Introduction

The following are notes from Partial Differential Equations for Scientists and Engineers written by Farlow.

Introduction to Partial Differential Equations

• What are PDE's
• PDE's are Differential equations where the unknown function depends on more than one variable
• examples
• u_t = u_xx (heat equation in one dimension)
• u_t = u_xx + u_yy (heat equation in two dimensions)
• u_rr + 1/r u_r  + 1/r₂ u _θθ (Laplace's Equation in Polar Coordinates)
• Why are PDE's Useful?
• relate space and time 
• used to describe laws of physics
• Solving Partial Differential Equations
• Separation of Variables
• Integral Transforms
• Change of Coordinates
• Transformation of the Dependent Variable
• Numerical Methods
• Perturbation Methods
• Impulse Response
• Integral Equations
• Calculus of Varaiations
• 6 basic classifications of PDEs
• order of equation
• u_t = u_xx (second order)
• Number of variables
• u_t = u_xx (two variables: x and t)
• Linearity
• Linear equations are when all the dependent variables u and its derivates are not multiplied against one another or squared
• A u_xx +B u_xy +C u_yy +D u_x +E u_y +F u = G
• Where A,B,C,D,E,F,G are constants or given functions of x and y
• Homogeneity
• equations are homogeneous if G is 0 for all x and y, if it is nonzero it is non-homogeneous
• Kinds of Coefficients
• constant coefficients versus variable coefficients
• Three basic types of linear equations
• parabolic
• describes heat flow and diffusion
• satsify B² - 4AC = 0
• hyperbolic
• describe vibrating systems
• B² - 4AC > 0
• elliptic
• steady state phenomena
• B² - 4AC < 0

Back to ToC

Farlow's Partial Differential Equations ToC

Table of Contents

The following is a table of contents for notes from Farlow's Partial Differential Equations for Scientists and Engineers.

1. Introduction
2. Diffusion Type Problems
3. Diffusion Type Problems (Parabolic Equations)
4. Boundary Conditions for Diffusion Type Problems
5. Derivation of the Heat Equation
6. Separation of Variables
7. Transforming Nonhomogeneous BCs into Homogeneous BC
8. Solving More Complicated Problems by Separation of Variables
9. Transforming Hard Equations into Easier ones
10. Solving Nonhomogenous PDEs (Eigenfunction Expansions)
11. Integral Transforms (Sine and Cosine Transforms)
12. The Fourier Series and Transform
13. The Fourier Transform and its Applications to PDEs
14. The Laplace Transform
15. Duhamel's Principle
16. The Convection Term in Diffusion Problems
17. The One-Dimensional Wave Equation (Hyperbolic Equation)
18. The D'Alembert Solution of the Wave Equation
19. More on the D'Alembert Solution
20. Boundary Conditions associated with the Wave Equation
21. The Finite Vibrating String (Standing Waves)
22. The Vibrating Beam (Fourth-Order PDE)
23. Dimensionless Problems
24. Classification of PDEs (Canonical Form of the Hyperbolic Equation)
25. The Wave Equation in Two and Three Dimensions (Free space)
26. The Finite Fourier Transforms (Sine and Cosine Transforms)
27. Superposition
28. First order Equations (Methods and Characteristics)
29. Nonlinear First-Order Equations (Conservation Equations)

Back to ToC