JetCalculus[DifferentialEquationData] - create a data structure for a system of differential equations
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
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 Rk of a jet space JkE, where π :E →M is a fiber bundle. A solution to the differential equation defined by the sub-manifold Rk is a section s:M → E whose jets jksx take values in Rk.
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 φ:Rk→JkEcan 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(...).
Create a space of 2 independent variables and 1 dependent variable.
Define the differential equation data structure for the Sine-Gordon equation uxy = sinu.
DE ≔ DifferentialEquationData⁡u1,2=sin⁡u,u1,2
Let's prolong the equation once. We get 3 equations which can be solved for the derivatives uxy =u1,1, uyyx = u2, 1, uxyy =u1,2.
DE1 ≔ Prolong⁡DE,1
Now we can turn DE1 into a transformation.
ι ≔ Transformation⁡DE1
Show that the function f vanishes on solutions to the Sine-Gordon equation.
f ≔ −cos⁡u⁢u1⁢u2+u1,22−u1,2⁢sin⁡u+u2⁢u1,1,2
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