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Di

erential Equations

Lecture 40: Separation of Variables and Heat Equation IVPs

Taking finite linear combinations of these work similarly to the Dirichlet case (and is the solution to (40.5) when f(x) is a finite linear combination of constants and cosines, in direct analogy to (40.3)), but in general, we’re interested in knowing when we can take infinite sums, i.e.

u(x, t) =

1 2

A+

X

Ane

(

n l

) kt

cos

nπx l

.

n=1

N o t i c e h o w w e w r o t e t h e n = 0 c a s e , a s 1 2 A ; t h e r e a s o n f o r t h i s w i l l b e m a d e c l e a r i n t h e f u t u r e . The initial condition (40.5c) means that we need

f(x) =

1 2

A+

X

An cos

nπx l

.

(40.6)

n=1

An expression of the form (40.6) is called the Fourier cosine series of f(x). 2.2. Other boundary conditions. It’s also possible for certain boundary conditions to re-

quire the “full” Fourier series of the initial data; this is an expression of the

form

f(x) =

1

2

A+

X

An cos

nπx

l

+ B n s i n

nπx

l



,

(40.7)

n=1

but for most of our purposes, we’ll want to solve problems with Dirichlet or Neumann conditions. However, in the process of learning about Fourier sine and cosine series, we’ll also learn how to compute the full Fourier series of a function.

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