Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

The differential_ring command returns a table representing the differential polynomial ring R = F(D){Y} endowed with a ranking.

If present, the field of constants F should be defined by field_extension. If omitted, it is the field of the rational numbers.

The parameter derivations = D sets the derivation variables.  The order of its elements enters in the definition of the ranking of R.

The parameter ranking = L defines both the differential indeterminates  of R and the ranking. Differential indeterminates must have the Maple type symbol.

The available sub-rankings are lex, grlexA, grlexB, degrevlexA, and degrevlexB.

Then L shall be given as L = [ a non-empty sequence of blocks ], where a block is either a single differential indeterminate or  `a sub-ranking` (a non-empty sequence of differential indeterminates). If  `a sub-ranking` is omitted, grlexA is chosen.

An elimination ranking is set between the differential indeterminates that belong to different blocks. The indeterminates that appear in the leftmost blocks are greater than the indeterminates in rightmost ones.

The derivatives of differential indeterminates that belong to the same block are ordered according to the specified sub-ranking.

The parameter notation = N sets the notation to be used in the inputs and the outputs of the functions of the diffalg package receiving R as one of their parameter.

The notation N = diff or Diff refers to the  Maple diff or Diff functionalities.

The notation N = jet is a more compact notation specific to this package. It is the default notation. A derivative is entered as u[a sequence of derivation variables], where u is one of the differential indeterminates.

The function denote makes the conversions between the three notations.

The  pair of parameters indeterminates = Y, leaders_of (P) = V provides an alternative way to define a ranking.

Y is a list or a set of symbols and defines the differential indeterminates of R.

The lists P and V must have the same length, say n. For each i (i = 1 .. n), V[i] must be a derivative occurring in the differential polynomial P[i]. The function associates weights to differential indeterminates and derivations so that the leader of P[i] is V[i]. The simplex algorithm is used for determining weights.

The description of the ranking defined on R can be obtained  by using the command print_ranking.

The parameters of differential_ring can actually appear in any order.

The command with(diffalg,differential_ring) allows the use of the abbreviated form of this command.

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

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\right]\,\mathrm{ranking}=\left[v\,u\right]\,\mathrm{notation}=\mathrm{diff}\right)$

$R≔\mathrm{ODE\_ring}$

$\mathrm{Ders}≔\left[v\left(x\right)\,u\left(x\right)\,\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{diff}\left(U\left(x\right)\,\mathrm{`\$`}\left(x\,i\right)\right)\,i=1..3\right)\,U=\left[v\,u\right]\right)\right]\:$

$\mathrm{derivatives}\left(\mathrm{Ders}\,R\,'\mathrm{increasingly}'\right)$

$\left[u\left(x\right)\,\frac{\ⅆ}{\ⅆx}u\left(x\right)\,\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\,\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)\,v\left(x\right)\,\frac{\ⅆ}{\ⅆx}v\left(x\right)\,\frac{{\ⅆ}^2}{\ⅆx^2}v\left(x\right)\,\frac{{\ⅆ}^3}{\ⅆx^3}v\left(x\right)\right]$

$\mathrm{R1}≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\left[u\,v\right]\right]\right)$

$\mathrm{R1}≔\mathrm{PDE\_ring}$

$\mathrm{R2}≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[y\,x\right]\,\mathrm{ranking}=\left[\left[u\,v\right]\right]\right)$

$\mathrm{R2}≔\mathrm{PDE\_ring}$

$\mathrm{denote}\left(v\left[x,y,y\right]\,\mathrm{diff}\,\mathrm{R1}\right)$

$\frac{{\partial }^3}{\partial x\partial y^2}v\left(x\,y\right)$

$n≔3\:$$\mathrm{Ders}≔\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(U\left[\mathrm{`\$`}\left(x\,i\right),\mathrm{`\$`}\left(y\,j\right)\right]\,i=0..n-j\right)\,j=0..n\right)\,U=\left[u\,v\right]\right)\right]\:$

$\mathrm{derivatives}\left(\mathrm{Ders}\,\mathrm{R1}\,'\mathrm{increasingly}'\right)$

$\left[v\left[\right]\,u\left[\right]\,v_y\,v_x\,u_y\,u_x\,v_{y,y}\,v_{x,y}\,v_{x,x}\,u_{y,y}\,u_{x,y}\,u_{x,x}\,v_{y,y,y}\,v_{x,y,y}\,v_{x,x,y}\,v_{x,x,x}\,u_{y,y,y}\,u_{x,y,y}\,u_{x,x,y}\,u_{x,x,x}\right]$

$\mathrm{derivatives}\left(\mathrm{Ders}\,\mathrm{R2}\,'\mathrm{increasingly}'\right)$

$\left[v\left[\right]\,u\left[\right]\,v_x\,v_y\,u_x\,u_y\,v_{x,x}\,v_{y,x}\,v_{y,y}\,u_{x,x}\,u_{y,x}\,u_{y,y}\,v_{x,x,x}\,v_{y,x,x}\,v_{y,y,x}\,v_{y,y,y}\,u_{x,x,x}\,u_{y,x,x}\,u_{y,y,x}\,u_{y,y,y}\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[z\,y\,x\right]\,\mathrm{ranking}=\left[\mathrm{lex}\left[u\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$n≔3\:$$\mathrm{Ders}≔\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(u\left[\mathrm{`\$`}\left(x\,i\right),\mathrm{`\$`}\left(y\,j\right),\mathrm{`\$`}\left(z\,k\right)\right]\,i=0..n-k-j\right)\,j=0..n-k\right)\,k=0..n\right)\right]\:$

$\mathrm{derivatives}\left(\mathrm{Ders}\,R\,'\mathrm{increasingly}'\right)$

$\left[u\left[\right]\,u_x\,u_{x,x}\,u_{x,x,x}\,u_y\,u_{y,x}\,u_{y,x,x}\,u_{y,y}\,u_{y,y,x}\,u_{y,y,y}\,u_z\,u_{z,x}\,u_{z,x,x}\,u_{z,y}\,u_{z,y,x}\,u_{z,y,y}\,u_{z,z}\,u_{z,z,x}\,u_{z,z,y}\,u_{z,z,z}\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\,z\right]\,\mathrm{ranking}=\left[\mathrm{lex}\left[w\right]\,\mathrm{degrevlexB}\left[v,u\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$\mathrm{print\_ranking}\left(R\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [w] are ordered by lex:
_U [tau] > _V [phi] when
    tau > phi for the lex. order [x, y, z] or
    tau = phi and _U > _V w.r.t. the list of indeterminates
Any derivative of [w] is greater than any derivative of [v, u]
The derivatives of [v, u] are ordered by degrevlexB:
_U [tau] > _V [phi] when
    |tau| > |phi| or
    |tau| = |phi| and tau < phi w.r.t. [z, y, x] or
    tau = phi and _U > _V w.r.t. the list of indeterminates

$n≔2\&colon;$$\mathrm{Ders}≔\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(U\left[\mathrm{`\$`}\left(x\&comma;i\right),\mathrm{`\$`}\left(y\&comma;j\right),\mathrm{`\$`}\left(z\&comma;k\right)\right]\&comma;i=0..n-k-j\right)\&comma;j=0..n-k\right)\&comma;k=0..n\right)\&comma;U=\left[w\&comma;v\&comma;u\right]\right)\right]\&colon;$

$\mathrm{derivatives}\left(\mathrm{Ders}\&comma;R\&comma;'\mathrm{increasingly}'\right)$

$\left[u\left[\right]\&comma;v\left[\right]\&comma;u_z\&comma;v_z\&comma;u_y\&comma;v_y\&comma;u_x\&comma;v_x\&comma;u_{z,z}\&comma;v_{z,z}\&comma;u_{y,z}\&comma;v_{y,z}\&comma;u_{x,z}\&comma;v_{x,z}\&comma;u_{y,y}\&comma;v_{y,y}\&comma;u_{x,y}\&comma;v_{x,y}\&comma;u_{x,x}\&comma;v_{x,x}\&comma;w\left[\right]\&comma;w_z\&comma;w_{z,z}\&comma;w_y\&comma;w_{y,z}\&comma;w_{y,y}\&comma;w_x\&comma;w_{x,z}\&comma;w_{x,y}\&comma;w_{x,x}\right]$

$p≔u\left[x,x\right]+v\left[y\right]\&colon;$

$q≔v\left[x,y\right]+v\left[y,y\right]\&colon;$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\&comma;y\right]\&comma;\mathrm{indeterminates}=\left\{u\&comma;v\right\}\&comma;\mathrm{leaders\_of}\left(\left[p\&comma;q\right]\right)=\left[v\left[y\right]\&comma;v\left[x,y\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$\mathrm{print\_ranking}\left(R\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [u, v] are ordered by weights:
Weights are [u = 0, v = 4, x = 2, y = 1]
_U [tau] > _V [phi] when
    weight (_U [tau]) > weight (_V [phi]) or
    weights are equal and _U > _V w.r.t. the list of indeterminates or
    weights and indeterminates are equal and
        tau > phi for the lex. order [x, y]

$n≔3\&colon;$$\mathrm{Ders}≔\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(U\left[\mathrm{`\$`}\left(x\&comma;i\right),\mathrm{`\$`}\left(y\&comma;j\right)\right]\&comma;i=0..n-j\right)\&comma;j=0..n\right)\&comma;U=\left[v\&comma;u\right]\right)\right]\&colon;$

$\mathrm{derivatives}\left(\mathrm{Ders}\&comma;R\&comma;'\mathrm{increasingly}'\right)$

$\left[u\left[\right]\&comma;u_y\&comma;u_{y,y}\&comma;u_x\&comma;u_{y,y,y}\&comma;u_{x,y}\&comma;v\left[\right]\&comma;u_{x,y,y}\&comma;u_{x,x}\&comma;v_y\&comma;u_{x,x,y}\&comma;v_{y,y}\&comma;v_x\&comma;u_{x,x,x}\&comma;v_{y,y,y}\&comma;v_{x,y}\&comma;v_{x,y,y}\&comma;v_{x,x}\&comma;v_{x,x,y}\&comma;v_{x,x,x}\right]$