Cheng Cosine science forum Guru Wannabe
Joined: 26 May 2005
Posts: 168

Posted: Fri Jul 21, 2006 8:54 am Post subject:
? break one integral into parts



Hi:
Given an integral I(t) = Integral( g(s), s = 0 to t ), g(s) = g1(s) when s
= 0 to t1
and g(s) = g2(s) when s = t1 to t, then this integral can also be evaluated
as:
I(t) = I1(t)+I2(t) = Integral( g1(s), t = 0 to t1 )+Integral( g2(s), t = t1
to t ).
Now suppose one has a diffusion equation: pdiff(u)/pdiff(t) =
laplace(u,x)+g(t,x)
where x lis in whole real liine and where g(s,x) = g1(s,x) when s = 0 to t1
, and g(s,x) = g2(s,x) when s = t1 to t.
Solution in Green's function is
u(t,x) = Intergal( G(ts, xz)*g(s,z), s = 0 to t and z is whole real line)
here G is Green's function. Like what we have at very begining, this can
be exressed as:
u(t,x) = Intergal( G(ts, xz)*g1(s,z), s = 0 to t1 and z is whole real
line)
+Intergal( G(ts, xz)*g2(s,z), s = t1 to t and z is whole real
line)
= u11(t,x)+u12(t,x)
But if we start with PDE then we have 2 subproblems:
from t = 0 to t1:
pdiff(u)/pdiff(t) = laplace(u,x)+g1(t,x) with zeros IC
and soln is:
u(t,x) = Intergal( G(ts, xz)*g1(s,z), s = 0 to t1 and z is whole real
line)
= u1(t1,x) when t = t1
BUT from t = t1 to t
pdiff(u)/pdiff(t) = laplace(u,x)+g1(t,x) with IC to be u1(t1,x)
and soln is:
u(t,x) = Intergal( G(t=t1, xz)*u1(t1,z), z is whole real line)
+Intergal( G(ts, xz)*g2(s,z), s = t1 to t and z is whole real
line)
= u21(t,x)+u22(t,x)
For the above we see the 2nd integral is okay, but how does one see the 1st
integral
is the same as 1st term of the 1stapproach? That is:
u21(t,x) = Intergal( G(t=t1, xz)*u1(t1,z), z is whole real line)
= Intergal( G(ts, xz)*g1(s,z), s = 0 to t1 and z is whole real line)
= u11(t,x)
Especially u1(t1,z) is expressed by Intergal( G(ts, xz)*g1(s,z), s = 0 to
t1 and z is whole real line)
so u21(t,x) actually has TWO integrals in its expression.
Thanks,
by Cheng Cosine
Jul/21/2k6 NC 
