? break one integral into parts

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

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(t-s, x-z)*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(t-s, x-z)*g1(s,z), s = 0 to t1 and z is whole real
line)
+Intergal( G(t-s, x-z)*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 sub-problems:

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(t-s, x-z)*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, x-z)*u1(t1,z), z is whole real line)
+Intergal( G(t-s, x-z)*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 1st-approach? That is:

u21(t,x) = Intergal( G(t=t1, x-z)*u1(t1,z), z is whole real line)
= Intergal( G(t-s, x-z)*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(t-s, x-z)*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

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