#hw26.txt.  Katie McKeon.  May 1, 2014.  

Help26:=proc()
 print(`ExpD2(M,N)`):
end:

#ExpD2(M,N): Let L(i,j) be the expected duration of the
#walk starting at i,j on the MxN grid and ending when
#(-,N), (-,0), (M,-) or (0,-) is reached, outputs the list
#of lists [[L[1,1],...,L[1,N-1]],...[L[M-1,1],...L[M-1,N-1]]]
ExpD2:=proc(M,N) local eq,var,i,j,var1, L,p:
p:=1/4:
var:=[seq([seq(L[i,j],j=1..N-1)],i=1..M-1)]:

eq:={seq(seq(L[i,j]=p*L[i-1,j]+p*L[i+1,j]+p*L[i,j-1]+p*L[i,j+1]+1,
	j=1..N-1),i=1..M-1)}:

eq:=subs({seq(L[i,0]=0,i=0..M),seq(L[i,N]=0,i=0..M),
	seq(L[0,j]=0,j=0..N),seq(L[M,j]=0,j=0..N)},eq):

var1:=solve(eq,{seq(op(var[i]),i=1..M-1)}):
subs(var1,var):

end:


#Data
#ExpD2(2,2);
#	[[1]]
#ExpD2(2,3);
#	[[4/3, 4/3]]
#ExpD2(5,5);
#	[[10/3, 14/3, 14/3, 10/3], 
#	[14/3, 20/3, 20/3, 14/3], 
#	[14/3, 20/3, 20/3, 14/3], 
#	[10/3, 14/3, 14/3, 10/3]]