Numerical Methods for Differential Equations

Numerical Basics

1. Interpolation

• Polynomial Interpolation with Vandermonde Matrix
• Lagrange interpolation
• Newton Basis Polynomial
• Spline Interpolation

2. Approximation and Errors

• Polynomial Interpolation Error
• Approximation through Regression
• Minimum Quadratic Error

3. Numerical Integration

• Quadrature - Derivations and Formulae
• Quadrature Error
• Quadratures - Accuracy and Error Table
• Difference between Orders of Error, Accuracy & Convergence
• Composite Quadrature
• Romberg’s Extrapolation Method

4 . Nonlinear Equations

• Fixed point or Banach’s iteration
• Banach’s fixed point theorem
• Newton-Raphson or Newton’s Method
• Relationship between Fixed Point Iteration & Newton’s Method

Numerical Solution of ODEs

1. Simple One Step Methods

• Overview
• Euler-Heun Method

2. Error, consistency and convergence

• Order of Consistency
• Order of Consistency of the Implicit Midpoint Rule

3. Classical Runge-Kutta Method

• Butcher Table
• Runge-Kutta Method Procedure

Heat Conduction Equation: Discretization of a PDE

1. Discretization in One Dimension

• Method of Lines

Misc

• Numerical Methods Test Prep