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PDEtools

 SymmetrySolutions
 computes the group invariant (point symmetry) solutions for a given PDE system

 Calling Sequence SymmetrySolutions(Solution, S, 'options'='value')

Parameters

 Solution - an equation, a set (list) of equations, or a set (list) of sets (lists) of equations, representing the solution to some system of equations (itself not required) S - the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of them representing an n-dimensional group of symmetries simplifier = ... - optional - indicates the simplifier to be used instead of the default simplify/size

Description

 • Given the Solution to a problem, as an equation or a set (list) of equations, or given a set (list) of these solutions, and given the infinitesimal generators of a symmetry S, or a list of them, SymmetrySolutions computes other solutions obtained by exploring the given symmetries. This command complements Infinitesimals and InvariantSolutions, which respetively compute the symmetry infinitesimals and the group (symmetry) invariant solutions of a PDE system.
 • The dependent variables of the problem do not need to be indicated: SymmetrySolutions will automatically take the left-hand-sides of the given solutions as dependent variables.
 • What SymmetrySolutions does can be decomposed into two steps that can be reproduced interactively:
 1 compute the finite form of the transformation associated to all the given symmetries (can be performed using SymmetryTransformation);
 2 apply this transformation to each of the given solutions (can be performed using dchange).
 You can optionally specify a simplifier to be applied to the result using the optional argument simplifier = ...
 • 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

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

 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{SymmetrySolutions},\mathrm{Infinitesimals}\right):$

Consider for instance the wave equation in two dimensions

 > ${\mathrm{pde}}_{1}≔\frac{{\partial }^{2}}{\partial {x}^{2}}u\left(x,y,t\right)+\frac{{\partial }^{2}}{\partial {y}^{2}}u\left(x,y,t\right)=0$
 ${{\mathrm{pde}}}_{{1}}{:=}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{y}{,}{t}\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}{}{u}{}\left({x}{,}{y}{,}{t}\right){=}{0}$ (1)

A solution for pde[1] obtained separating the variables by product

 > ${\mathrm{sol}}_{1}≔\mathrm{pdsolve}\left({\mathrm{pde}}_{1},\mathrm{HINT}=\mathrm{*},\mathrm{build}\right)$
 ${{\mathrm{sol}}}_{{1}}{:=}{u}{}\left({x}{,}{y}{,}{t}\right){=}{\mathrm{_F4}}{}\left({t}\right){}{\mathrm{_C1}}{}{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}{}{\mathrm{_C3}}{}{\mathrm{sin}}{}\left(\sqrt{{{\mathrm{_c}}}_{{1}}}{}{y}\right){+}{\mathrm{_F4}}{}\left({t}\right){}{\mathrm{_C1}}{}{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}{}{\mathrm{_C4}}{}{\mathrm{cos}}{}\left(\sqrt{{{\mathrm{_c}}}_{{1}}}{}{y}\right){+}\frac{{\mathrm{_F4}}{}\left({t}\right){}{\mathrm{_C2}}{}{\mathrm{_C3}}{}{\mathrm{sin}}{}\left(\sqrt{{{\mathrm{_c}}}_{{1}}}{}{y}\right)}{{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}}{+}\frac{{\mathrm{_F4}}{}\left({t}\right){}{\mathrm{_C2}}{}{\mathrm{_C4}}{}{\mathrm{cos}}{}\left(\sqrt{{{\mathrm{_c}}}_{{1}}}{}{y}\right)}{{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}}$ (2)

Some Infintesimals of polynomial type for pde[1] are

 > ${S}_{1}≔\left[{\mathrm{Infinitesimals}\left({\mathrm{pde}}_{1},\mathrm{typeofsymmetry}=\mathrm{polynomial},\mathrm{displayfunctionality}=\mathrm{false}\right)}_{-2..-1}\right]$
 ${{S}}_{{1}}{:=}\left[\left[{{\mathrm{_ξ}}}_{{x}}{=}{-}{2}{}{y}{}{x}{,}{{\mathrm{_ξ}}}_{{y}}{=}{{x}}^{{2}}{-}{{y}}^{{2}}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{=}\frac{{1}}{{2}}{}{{x}}^{{2}}{-}\frac{{1}}{{2}}{}{{y}}^{{2}}{,}{{\mathrm{_ξ}}}_{{y}}{=}{y}{}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}{0}\right]\right]$ (3)

New solutions for pde[1] exploring these infinitesimals can be computed using SymmetrySolutions as follows

 > ${\mathrm{newsol}}_{1}≔\mathrm{SymmetrySolutions}\left({\mathrm{sol}}_{1},{S}_{1}\right)$
 ${{\mathrm{newsol}}}_{{1}}{:=}\left\{{u}{}\left({x}{,}{y}{,}{t}\right){=}{\mathrm{_F4}}{}\left({t}\right){}\left({\mathrm{cos}}{}\left(\frac{\left({\mathrm{_ε}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){-}{y}\right){}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{1}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{-}{2}{}{y}{}{\mathrm{_ε}}}\right){}{\mathrm{_C4}}{-}{\mathrm{sin}}{}\left(\frac{\left({\mathrm{_ε}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){-}{y}\right){}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{1}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{-}{2}{}{y}{}{\mathrm{_ε}}}\right){}{\mathrm{_C3}}\right){}\left({{ⅇ}}^{\frac{{2}{}{x}{}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{1}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{-}{2}{}{y}{}{\mathrm{_ε}}}}{}{\mathrm{_C1}}{+}{\mathrm{_C2}}\right){}{{ⅇ}}^{{-}\frac{{x}{}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{1}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{-}{2}{}{y}{}{\mathrm{_ε}}}}\right\}{,}\left\{{u}{}\left({x}{,}{y}{,}{t}\right){=}{\mathrm{_F4}}{}\left({t}\right){}\left({{ⅇ}}^{\frac{{4}{}\left({\mathrm{_ε}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){+}{2}{}{x}\right){}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{4}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{+}{4}{}{\mathrm{_ε}}{}{x}}}{}{\mathrm{_C1}}{+}{\mathrm{_C2}}\right){}\left({\mathrm{cos}}{}\left(\frac{{4}{}{y}{}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{4}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{+}{4}{}{\mathrm{_ε}}{}{x}}\right){}{\mathrm{_C4}}{+}{\mathrm{sin}}{}\left(\frac{{4}{}{y}{}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{4}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{+}{4}{}{\mathrm{_ε}}{}{x}}\right){}{\mathrm{_C3}}\right){}{{ⅇ}}^{{-}\frac{{2}{}\left({\mathrm{_ε}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){+}{2}{}{x}\right){}\sqrt{{{\mathrm{_c}}}_{{1}}}}{{4}{+}\left({{x}}^{{2}}{+}{{y}}^{{2}}\right){}{{\mathrm{_ε}}}^{{2}}{+}{4}{}{\mathrm{_ε}}{}{x}}}\right\}$ (4)

Note the introduction of the Lie group parameter $\mathrm{_ϵ}$; the new solutions are valid for any value (not depending on $x$, $y$ or $t$) of it. These solutions can be verified using pdetest

 > $\mathrm{map}\left(\mathrm{pdetest},\left[{\mathrm{newsol}}_{1}\right],{\mathrm{pde}}_{1}\right)$
 $\left[{0}{,}{0}\right]$ (5)