The complex exponential function REVIEW

Hyperbolic functions

Newton’s 2 nd Law for Small Oscillations =0 ~0

Differential eigenvalue problems

Partial derivatives Increment: x part y part

Total derivatives x part y part t part

The material derivative: derivative “following the motion” x part y part t part u v

Multivariate Calculus 2: partial integration separation of variables Fourier methods

Partial differential equations Algebraic equation: involves functions; solutions are numbers. Ordinary differential equation (ODE): involves total derivatives; solutions are univariate functions. Partial differential equation (PDE): involves partial derivatives; solutions are multivariate functions.

Notation

Classification

Order =order of highest derivative with respect to any variable.

Partial integration Instead of constant, add function of other variable(s)

Partial integration

Solution by separation of variables Try to reduce PDE to two (or more) ODEs.

Is it possible for functions of two different variables to be equal?

The Plan

Example 1

Thermal diffusion in a 1D bar Boundary conditions: Initial condition:

Applications Diffusion of: Heat Salt Chemicals, e.g. O 2, CO 2, pollutants Critters, e.g. phytoplankton Diseases Galactic civilizations Money (negative diffusion)

Thermal diffusion in a 1D bar Boundary conditions: Initial condition:

Solution by separation of variables

First try

Second try

Characteristics of time dependence: T  0 as t  ∞, i.e. temperature equalizes to the temperature of the endpoints. Higher  (diffusivity) leads to faster diffusion. Higher n (faster spatial variation) leads to faster diffusion.

In fact: Time scale is proportional to (length scale) 2.

In fact: E.G. Water 1/2 as deep takes 1/4 as long to boil!

Sharp gradients diffuse rapidly.  This observation is surprising.

Now what about the initial condition? Boundary conditions: Initial condition: ?

?

Fourier’s Theorem

To find the constants:

Problem solved Boundary conditions: Initial condition:

Homework clarification 3.3 partial integration 3.4 separation of variables 3.5 Fourier series solution for guitar string 1. Solve for a single Fourier mode Separate Choose sign of separation constant Satisfy boundary conditions and initial condition h t =0 Write down the general solution satisfying h(x,0)=h 0 (x), but don’t derive the coefficients for the Fourier series. 2. Interpret Time dependence: exponential or …? Describe in physical terms. 3. Time scale Is the time scale for a mode proportional to the length scale squared? If not, what?

Isocontours