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X

X[order, ivs, dvs, JetOrder, ig][expr]
gives the action of the prolongation of order order of the infinitesimal generator ig, with independent variables ivs and dependent variables dvs in the Jet space of order JetOrder, on the expression expr.
X[order, ivs, dvs, JetOrder, ig][]
gives the prolonged infinitesimal generator of order order of the infinitesimal generator ig, with independent variables ivs and dependent variables dvs in the Jet space of order JetOrder.
  • order can be an integer denoting the order of the prolongation or, a set of lists of integers of the same dimension as the set of independent variables for defining specific coefficients of the prolongation.
  • order is an optional argument when an expression is provided. When not specified, the appropriate prolongation will be choosen automatically based on the derivatives of the given expression.
  • ivs is a set of symbols defining the space of the independent variables.
  • dvs is a set of symbols defining the space of the dependent variables. If instead of the names of the functions they are given as functions then, the set ivs can be empty.
  • JetOrder is an integer or a list of integers defining the Jet space of the infinitesimal generator. In the case of a list, its dimension must be equal to the dimension of the independent variable space.
  • JetOrder is an optional argument. When not specified, it is assumed to be 0. When specified, the infinitesimal generator is given in its canonical form.
  • When it is more desirable that the coefficients of the canonical form of the infinitesimal generator to depend on specific derivatives of the depending variables, instead of defining a JetOrder those derivatives can be included into the dvs set.
  • ig is the actual infinitesimal generator used in the prolongation.
  • ig is an optional argument. When not specified, the general form of the infinitesimal generator, according to its Jet space, will be used.
  • expr can be a system of differential equations, a differential equation or an expression containing the independent, dependent variables and derivatives of them.
  • For the prolonged generator, a special notation is given for the partial derivatives with respect to a partial derivative of one of the independent variables. This is denoted by the operator ∂_(_[...]), where _ is the name of the independent variable and inside the square brackets brackets we give the number of derivations for each of the independent variables. For example, u[2, 1] denotes the operator uxxt, for u=u(x, t).
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Basic Examples   (0)
Constructing Infinitesimal Generators   (8)
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Infinitesimal generator for the case of a scalar function depending on one variable.
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Infinitesimal generator for scalar a function f(x,y).
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Infinitesimal generator for a pair of functions f(x,y), g(x,y).
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The set of independent variables can be empty if in the set of the dependent variables functions are given instead.
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Infinitesimal generator, when a pair of functions with different dependence, f(x,y), g(s,t), and an additional continuous parameter k are involved.
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Infinitesimal generator, in characteristic form, including the Jet space of order 2.
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The symbol Q is used for denoting the characteristic form of the infinitesimal generator.
 
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Infinitesimal generator, in characteristic form, including the jetspace up to order 3 for the first variable and up to order 1 for the second.
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Infinitesimal generator, in characteristic form, with custom dependence.
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Again, in these situations the ivs set can be empty.
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