Solving Heat equation by Boundary Element Methods

Sorry to post this message in English but my computer I am using now doesn't have the Thai font.

I have a problem on solving the Heat equation using Boundary Element Method (BEM), which I can?t find it in any texts/ journals. So any of you who arespecialised with this kind of math technique please help.

The trouble is that Volterra equation (of the first kind) with singularity (Abel equation) eventually comes into play after BEM is first introduced into this heat equation.

Given the one-dimensional heat equation
du/dt = du^2/dx^2
or in the homogeneous form
du/dt - du^2/dx^2 = 0
Solving for
u(x, t; x?, t?) where x? > x and t? > t
Its initial condition is
u(x, 0) = f(x) = 1 for all 0<= x? <= 1 (let us assume that f(x) = 1 for simplicity)
and the boundary conditions are
u(0, t?) = u(1, t?) = 0 where t? > 0
Introducing Dirac delta function
g(x, t; x?, t?)
which satisfies the equation
du/dt + du^2/dx^2 = 0
and that by taking limit t? &agrave; t
g(x, t; x?) = delta_(x-x?)
Hence
g(x, t; x?, t?) = 1/(sqrt(4.pi.(t?-t)) . exp(-(x-x?)^2/4(t?-t))
Multiplying the function g with du/dt - du^2/dx^2 , integrating twice with respect to x (from 0 to 1) and with respect to t (from 0 to t? ) result in

u(x?, t?) = INT with respect to x from 0 to 1 on g . f(x) given that t = 0
+ INT with respect to t from 0 to t? on g. du/dx given that x = 1
- INT with respect to t from 0 to t? on g. du/dx given that x = 0

The derivation of u(x?, t?) may be a bit lengthy but quite straightforward though.
Let denote the du/dx on both boundaries at x = 0 and x = 1 by F_0 and F_1, respectively. Notice that F_0 and F_1 can?t be taken out of the integral since they are not constant. One of the mechanical engineering paper published in the paper-collection text, edited by Brebbia, C.A., called ?New Developments in Boundary Element Methods: proceedng of the second international seminar on recent advances in Boundary Element methods held at University of Southampton March (sponsored by the International Society for Computational Methods in Engineering? got it wrong by assuming that F_0 and F_1 does not vary much, and therefore can be removed out of the integral signs. This is in fact a misleading assumption really. In my opinion unless we use numerical method and assume that by Mean value theorem we can then take them out of from the first small time step, we can?t do that because F_0 and F_1 represents heat fluxes which of course vary through time.

However setting x? = 0 and x? = 1 on both ends, the equation equals zero which is subject to boundary conditions.

At x? = 0, we have
0 = INT with respect to x from 0 to 1 on exp(-(x-0?)^2/4(t?-0))/(sqrt(4.pi.(t?-0))

+ INT with respect to t from 0 to t? on exp(-(1-0)^2/4(t?-t))/(sqrt(4.pi.(t?-t)) . F_1
- INT with respect to t from 0 to t? on exp(-(0-0)^2/4(t?-t))/(sqrt(4.pi.(t?-t)) . F_0

= INT with respect to x from 0 to 1 on exp(-(x-0?)^2/4(t?))/(sqrt(4.pi.(t?))

+ INT with respect to t from 0 to t? on exp(-(1-0)^2/4(t?-t))/(sqrt(4.pi.(t?-t)) . F_1
- INT with respect to t from 0 to t? on 1/(sqrt(4.pi.(t?-t)) . F_0

At x? = 1, we then have
0 = INT with respect to x from 0 to 1 on exp(-(x-0?)^2/4(t?))/(sqrt(4.pi.(t?))

+ INT with respect to t from 0 to t? on 1/(sqrt(4.pi.(t?-t)) . F_1
- INT with respect to t from 0 to t? on exp(-(-1)^2/4(t?))/(sqrt(4.pi.(t?-t)) . F_0

My question is that how can we solve for the unknown F_0 and F_1 which can lead to solving for u(x?, t?), analytically.

The third integral (when x? = 0) and the second integral (when x? = 1), where both kernels vanish), are multiplied by 4 Pi in order to get rid of it for the moment

INT with respect to t from 0 to t? on F_0/(sqrt((t?-t)
and
INT with respect to t from 0 to t? on F_1/(sqrt((t?-t)

can be solved explicitly as shown in ?Analytical and Numerical Solutions for Volterra Equations?, by Peter Linz, a SIAM applied math text. These Abel equations have singularity when integrating on the denominator: sqrt(t? t).

Note that we can?t differentiate both equations on the boundaries to get rid of the integral with respect to t? since the result would be zero according to the kernels. I know that this is really an awkward situation whereby nothing much can be done explicitly. But on the other hand, I hope that some of you, good Thai mathematicians, would help me sorting this problem out analytically. So please help.

By the way, another alternative as far as I could think of for the moment is Laplace transform. Nevertheless, using either approach to solve this time-dependent equation should be cumbersome.

Thank you.