Some PDE's like the heat or wave equation
f_{t}(t,x) = f_{xx}(t,x)
f_{tt}(t,x) = f_{xx}(t,x)


can be solved similarly as ODE's.
x_{t }(t) = A x(t)
x_{tt}(t) = A x(t)


Diagonalizing A
A v=L v
A v=n^{2} v
A v=0


on eigenspaces led to
v'=L v
v''=n^{2} v
v''=0


which are solved by
v(t)=exp(Lt) v
v(t)=v(0) cos(nt)+v'(0) sin(nt)/n
v(t)=v(0)+t v'(0)


Diagonalizing T=D^{2}
T sin(nx)=n^{2}sin(nx),
T cos(nx)=n^{2}cos(nx),
T 2^{1/2} = 0.


on eigenspaces leads to
f'(t)=n^{2}f(t)
f''(t)=n^{2}f(t)
f''(t)=0


which are solved by
f(t)=exp(n^{2} t) f(0)
f(t)=cos(nt) f(0)+sin(nt)/n f'(0)
f(t)=f(0)+t f'(0)


For f written in the
Fourier basis
f(0,x) = a_{0}
+ a_{n} cos(n x)
+ b_{n} sin(n x)


we obtain a solution to the heat equation
f(t,x) = a_{0}
+ a_{n} exp(n^{2} t) cos(n x)
+ b_{n} exp(n^{2} t) sin(n x)


For f,f' written in the Fourier basis as
f(0,x) = a_{0}
+ a_{n} cos(nx) + b_{n} sin(nx)
f' (0,x) = a'_{0}
+ a'_{n} cos(nx) + b'_{n} sin(nx)


we obtain a solution to the wave equation
f(t,x) = (a_{0}+a'_{0} t)
+(a_{n} cos(nt)+a'_{n} sin(nt)/n) cos(nx)
+(b_{n} cos(nt) + b'_{n} sin(nt)/n) sin(nx)


Fourier decomposition
In this particular case, the Fourier coefficients b_{n} belonging
to sin(nx) are zero because the function is an even function. Also
a_{n} is zero here if n is even.


= 

+ 

+ 

+ 

+ 

+ 
f(x) 
= 
a_{1} cos(x) 
+ 
a_{3} cos(3x) 
+ 
a_{5} cos(5x) 
+ 
a_{7} cos(7x) 
+ 
a_{9} cos(9x) 
+ 
Heat evolution
The heat evolution is simple on each eigenfunction f=cos(nx) of D^{2}.
Since f' = n^{2} f for such functions, the decay is fast, if n
is large.

Wave evolution
The wave evolution is simple on each eigenfunction f=cos(nx) of D^{2}.
Since f'' = n^{2} f for such functions, the waves oscillate fast for
small wavelength.

Using Symmetry
Remarks. In the case, when we are interested in the evolution on an interval like
[0,], one can flip the graph at the y axes to obtain an
even function f(x)=f(x) on [,]
which has a pure cos series.

For functions which are zero at 0 and ,
it makes also sense to continue on the left to an odd function f(x)=f(x) which has a
sin Fourier series. This simplifies the formulas.



2D heat and wave equation
