# beq.song: Song et al.'s analytic solution to Boussinesq equation in a... In boussinesq: Analytic Solutions for (ground-water) Boussinesq Equation

## Description

Song et al.'s analytic solution to Boussinesq equation in a 1D semi-infinite domain with a Dirichlet boundary condition

## Usage

 ```1 2``` ``` beq.song(t = 0.5, x = 1, s = 0.4, h1 = 1, ks = 0.01, nmax = 4, alpha = 1) ```

## Arguments

 `t` time coordinate. `x` spatial coordinate. Default is `seq(from=0,to=L,by=by)`. `h1` water surface level or boundary condition coefficient at `x=0`. Left Dirichlet Bounday Condition. `ks` Hydraulic conductivity `s` drainable pororosity (assumed to be constant) `nmax` order of power series considered for the analytic solution solution. Default is 4. `alpha` α exponent see Song at al, 2007

## Value

The water surface eletion vs time and space obtained by the analytic solution of Boussinesq Equation

## Note

For major details, see Song at al, 2007

Emanuele Cordano

## References

Song, Zhi-yao;Li, Ling;David, Lockington. (2007), "Note on Barenblatt power series solution to Boussinesq equation",Applied Mathematics and Mechanics, http://www.springerlink.com/content/w0u8667772712801/ ,http://dx.doi.org/10.1007/s10483-007-0612-x

`beq.song.dimensionless`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```L <- 1000 x <- seq(from=0,to=L,by=L/100) t <- c(4,5,20) # days h_sol1 <- beq.song(t=t*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0) h_sol2 <- beq.song(t=t*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0) h_sol3 <- beq.song(t=t*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0) plot(x,h_sol1,type="l",lty=1,main="Water Surface Elevetion (Song at's solution) ",xlab="x[m]",ylab="h[m]") lines(x,h_sol2,lty=2) lines(x,h_sol3,lty=3) legend("topright",lty=1:3,legend=paste("t=",t,"days",sep=" ")) ```