Solving More Difficult Partial Differential Equations (Part 5, Sec. iii)

 Look at the condition u(x,y,0) =

 T _o  xy (x - a) (y - b)

Then:

 å^¥ _n=1  å^¥ _m= 1 a _nm sin (np/ a) x sin  (mp/ b)

Suppose a function f(x, y) which is periodic of period 2p 

Consider y fixed and suppose the function has the Fourier Series expansion;

f(x, y) =   å^¥ _n=o [a (y) cos nx + b(y) sin nx]

Suppose in addition that a(y) and b(y) have Fourier series expansions, e.g.

a(y) = å^¥ _n=o [a _nm cos ny + b _nm sin ny]

And:

b(y) = å^¥ _m=o  (g _nm cos my + d _nm sin ny]

Substitution of a _n (y),  b _n (y) into series for f(x,y) results in terms of the form:

cos nx cos my, cos nx sin my, sin nx cos my, sin nx sin my

Hence:

f(x, y) = å^¥ _n=o å^¥ _m=o  a _nm (cos nx cos my) + b _nm (cos nx sin my)  +

c _nm (sin nx cos my) +   d _nm (sin nx sin my)

Note that the preceding is a double Fourier series.

This brief diversion enables us to continue our treatment as follows, so writing:

ò ^p _-p  f(x,y) sin ax dx = p å^¥ _m=1   c _am cos my + p å^¥ _m=1  d _am sin my

Next, multiply both sides by sin by and integrate, i.e.

Which yields:

ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax dx] sin by dy

Þ

å _m  ò ^p _-p  d _am sin by  sin my dy   =  p d _ab

Which tells us:

 d _ab   =   1/ p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax sin by dx dy

Similarly:  

a _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax cos by dx dy

b _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax sin by dx dy

c _ab   = 1/ p ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax cos by dx dy

And further:  

d _n0 =  d _0m   =    c _0m  =    b _n0

_{Now take:}

_{ò ^p -p   [ò ^p -p  f(x,y) sin ax dx dy =}  p (2 p c _a0)

c _a0   = 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin ax dx dy

And:

b _0b   =   1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin by dx dy

Using the preceding we may write:

 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) cos ax dx dy =  a _a0

Similarly:

a _ab   = 1/ 2p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) cos by dx dy 

To get  a _a0:    Take:  ò ^p _-p  f(x,y)  dx = 

2p å^¥ _m=0   a _0m cos my  + 2p å^¥ _m=0   b _0m sin my  etc.

If f(x,y) is an odd function in x and y then:

a _oo =  a _0m   =  a _nm  =    b _0m  =    b _nm=    c _no  =  c _nm  = 0

The series then reduces to:

å^¥ _n=1  å^¥ _{m= 1   d nm} sin nx sin my

And:

d _nm   = 4/p ²  ò ^p _{-p   [}ò ^p _-p  f(x,y) sin nx sin my dx dy   {if f(x,y) odd