 create a data structure for a system of differential equations - Maple Programming Help

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JetCalculus[DifferentialEquationData] - create a data structure for a system of differential equations

Calling Sequences

DifferentialEquationData(EqList, VarList)

Parameters

EqList    - a list of functions on a jet space defining a system of differential equations

VarList   - a list of jet space coordinates such that the equations in EqList may be solved algebraically for these coordinates

Description

 • In the jet bundle approach to the geometric theory of differential equations, a system of $k$-th order differential equations is represented by a sub-manifold ${R}^{k}$ of a jet space ${J}^{k}\left(E\right)$, where  is a fiber bundle. A solution to the differential equation defined by the sub-manifold is a section whose jets take values in ${R}^{k}$.
 • The DifferentialEquationData command creates an internal data structure which allows for the subsequent manipulation of the system of differential equations. The differential equations can be formally prolonged to higher order jet spaces using the Prolong command. The imbedding $\mathrm{φ}:{R}^{k}\to {J}^{k}\left(E\right)$can be constructed using the Transformation command.
 • The command DifferentialEquationData is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form DifferentialEquationData(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-DifferentialEquationData(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

Example 1.

Create a space of 2 independent variables and 1 dependent variable.

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],E,1\right):$

Define the differential equation data structure for the Sine-Gordon equation .

 E > $\mathrm{DE}≔\mathrm{DifferentialEquationData}\left(\left[u\left[1,2\right]=\mathrm{sin}\left(u\left[\right]\right)\right],\left[u\left[1,2\right]\right]\right)$
 ${\mathrm{DE}}{≔}\left[\left\{{{u}}_{{1}{,}{2}}\right\}{,}\left[{{u}}_{{1}{,}{2}}{-}{\mathrm{sin}}{}\left({{u}}_{\left[\right]}\right)\right]\right]$ (2.1)

Let's prolong the equation once. We get 3 equations which can be solved for the derivatives  .

 E > $\mathrm{DE1}≔\mathrm{Prolong}\left(\mathrm{DE},1\right)$
 ${\mathrm{DE1}}{≔}\left[\left\{{{u}}_{{1}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{2}{,}{2}}\right\}{,}\left[{{u}}_{{1}{,}{2}}{-}{\mathrm{sin}}{}\left({{u}}_{\left[\right]}\right){,}{-}{\mathrm{cos}}{}\left({{u}}_{\left[\right]}\right){}{{u}}_{{1}}{+}{{u}}_{{1}{,}{1}{,}{2}}{,}{-}{\mathrm{cos}}{}\left({{u}}_{\left[\right]}\right){}{{u}}_{{2}}{+}{{u}}_{{1}{,}{2}{,}{2}}\right]\right]$ (2.2)

Now we can turn DE1 into a transformation.

 E > $\mathrm{\iota }≔\mathrm{Transformation}\left(\mathrm{DE1}\right)$
 ${\mathrm{\iota }}{≔}\left[{x}{=}{x}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{=}{{u}}_{{1}}{,}{{u}}_{{2}}{=}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{=}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{=}{\mathrm{sin}}{}\left({{u}}_{\left[\right]}\right){,}{{u}}_{{2}{,}{2}}{=}{{u}}_{{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}}{=}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{2}}{=}{\mathrm{cos}}{}\left({{u}}_{\left[\right]}\right){}{{u}}_{{1}}{,}{{u}}_{{1}{,}{2}{,}{2}}{=}{\mathrm{cos}}{}\left({{u}}_{\left[\right]}\right){}{{u}}_{{2}}{,}{{u}}_{{2}{,}{2}{,}{2}}{=}{{u}}_{{2}{,}{2}{,}{2}}\right]$ (2.3)

Show that the function f vanishes on solutions to the Sine-Gordon equation.

 E > $f≔-\mathrm{cos}\left(u\left[\right]\right)u\left[1\right]u\left[2\right]+{u\left[1,2\right]}^{2}-u\left[1,2\right]\mathrm{sin}\left(u\left[\right]\right)+u\left[2\right]u\left[1,1,2\right]$
 ${f}{≔}{-}{\mathrm{cos}}{}\left({{u}}_{\left[\right]}\right){}{{u}}_{{1}}{}{{u}}_{{2}}{+}{{u}}_{{1}{,}{2}}^{{2}}{-}{{u}}_{{1}{,}{2}}{}{\mathrm{sin}}{}\left({{u}}_{\left[\right]}\right){+}{{u}}_{{2}}{}{{u}}_{{1}{,}{1}{,}{2}}$ (2.4)
 E > $\mathrm{Pullback}\left(\mathrm{\iota },f\right)$
 ${0}$ (2.5)