 6.1. Continuity and Differentiability of Mappings and Their Inverses.- 6.2. Stability and Inverse Stability of Sequences of Mappings.- 6.3. Consistency and Discrete Convergence of Mappings.- 6.4. Discrete Convergence of Solutions and Biconvergence.- References.- 7. Compactness Criteria for Discrete Convergence.- 7.1. Discrete Compact Sequences of Elements.- 7.2. A-Regular and Regularly Convergent Mappings.- 7.3. Discrete Compact Sequences of Mappings and Biconvergence for Equations of the Second Kind.- 7.4. Projection Methods for the Approximate Solution of Nonlinear Fixed Point Equations.- References.- III: Convergence Analysis for Approximate Solutions of Boundary-Value Problems and Integral Equations.- 8. Convergence of Finite-Difference Methods for Boundary-Value Problems.- 8.1. Convergence of Difference Methods for Ordinary Differential Equations Via Maximum Principles.- 8.2. Convergence of Difference Methods for Ordinary Differential Equations Via Compactness Arguments.- 8.3. Convergence of the Five-Point Difference Approximation for Poisson's Equation.- References.- 9. Biconvergence for Projection Methods Via Variational Principles.- 9.1. Approximability.- 9.2. Stability, Inverse Stability, and Biconvergence for Linear, Problems.- 9.3. Biconvergence for Nonlinear Problems.- References.- 10. Convergence of Perturbations of Integral Equations of The Second Kind.- 10.1. Statement of the Problem and Consistency.- 10.2. Equidifferentiability.- 10.3. Biconvergence.- References.- IV: Inverse Stability, Consistency and Convergence for Initial Value Problems in Partial Differential Equations.- 11. Inverse Stability and Convergence for General Discrete-Time Approximations of Linear and Nonlinear Initial Value Problems.- 11.1. Statement of the Problem and Differentiability.- 11.2. Inverse Stability.- 11.3. Consistency and Convergence.- References.- 12. Special Criteria for Inverse Stability.- 12.1. Linear Finite-Difference Methods with Positivity Properties.- 12.2. The von Neumann Condition.- 12.3. Inverse Stability of Galerkin Methods.- 12.4. Inverse Stability of Nonlinear Methods.- References.- 13. Convergence Analysis of Special Methods.- 13.1. Consistency and Convergence of Finite-Difference Approximations.- 13.2. Consistency and Error Analysis of Discrete-Time Galerkin Methods.- References.- Glossary of Symbols.