Dear MathStackExchange i need your help. I'm trying to find the computation of this differential equation (from Gibson and Lo) regarding the settlement of soil over time. Consider the following differential equation
$$\frac{k}{\gamma_w}\frac{\partial^2\sigma}{\partial z^2}=a\frac{\partial \sigma}{\partial t}+\lambda\sigma-\frac{\lambda^2}{b}\int^{0}_{t}\sigma e^{-\frac{\lambda}{b}(t-\tau)}d\tau$$
given the following boundary conditions $\sigma(z=0,t)=q_0$ and $\frac{\partial \sigma}{\partial t}=0$ at $z=H$.
In the text is said that Gibson and Lo applied Laplace transforms, convolution and countour integral, to obtain the following solution only dependent on time $t$
$$(a+b)q_{0}H \left(1+\frac{8}{\pi^2}\sum_{n=0}^{\infty}\frac{1}{n^2} \left[\frac{K-x_1}{x_1+x_2}e^{-x_2t}-\frac{K-x_2}{x_1-x_2}e^{-x_1t} \right] \right)$$
where
- $K=\frac{a}{a+b}K_1$
- $K_1=\frac{n^2\pi^2k}{4\gamma_wH^2}$
- $\alpha=\lambda(\frac{1}{a}+\frac{1}{b})$
- $\beta=\frac{\lambda}{b}$
- $x_1=\frac{1}{2}\left((\alpha+K_1)+\sqrt{(\alpha + K_1)^2-4\beta K_1}\right)$
- $x_2=\frac{1}{2}\left((\alpha+K_1)-\sqrt{(\alpha + K_1)^2-4\beta K_1}\right)$
I tried to find the step by step solution, even in articles (I cannot find the original one from Gibson and Lo 1961) but they only refer to the techniques and present the latter solution. Can someone please give me an advice? Thanks!