JetDifferentialEquation(JB, D)¶

JB: JetBundleCategory
D: JetBundleFunctionCategory JB

JetDifferentialEquation provides the basic data structures and procedures for differential equations as needed in the geometric theory. Differential equation means here always a submanifold in the jet bundle. The concrete equations which define this submanifold are called system. In an object of the type JetDifferentialEquation much more than only the system is stored. D denotes the class of functions allowed as equations. It is assumed that the simplify procedure of D returns only independent equations and a system with symbol in row echelon form.

analyseSymbol: SparseEchelonMatrix(JB, D) -> Record(Rank: NonNegativeInteger, NumMultVar: NonNegativeInteger, Betas: List NonNegativeInteger): analyseSymbol(symb) computes the multiplicative variables of the symbol symb.

coerce: % -> OutputForm: coerce(de) transforms the differential equation de to OutputForm.

copy: % -> %: copy(De) returns a copy of the equation De.

dimension: (%, NonNegativeInteger) -> NonNegativeInteger: dimension(de, q) computes the dimension of the differential equation de as a submanifold of the q-th order jet bundle. The result is correct only, if de is simplified.

display: % -> Void: display(de) prints all information stored about the differential equation de. This comprises the system ordered by the order of the equations, the Jacobi matrices separately for each order and the index of the independent variable with respect to which the equation was lastly differentiated (1 for not prolonged equations).

extractSymbol: (%, Boolean) -> SparseEchelonMatrix(JB, D): extractSymbol(de, solved?) computes the symbol of the differential equation de. If solved? is true, the row echelon form of the symbol is computed at once.

insert: (List D, %) -> %: insert(sys, de) adds the system sys = 0 to the differential equation de.

jacobiMatrix: % -> List SparseEchelonMatrix(JB, D): jacobiMatrix(De) returns a list of Jacobi matrices sorted by the order of the equations.

join: (%, %) -> %: join(de1, de2) combines de1 and de2 to a single differential equation.

makeSystem: List D -> %: makeSystem(sys) creates a differential equation from a system.

order: % -> NonNegativeInteger: order(de) yields the order of the differential equation de.

printSys: List D -> OutputForm: printSys(sys) writes a list of functions as a vector of equations (with right hand side 0) and coerces the result to OutputForm.

project: (%, NonNegativeInteger) -> %: project(de, q) projects the differential equation de of order higher than q into the q-th order jet bundle.

prolong: % -> Record(SDe: %, IC: List D): prolong(de) prolongs the differential equation de. Additionally the arising integrability conditions are returned.

prolong: (%, NonNegativeInteger) -> Record(SDe: %, IC: List D): prolong(de, q) is like prolong(de). However, only equations of lower order than q are prolonged.

prolongMV: Record(Rank: NonNegativeInteger, NumMultVar: NonNegativeInteger, Betas: List NonNegativeInteger) -> Record(Rank: NonNegativeInteger, NumMultVar: NonNegativeInteger, Betas: List NonNegativeInteger): prolongMV(mv) calculates the number of multiplicative variables for the prolongation of an involutive symbol.

prolongSymbol: SparseEchelonMatrix(JB, D) -> SparseEchelonMatrix(JB, D): prolongSymbol(symb) prolongs directly the symbol symb.

retract: % -> List D: retract(de) returns the system defining the differential equation de.

setSimpMode: NonNegativeInteger -> NonNegativeInteger: setSimpMode(i) sets the flag controlling the used simplifications and returns the old value. Current values are: i = 0 -> No simplification modulo lower order equations. i = 1 -> Simplification modulo lower order equations. Default is 0.

simplify: % -> Record(SDe: %, IC: List D): simplify(de) simplifies the equations of each order separately using the procedure simplify from D. Found integrability conditions are also returned separately.

tableau: (SparseEchelonMatrix(JB, D), JetDifferential(JB, D)) -> SparseEchelonMatrix(JB, D): tableau(symb, chi) computes the tableau parametrized by a given one-form.

tableau: (SparseEchelonMatrix(JB, D), List JetDifferential(JB, D)) -> SparseEchelonMatrix(JB, D): tableau(symb, lchi) computes the extended tableau parametrized by a given list of one-forms.