computes the group invariant (symmetry) solutions for a given PDE system - Maple Help

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PDEtools[InvariantSolutions] - computes the group invariant (symmetry) solutions for a given PDE system

 Calling Sequence InvariantSolutions(PDESYS, DepVars, S, 'options'='value')

Parameters

 PDESYS - a PDE or a set or list of them; it can include ODEs and not differential equations DepVars - optional - may be required; a function or a list of them indicating the dependent variables of the problem S - optional - a list with the functional form of the infinitesimals of a symmetry generator, or a list of lists of them checkconsistency = ... - optional - can be true or false (default), to check the consistency of PDESYS before proceeding closesystem = ... - optional - can be true or false (default), to derive and then include in PDESYS, explicitly, all its integrability conditions before computing the determining system for its symmetries degreeofinfinitesimals = ... - optional - related to the option typeofsymmetry = polynomial, indicates the upper bound degree of the polynomial infinitesimals with respect to all or each of the dependent variables dependency = ... - optional - indicates the dependency of the invariant solutions, as a number (of jet variables), a range of them, a name, or a set of dependencies each of which can be a set or list of variables or numbers dependencyofinfinitesimals = ... - optional - same as dependency but for the infinitesimals with which the invariant solutions are constructed discardinfinitesimalswithintegrals = ... - optional - can be true or false (default), to discard infinitesimals that contain uncomputed integrals display = ... - optional - can be true or false (default), to display or not the solutions on the screen as they are computed HINT = ... - optional - a list with the functional form of the infinitesimals of a symmetry generator invariants = ... - optional - a list of lists, where each inner list contains invariants to be used to compute group invariant solutions numberofsolutions = ... - optional - can be a non-negative integer or the keyword all, to specify the maximum number of invariant solutions desired; default is 10 onlythetransformation = ... - optional - can be true or false, to return or not just the different transformations used to reduce the number of independent variables of PDESYS reducedsystemsolver = ... - optional - can be pdsolve or InvariantSolutions, specifies the command to be used to solve the system resulting from reducing the number of independent variables of PDESYS reduceinonego = ... - optional - a positive integer up to infinity, specifies how many independent variables are reduced in one go; default is as much as possible removeredundant = ... - optional - can be true or false (default), to remove or not, redundant solutions which happen to be particular cases of more general solutions being returned together simplifier = ... - optional - indicates the simplifier to be used instead of the default simplify/size specialize_Fn = ... - optional - can be true, false (default), or a set of procedures, to specialize any arbitrary functions that may appear in the infinitesimals split = ... - optional - can be true (default) or false, to split or not the list of infinitesimals into cases by specializing the integration constants $\mathrm{_Cn}$ typeofsymmetry = ... - optional - can be any of pointlike (default), contact, evolutionary, or general, or any of polynomial or functionfield, or And(, ) where kind is any of the first three types mentioned and functionality is any of polynomial or functionfield, indicates the type of symmetry to be computed usestandardinvariants = ... - optional - can be true or false (default), to use the standard Invariants instead of the CharacteristicQInvariants for the purpose of computing the invariant solutions

Description

 • Given a PDE problem (PDESYS), as an equation or a set or list of them, the InvariantSolutions command computes the so-called group invariant solutions of PDESYS; these are solutions derived from n-dimensional symmetry groups admitted by PDESYS by reducing the number of independent variables by n in one go. From a practical point of view, this is an important difference with regards to the so-called similarity solutions (also implemented in PDEtools - see SimilaritySolutions) where the number of independent variables are reduced one at a time and to arrive at a solution requires computing further symmetries for the reduced equation at each step. A "similarity solutions" approach can also be performed using InvariantSolutions using the optional argument reduceinonego = 1. InvariantSolutions also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.
 • The group-invariant solutions for PDESYS returned in one step by InvariantSolutions, can also be computed step by step, by first computing the determining PDE system, next solving this system to obtain the infinitesimals. With the infinitesimals, you compute an invariant transformation to reduce the number of independent variables of PDESYS. Finally you solve the reduced system to change variables back, resulting in the solutions for PDESYS. You can compute any of these steps directly departing from PDESYS, respectively using the commands: DeterminingPDE, Infinitesimals, InvariantTransformation, dchange and InvariantSolutions.
 • If DepVars is not given, InvariantSolutions will consider all the differentiated unknown functions in PDESYS as unknowns of the problems. Specifying DepVars however permits not only restricting the unknowns in different ways but also specifying unknowns of the problems which do not appear differentiated in PDESYS.
 • It is possible to indicate the functional form of the symmetries, as shown in the examples, otherwise InvariantSolutions will compute only the solutions that can be derived from the point symmetries admitted by PDESYS.
 • To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

Examples

InvariantSolutions is programmed to automatically return invariant solutions (up to 10) reducing as many independent variables as possible in one go. So invoking it as in InvariantSolutions(PDE) would suffice for that purpose. Most of the time, however, we are interested in either the most general invariant solutions, or those that can be computed faster, or that depend only on certain variables and so on. The following examples aim at illustrating the different ways of computing all these solutions.

Consider the wave equation in four dimensions; to avoid redundant typing on input and on the display use diff_table and declare

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $U:=\mathrm{diff_table}\left(u\left(x,y,z,t\right)\right):$
 > $\mathrm{declare}\left({U}_{[]}\right)$
 ${u}{}\left({x}{,}{y}{,}{z}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{u}$ (1)
 > ${\mathrm{pde}}_{1}:={U}_{x,x}+{U}_{y,y}+{U}_{z,z}-{U}_{t,t}=0$
 ${{\mathrm{pde}}}_{{1}}{:=}{{u}}_{{x}{,}{x}}{+}{{u}}_{{y}{,}{y}}{+}{{u}}_{{z}{,}{z}}{-}{{u}}_{{t}{,}{t}}{=}{0}$ (2)

Compute just two solutions, both depending on the four variables {x, y, z, t}, reducing the number of independent variables only by three in one go (so that the problem becomes an ODE)

 > $\mathrm{InvariantSolutions}\left({\mathrm{pde}}_{1},\mathrm{numberofsolutions}=2,\mathrm{dependency}=4,\mathrm{reduceinonego}=3\right)$
 ${u}{=}\frac{{\mathrm{_C1}}{}\left({{t}}^{{2}}{-}{{x}}^{{2}}{-}{{y}}^{{2}}{-}{{z}}^{{2}}\right){+}{\mathrm{_C2}}}{{{t}}^{{2}}{-}{{x}}^{{2}}{-}{{y}}^{{2}}{-}{{z}}^{{2}}}{,}{u}{=}\frac{{\mathrm{_C1}}{}\sqrt{\frac{{}^{}}{}}}{}$