Together, the lists of independent and dependent variables provide the ranking of R, that is, an ordering for the derivatives of the dependent variables with respect to the independent variables. In brief, the elements in the second list (dependent variables or sublists of them) are ranked between each other using an elimination raking and those within a sublist are ranked used an orderly ranking.

An elimination ranking is one where derivatives of leftmost dependent variables, or of all the dependent variables in a leftmost sublist within blocks, rank higher than derivatives of variables to their right. An orderly ranking is one where derivatives of higher differential order rank higher than derivatives of lower differential order, and if the order is the same, then derivatives of leftmost variables rank higher. Hence for example if the ranking passed consists of these two lists, [x,t], [u, [v, w]]

$\mathrm{ring}≔\mathrm{DifferentialThomas}:-\mathrm{Ranking}\left(\left[x\,t\right]\,\left[u\,\left[v\,w\right]\right]\right)\:$

the list of second and first order derivatives $\left[u_{t,t}\,u_{x,x}\,u_{t,x}\,v_{t,t}\,v_{x,x}\,v_{t,x}\,w_{t,t}\,w_{x,x}\,w_{t,x}\,u_t\,u_x\,v_t\,v_x\,w_t\,w_x\right]\,$rank according to $\left[u_{x,x}\,u_{x,t}\,u_{t,t}\,u_x\,u_t\,v_{x,x}\,v_{x,t}\,v_{t,t}\,w_{x,x}\,w_{x,t}\,w_{t,t}\,v_x\,v_t\,w_x\,w_t\right]$ (see also DifferentialAlgebra:-Tools:-SortByRank)

When a system of differential polynomials is handled by DifferentialThomas commands, rankings are typically used to decide which functions and derivatives are expressed in terms of which other ones. When the system is nonlinear and handled by ThomasDecomposition, taking into account the ranking will also frequently result in a splitting of the system into cases (simple differential systems), that then naturally depend on the ordering of variables within the ranking.

For further information on rankings see the Optional arguments section of PDEtools:-casesplit and the Rankings section of the DifferentialAlgebra Glossary.

Consider the 3 following equations.

$\mathrm{p1}≔\mathrm{diff}\left(u\left(x\,y\right)\,x\right)+v\left(x\,y\right)-y$

$\mathrm{p1}≔\frac{\partial }{\partial x}u\left(x\,y\right)+v\left(x\,y\right)-y$

$\mathrm{p2}≔\mathrm{diff}\left(u\left(x\,y\right)\,y\right)+v\left(x\,y\right)$

$\mathrm{p2}≔\frac{\partial }{\partial y}u\left(x\,y\right)+v\left(x\,y\right)$

$\mathrm{p3}≔\mathrm{diff}\left(v\left(x\,y\right)\,x\right)-\mathrm{diff}\left(v\left(x\,y\right)\,y\right)$

$\mathrm{p3}≔\frac{\partial }{\partial x}v\left(x\,y\right)-\frac{\partial }{\partial y}v\left(x\,y\right)$

Before any manipulation of these equations, indicate the dependent and independent variables, and an ordering (ranking) for them variables all inside a list within a list, as in $\left[\left[u\,v\right]\right]$

$R≔\mathrm{Ranking}\left(\left[x\,y\right]\,\left[\left[u\,v\right]\right]\right)$

$R≔\mathrm{ranking}$

Call now the ThomasDecomposition to simplify the system taking into account all the integrability conditions implied in it (note you can pass the ranking R as a second argument, or omit it completely in which case the last ranking computed using Ranking will be used)

$\mathrm{ThomasDecomposition}\left(\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\right)$

This result indicates that the system bears a contradiction; $\mathrm{p1}$, $\mathrm{p2}$, and $\mathrm{p3}$ have no common solution, the system is inconsistent; see details in the Examples section of ThomasDecomposition.

To perform a chain resolution of a system of ordinary differential equations, use an elimination ranking. Consider the differential system defined by the following set $S$ of differential polynomials in the unknown functions $x\left(t\right),y\left(t\right),z\left(t\right)$.

$S≔\left[\mathrm{diff}\left(x\left(t\right)\,t\right)-x\left(t\right)\left(x\left(t\right)+y\left(t\right)\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)+y\left(t\right)\left(x\left(t\right)+y\left(t\right)\right)\,{\mathrm{diff}\left(x\left(t\right)\,t\right)}^2+{\mathrm{diff}\left(y\left(t\right)\,t\right)}^2+{\mathrm{diff}\left(z\left(t\right)\,t\right)}^2-1\right]$

$S≔\left[\frac{\ⅆ}{\ⅆt}x\left(t\right)-x\left(t\right)\left(x\left(t\right)+y\left(t\right)\right)\,\frac{\ⅆ}{\ⅆt}y\left(t\right)+y\left(t\right)\left(x\left(t\right)+y\left(t\right)\right)\,{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)}^2-1\right]$

Using an elimination ranking where $z\left(t\right)>y\left(t\right)>x\left(t\right)$, you rewrite the system in such a way that there is one equation in $x\left(t\right)$ alone, an equation determining $y\left(t\right)$ in terms of $x\left(t\right)$, and finally an equation determining $z\left(t\right)$ in terms of $y\left(t\right)$ and $x\left(t\right)$. To achieve this elimination use an elimination ranking, that is one where all the dependent variables are enclosed in a list in equal footing, and first the variables that you want to be expressed in terms of the next ones, so as in $\left[z\,y\,x\right]$.

$R≔\mathrm{Ranking}\left(\left[t\right]\,\left[z\,y\,x\right]\right)\:$

$G≔\mathrm{ThomasDecomposition}\left(S\,R\right)$

$G≔\left[\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\right]$

In the result above we see three cases. The equations of the three cases satisfying this elimination ranking mentioned are

$\mathrm{Equations}\left(G\left[1\right]\right)$

$\left[2{x\left(t\right)}^4{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{x\left(t\right)}^4{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)}^2-2{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^3{x\left(t\right)}^2-{x\left(t\right)}^4+{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^4\,{x\left(t\right)}^2+x\left(t\right)y\left(t\right)-\frac{\ⅆ}{\ⅆt}x\left(t\right)\,2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)x\left(t\right)-\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right]$

$\mathrm{Equations}\left(G\left[2\right]\right)$

$\left[{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)}^2-1\,x\left(t\right)+y\left(t\right)\,8{x\left(t\right)}^8-13{x\left(t\right)}^4+16\right]$

$\mathrm{Equations}\left(G\left[3\right]\right)$

$\left[{y\left(t\right)}^4+{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)}^2-1\,{y\left(t\right)}^2+\frac{\ⅆ}{\ⅆt}y\left(t\right)\,x\left(t\right)\right]$