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detools::ncDetSys -- determining system for non-classical Lie symmetries

Introduction

detools::ncDetSys sets up the determining system for the generators of non-classical Lie point symmetries of a given system of differential equations. As for most methods in the detools library there exist several possibilities for entering the differential equations. The precise working of the method can be controlled by a number of options; especially it is possible to prescribe a special ansatz for the symmetry generators.

Call(s)

detools::ncDetSys(de, indl, depl <, Ansatz = ans, Param = paraml> <, Expr = ebool> <, Interactive = bool> <, Autoreduced = bool> <, Steps = n>)
detools::ncDetSys(df <, Ansatz = ans, Param = paraml> <, Expr = ebool> <, Interactive = bool> <, Autoreduced= bool> <, Steps = n>)

Parameters

de - the differential equation(s): either a single expression or a list of expressions.
indl - the independent variable(s): a list of (indexed) identifiers.
depl - the dependent variable(s): a list of (indexed) identifiers.
df - the differential equation(s): either an element of a domain DF in Cat::DifferentialFunction or a list of such elements.

Options

Ansatz - prescribes an ansatz for the generators.
Param - lists the names of the parameters (functions or constants) contained in the ansatz.
Expr - determines the type of the output of detools::ncDetSys.
Interactive - controls the behaviour, if detools::ncDetSys has problems with solving the differential equations for their leading derivatives.
Autoreduced - controls whether the equations of the determining system are automatically simplified (autoreduce) by detools::ncDetSys. If bool=FALSE, no simplifications are performed.
Steps - determines the level at which non-classical symmetries are sought.

Returns

The determining system is returned as a list. The type of the list element is controlled by the option Expr.

Side Effects

detools::ncDetSys reads and writes some entries of the table detools::data. This includes especially further information about the used data types.

Related Functions

detools::detSys

Details

Option: Ansatz=ans

Option: Param=paraml

Option: Expr=ebool

Option: Interactive=bool

Option: Steps=n

Example 1

The first example for a non-classical symmetry was found for the heat equation diff(u(t,x),t)-diff(u(t,x),x,x)=0. With the following command one sets up the determining equations for the first level of non-classical symmetries.

>> detools::ncDetSys(u([t]) - u([x, x]), [t, x], [u], Steps=1)
      [(-2) (XI1([u]) XI1), 2 (XI1([u]) PHI1) + 2 (XI1([x]) XI2),
      
                            2                  3
         XI1([u, u]) XI2 XI1  - XI2([u, u]) XI1 ,
      
                                2
         3 (XI2([u, u]) PHI1 XI1 ) - 2 (XI1([u, u]) PHI1 XI2 XI1) -
      
                             2                     2
         PHI1([u, u]) XI2 XI1  - 2 (XI1([x, u]) XI2  XI1) +
      
                               2                   2
         2 (XI2([x, u]) XI2 XI1 ) - 2 (XI2([u]) XI2  XI1),
      
                         2                          2
         XI1([u, u]) PHI1  XI2 - 3 (XI2([u, u]) PHI1  XI1) +
      
                                                                2
         2 (PHI1([u, u]) PHI1 XI2 XI1) + 2 (XI1([x, u]) PHI1 XI2 ) -
      
                                                           2
         4 (XI2([x, u]) PHI1 XI2 XI1) + 2 (PHI1([x, u]) XI2  XI1) +
      
                        3                  2
         XI1([x, x]) XI2  - XI2([x, x]) XI2  XI1 +
      
                             2                   3
         2 (XI2([u]) PHI1 XI2 ) + 2 (XI2([x]) XI2 ) -
      
                     3               2                      3
         XI1([t]) XI2  + XI2([t]) XI2  XI1, XI2([u, u]) PHI1  -
      
                          2                          2
         PHI1([u, u]) PHI1  XI2 + 2 (XI2([x, u]) PHI1  XI2) -
      
                                 2                        2
         2 (PHI1([x, u]) PHI1 XI2 ) + XI2([x, x]) PHI1 XI2  -
      
                         3                    2                3
         PHI1([x, x]) XI2  - XI2([t]) PHI1 XI2  + PHI1([t]) XI2 ]

If one compares with the determining system for the classical symmetries, one sees that the equations obtained here are not only considerably more involved, they are also non-linear. This is typical for non-classical symmetries and makes it much harder to solve the determining system. The problem becomes even more severe for higher values of Steps.

Background




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