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 A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained?
Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained?
In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1):
$$ y(\bar{r}) = 1 + n \int_{\bar{r}'<R} y(\bar{r}') d\bar{r}'  n \int_{\bar{r}'<R, \bar{r}  \bar{r}'<R} y(\bar{r}') y(\bar{r}  \bar{r}') d\bar{r}'$$
Where ##\bar{r}## and ##\bar{r}'## are 3vectors (x, y, z) and ##y(\bar{r})## is a function of one of those vectors. R and n are constants. Note that ##\bar{r}## and ##\bar{r}'## are two different vector variables, the prime does NOT indicate a derivative. He then says he uses a onesided Laplace transform to obtain the following equation (Eq. 3):
$$t[F(t) + G(t)] = t^{1}[1 + 24 \eta K ]  12 \eta [F(t)  F(t)]G(t)$$
where
$$ F(t) = R^{2} \int^R_0 ry(r) exp(sr) dr $$
$$ G(t) = R^{2} \int^\infty_R ry(r) exp(sr) dr $$
$$ K = R^{3} \int^R_0 r^2 y(r) dr $$
$$\eta = \dfrac{1}{6} \pi R^3 n$$
$$t=sR$$
A 3D Laplace transform in spherical coordinates seems rare enough that it is difficult to find resources to reach the latter equations from the first one. I've been able to convert the Laplace transform to spherical coordinates, integrate over the angles, and obtain some equations similar to the latter equations, but not exactly. Clearly, the convolution property of Laplace transforms are used to obtain the RHS of Eq. 3. Any further suggestions or guidance that anyone could provide would be much appreciated.
In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1):
$$ y(\bar{r}) = 1 + n \int_{\bar{r}'<R} y(\bar{r}') d\bar{r}'  n \int_{\bar{r}'<R, \bar{r}  \bar{r}'<R} y(\bar{r}') y(\bar{r}  \bar{r}') d\bar{r}'$$
Where ##\bar{r}## and ##\bar{r}'## are 3vectors (x, y, z) and ##y(\bar{r})## is a function of one of those vectors. R and n are constants. Note that ##\bar{r}## and ##\bar{r}'## are two different vector variables, the prime does NOT indicate a derivative. He then says he uses a onesided Laplace transform to obtain the following equation (Eq. 3):
$$t[F(t) + G(t)] = t^{1}[1 + 24 \eta K ]  12 \eta [F(t)  F(t)]G(t)$$
where
$$ F(t) = R^{2} \int^R_0 ry(r) exp(sr) dr $$
$$ G(t) = R^{2} \int^\infty_R ry(r) exp(sr) dr $$
$$ K = R^{3} \int^R_0 r^2 y(r) dr $$
$$\eta = \dfrac{1}{6} \pi R^3 n$$
$$t=sR$$
A 3D Laplace transform in spherical coordinates seems rare enough that it is difficult to find resources to reach the latter equations from the first one. I've been able to convert the Laplace transform to spherical coordinates, integrate over the angles, and obtain some equations similar to the latter equations, but not exactly. Clearly, the convolution property of Laplace transforms are used to obtain the RHS of Eq. 3. Any further suggestions or guidance that anyone could provide would be much appreciated.
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