Numerical Differential Equations

The following sections are included:

• Dedication
• Preface
• Foreword
• Contents

The following sections are included:

• Introduction
• Normed Linear Spaces
• A Classification of Order 2 PDEs
• Multivariate Polynomial Interpolation
• Numerical Integration for a Rectangle and a Triangle

The following sections are included:

• Introduction
• Flow about an Airfoil: Harmonic Conjugates and Analytic Functions
• Temperature Distribution along a Rod, the Derivation and Solution of the 1D Heat Equation
• The Heat Equation and FDM Stability

The following sections are included:

• Introduction
• Ideal Fluid Flow, the Laplace Equation via FEM and a Rectangular Partition
• Waves in an Enclosed Pool, Helmholtz Equation via FEM and a Triangular Partition

The following sections are included:

• Introduction
• Predator/Prey Models, Stability Analysis
• Visualizing Predator/Prey Population Models with the Runge–Kutta Method
• Population Model with Herding Instinct Using Stochastic Collocation Method

The following sections are included:

• Introduction
• Options Pricing, the Black–Scholes Equation, and Collocation
• Options Pricing, the Black–Scholes Equation, and the 1D Heat Equation
• River Pollution, the Transport–diffusion Equation via Collocation
• Traffic Congestion, a Different Flow Problem

The following sections are included:

• Introduction
• Cell Chemotaxis
• The NSE, the Stokes Flow, and the Bernoulli Equation
• Applying FEM to the Stokes Equation, TR Time Stepping
• Herd Formation Model, FEM
• A Remark on Numerical Processes, Convergence, and Existence Theory

The following sections are included:

• Introduction
• Convergence, Consistency, and Stability
• Extended Differences
• Difference Schemes for Hyperbolic PDE
• Remarks on Stability and Convergence
• Difference Schemes for Elliptical PDE

The following sections are included:

• Introduction
• The Model 1: Elements and Nodes
• The Model 2: Polynomial Basis Functions
• Serendipity
• The Linear System
• Boundary Values, Neumann, Dirichlet, and Robin

The following sections are included:

• Introduction
• The Weak Derivative
• Sobolev Spaces
• Boundary Values
• Weak Solutions of Elliptical Equations
• Sobolev Embedding
• Polynomial Interpolation on a Sobolev Space
• Convergence for Finite Element Method

The following sections are included:

• Introduction
• Interpolation with Legendre Polynomials
• The Collocation Inner Product
• Convergence for Gaussian Collocation
• Remarks on Collocation and Superconvergence

The following sections are included:

• Bibliography
• Index