separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE or system of them. In the case of a single PDE, a complete solution is defined to be a solution that depends on $\sum _{i=1}^n\left({\mathrm{diff\_ord}}_i+1\right)$ parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each of the independent variables. In the case of a system of PDEs where the integrability conditions where taken into account, the number of parameters is the sum of the number of parameters involved in the separable solution of each PDE.

By default (given only the two first arguments), the program will check separability by sum.

separability requires that the PDE be polynomial in the derivatives of the unknown function. If, in the process of checking separability, an independent variable is found to be of differentiation order zero, then an error message indicating that this constitutes a degenerate case will be displayed; the only exception is when the variable appears in the PDE only through derivatives of the function F(x,y,...).

If there is a complete solution of the PDE through separation of variables, separability returns 0, indicating that the separability conditions were identically satisfied. Otherwise, the program returns one or more expressions that must vanish for the PDE to become separable. In this way, separability can also be used to set up the equations that must be satisfied for a given PDE to become separable.

This function is part of the PDEtools package, and so it can be used in the form separability(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[separability](..).

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

$\mathrm{pde1}≔\frac{1}{2m}{\mathrm{diff}\left(S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,r\right)}^2+a\left(r\right)+\frac{1}{2m{r}^2}\left({\mathrm{diff}\left(S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,\mathrm{\theta }\right)}^2+2mb\left(\mathrm{\theta }\right)\right)+\frac{1}{2m{r}^2{\sin \left(\mathrm{\theta }\right)}^2}{\mathrm{diff}\left(S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,\mathrm{\phi }\right)}^2+\frac{c\left(\mathrm{\phi }\right)}{r^2{\sin \left(\mathrm{\theta }\right)}^2}-\mathrm{diff}\left(S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,t\right)=0$

$\mathrm{pde1}≔\frac{{\left(\frac{\partial }{\partial r}S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right)}^2}{2m}+a\left(r\right)+\frac{{\left(\frac{\partial }{\partial \mathrm{\theta }}S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right)}^2+2mb\left(\mathrm{\theta }\right)}{2m{r}^2}+\frac{{\left(\frac{\partial }{\partial \mathrm{\phi }}S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right)}^2}{2m{r}^2{\sin \left(\mathrm{\theta }\right)}^2}+\frac{c\left(\mathrm{\phi }\right)}{r^2{\sin \left(\mathrm{\theta }\right)}^2}-\frac{\partial }{\partial t}S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)=0$

$\mathrm{separability}\left(\mathrm{pde1}\,S\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right)$

$0$

$\mathrm{pde2}≔\left(y-z\right)\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\,x\right)+\left(z-x\right)\mathrm{diff}\left(u\left(x\,y\,z\right)\,y\,y\right)+\left(x-y\right)\mathrm{diff}\left(u\left(x\,y\,z\right)\,z\,z\right)=E$

$\mathrm{pde2}≔\left(y-z\right)\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,z\right)\right)+\left(z-x\right)\left(\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,z\right)\right)+\left(x-y\right)\left(\frac{{\partial }^2}{\partial z^2}u\left(x\,y\,z\right)\right)=E$

$\mathrm{separability}\left(\mathrm{pde2}\,u\left(x\,y\,z\right)\right)$

$\left(y-z\right)\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\,z\right)\right)+\left(z-x\right)\left(\frac{{\partial }^2}{\partial y^2}u\left(x\,y\,z\right)\right)+\left(x-y\right)\left(\frac{{\partial }^2}{\partial z^2}u\left(x\,y\,z\right)\right)$

$\mathrm{pde3}≔\mathrm{diff}\left(u\left(x\,y\right)\,x\,x\right)+\mathrm{diff}\left(u\left(x\,y\right)\,y\,y\right)+w^{2}u\left(x\,y\right)=0$

$\mathrm{pde3}≔\frac{{\partial }^2}{\partial x^2}u\left(x\,y\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\,y\right)+w^{2}u\left(x\,y\right)=0$

$\mathrm{separability}\left(\mathrm{pde3}\,u\left(x\,y\right)\right)$

$0$

$\mathrm{separability}\left(\mathrm{pde3}\,u\left(x\,y\right)\,\mathrm{`*`}\right)$

$0$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(u\,v\right)\left(x\,y\,z\right)\right)$

$u\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}u$

$v\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}v$

$\mathrm{sys}≔\left[\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\,y\,z\right)v\left(x\,y\,z\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\,y\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,z\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\,z\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,y\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,y\,z\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,y\,z\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,x\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,y\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,x\,z\right)+\mathrm{diff}\left(u\left(x\,y\,z\right)\,z\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,x\,y\right)+u\left(x\,y\,z\right)\mathrm{diff}\left(v\left(x\,y\,z\right)\,x\,y\,z\right)=0\,\mathrm{diff}\left(u\left(x\,y\,z\right)\,x\,y\,z\right)+\mathrm{diff}\left(v\left(x\,y\,z\right)\,x\,y\,z\right)=0\right]$

$\mathrm{sys}≔\left[u_{x,y,z}v+u_{x,y}{v}_z+u_{x,z}{v}_y+u_x{v}_{y,z}+u_{y,z}{v}_x+u_y{v}_{x,z}+u_z{v}_{x,y}+u{v}_{x,y,z}=0\,u_{x,y,z}+v_{x,y,z}=0\right]$

$\mathrm{separability}\left(\mathrm{sys}\right)$

$0$

$\mathrm{remain}≔\left[\mathrm{separability}\left(\mathrm{sys}\,\mathrm{`*`}\right)\right]$

$\mathrm{remain}≔\left[u_x{v}_x\left({u_z}^2{v}^2+{v_z}^2{u}^2\right)\left({u_y}^2{v}^2+{v_y}^2{u}^2\right)\left(u_x{u}_y{u}_z{v}^2+v_x{v}_y{v}_z{u}^2\right)\,u_y{v}_y\left({u_z}^2{v}^2+{v_z}^2{u}^2\right)\left({u_x}^2{v}^2+{v_x}^2{u}^2\right)\left(u_x{u}_y{u}_z{v}^2+v_x{v}_y{v}_z{u}^2\right)\,u_z{v}_z\left({u_y}^2{v}^2+{v_y}^2{u}^2\right)\left({u_x}^2{v}^2+{v_x}^2{u}^2\right)\left(u_x{u}_y{u}_z{v}^2+v_x{v}_y{v}_z{u}^2\right)\right]$

$\mathrm{restriction}≔\left[u\left(x\,y\,z\right)=u\left(x\,z\right)\,v\left(x\,y\,z\right)=v\left(y\,z\right)\right]$

$\mathrm{restriction}≔\left[u=u\left(x\,z\right)\,v=v\left(y\,z\right)\right]$

$\mathrm{eval}\left(\mathrm{remain}\,\mathrm{restriction}\right)$

$\left[0\,0\,0\right]$

$\mathrm{pdsolve}\left(\mathrm{sys}\,\mathrm{HINT}=\mathrm{`+`}\right)$

$\left\{u=\mathrm{f\_\_1}\left(x\right)+\mathrm{f\_\_2}\left(y\right)+\mathrm{f\_\_3}\left(z\right)\,v=\mathrm{f\_\_4}\left(x\right)+\mathrm{f\_\_5}\left(y\right)+\mathrm{f\_\_6}\left(z\right)\right\}$

$\mathrm{pdetest}\left(\,\mathrm{sys}\right)$

$\left[0\,0\right]$

$\mathrm{pdsolve}\left(\mathrm{sys}\,\mathrm{HINT}=\mathrm{`*`}\right)$

$\left\{u=\mathrm{f\_\_1}\left(x\right)\mathrm{f\_\_2}\left(y\right)\mathrm{c\_\_1}\,v=\mathrm{f\_\_4}\left(x\right)\mathrm{f\_\_5}\left(y\right)\mathrm{c\_\_2}\right\}$

$\mathrm{pdetest}\left(\,\mathrm{sys}\right)$

$\left[0\,0\right]$