Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

PDE's (Solution by diagonalization)

Some PDE's like the heat or wave equation
    ft(t,x) = fxx(t,x)
    ftt(t,x) = fxx(t,x)
can be solved similarly as ODE's.
    xt (t) = A x(t)    
    xtt(t) = A x(t)
Diagonalizing A
   A v=L v
   A v=-n2 v
   A v=0
on eigenspaces led to
    v'=L v    
    v''=-n2 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=D2
T sin(nx)=-n2sin(nx),
T cos(nx)=-n2cos(nx), 
T 21/2 = 0. 
on eigenspaces leads to
f'(t)=-n2f(t)
f''(t)=-n2f(t)
f''(t)=0  
which are solved by
f(t)=exp(-n2 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) = a0 + an cos(n x) + bn sin(n x)
we obtain a solution to the heat equation
f(t,x) = a0 + an exp(-n2 t) cos(n x) + bn exp(-n2 t) sin(n x)
For f,f' written in the Fourier basis as
f(0,x) = a0 + an cos(nx) + bn 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) = (a0+a'0 t)
+(an cos(nt)+a'n sin(nt)/n) cos(nx) +(bn cos(nt) + b'n sin(nt)/n) sin(nx)


Fourier decomposition

In this particular case, the Fourier coefficients bn belonging to sin(nx) are zero because the function is an even function. Also an is zero here if n is even.
= + + + + +
f(x) = a1 cos(x) + a3 cos(3x) + a5 cos(5x) + a7 cos(7x) + a9 cos(9x) +

Heat evolution

The heat evolution is simple on each eigenfunction f=cos(nx) of D2. Since f' = -n2 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 D2. Since f'' = -n2 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



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