splitsys - Maple Help

PDEtools

 splitsys
 split sets of (algebraic or differential) equations into uncoupled subsets

 Calling Sequence splitsys(sys, funcs)

Parameters

 sys - set of algebraic or differential equations funcs - set with the indeterminate functions

Description

 • The splitsys function returns the received set of equations split, as much as possible, into subsets of equations mutually coupled, but not coupled with the equations of the other subsets. This splitting of the set may be useful for solving the system.
 • This function is part of the PDEtools package, and so it can be used in the form splitsys(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[splitsys](..).

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $\mathrm{sys}≔\left\{k\left(-2\mathrm{q2}\left(t\right)+2\mathrm{q3}\left(t\right)\right)=-\mathrm{diff}\left(\mathrm{p3}\left(t\right),t\right),k\left(2\mathrm{q2}\left(t\right)-2\mathrm{q3}\left(t\right)\right)=-\mathrm{diff}\left(\mathrm{p2}\left(t\right),t\right),2k\mathrm{q1}\left(t\right)=-\mathrm{diff}\left(\mathrm{p1}\left(t\right),t\right),\mathrm{p3}\left(t\right)\mathrm{\alpha }=\mathrm{diff}\left(\mathrm{q2}\left(t\right),t\right),\mathrm{p2}\left(t\right)\mathrm{\alpha }+\frac{\mathrm{p3}\left(t\right)}{\mathrm{m3}}=\mathrm{diff}\left(\mathrm{q3}\left(t\right),t\right),\frac{\mathrm{p1}\left(t\right)}{\mathrm{m1}}=\mathrm{diff}\left(\mathrm{q1}\left(t\right),t\right)\right\}$
 ${\mathrm{sys}}{≔}\left\{{k}{}\left({-}{2}{}{\mathrm{q2}}{}\left({t}\right){+}{2}{}{\mathrm{q3}}{}\left({t}\right)\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p3}}{}\left({t}\right){,}{k}{}\left({2}{}{\mathrm{q2}}{}\left({t}\right){-}{2}{}{\mathrm{q3}}{}\left({t}\right)\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p2}}{}\left({t}\right){,}\frac{{\mathrm{p1}}{}\left({t}\right)}{{\mathrm{m1}}}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q1}}{}\left({t}\right){,}{\mathrm{p3}}{}\left({t}\right){}{\mathrm{\alpha }}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q2}}{}\left({t}\right){,}{2}{}{k}{}{\mathrm{q1}}{}\left({t}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p1}}{}\left({t}\right){,}{\mathrm{p2}}{}\left({t}\right){}{\mathrm{\alpha }}{+}\frac{{\mathrm{p3}}{}\left({t}\right)}{{\mathrm{m3}}}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q3}}{}\left({t}\right)\right\}$ (1)
 > $\mathrm{funcs}≔\mathrm{indets}\left(\mathrm{sys},\mathrm{Function}\right)$
 ${\mathrm{funcs}}{≔}\left\{{\mathrm{p1}}{}\left({t}\right){,}{\mathrm{p2}}{}\left({t}\right){,}{\mathrm{p3}}{}\left({t}\right){,}{\mathrm{q1}}{}\left({t}\right){,}{\mathrm{q2}}{}\left({t}\right){,}{\mathrm{q3}}{}\left({t}\right)\right\}$ (2)

The system of ODEs is split into two subsets.

 > $\mathrm{splitsys}\left(\mathrm{sys},\mathrm{funcs}\right)$
 $\left\{\frac{{\mathrm{p1}}{}\left({t}\right)}{{\mathrm{m1}}}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q1}}{}\left({t}\right){,}{2}{}{k}{}{\mathrm{q1}}{}\left({t}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p1}}{}\left({t}\right)\right\}{,}\left\{{k}{}\left({-}{2}{}{\mathrm{q2}}{}\left({t}\right){+}{2}{}{\mathrm{q3}}{}\left({t}\right)\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p3}}{}\left({t}\right){,}{k}{}\left({2}{}{\mathrm{q2}}{}\left({t}\right){-}{2}{}{\mathrm{q3}}{}\left({t}\right)\right){=}{-}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{p2}}{}\left({t}\right){,}{\mathrm{p3}}{}\left({t}\right){}{\mathrm{\alpha }}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q2}}{}\left({t}\right){,}{\mathrm{p2}}{}\left({t}\right){}{\mathrm{\alpha }}{+}\frac{{\mathrm{p3}}{}\left({t}\right)}{{\mathrm{m3}}}{=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{q3}}{}\left({t}\right)\right\}$ (3)

Split a mixed algebraic-differential system of equations.

 > $\mathrm{sys}≔\left\{x=r\left(x\right)\mathrm{cos}\left(\mathrm{\phi }\left(x\right)\right),\mathrm{diff}\left(f\left(x\right),x\right)=\mathrm{cos}\left(g\left(x\right)\right),\mathrm{diff}\left(g\left(x\right),x\right)=f\left(x\right),y\left(x\right)=r\left(x\right)\mathrm{sin}\left(\mathrm{\phi }\left(x\right)\right)\right\}$
 ${\mathrm{sys}}{≔}\left\{{x}{=}{r}{}\left({x}\right){}{\mathrm{cos}}{}\left({\mathrm{\phi }}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right){=}{\mathrm{cos}}{}\left({g}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}\right){=}{f}{}\left({x}\right){,}{y}{}\left({x}\right){=}{r}{}\left({x}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}{}\left({x}\right)\right)\right\}$ (4)
 > $\mathrm{splitsys}\left(\mathrm{sys},\left\{f,g,\mathrm{\phi },r\right\}\left(x\right)\right)$
 $\left\{{x}{=}{r}{}\left({x}\right){}{\mathrm{cos}}{}\left({\mathrm{\phi }}{}\left({x}\right)\right){,}{y}{}\left({x}\right){=}{r}{}\left({x}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}{}\left({x}\right)\right)\right\}{,}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right){=}{\mathrm{cos}}{}\left({g}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}\right){=}{f}{}\left({x}\right)\right\}$ (5)

Split a system of PDEs.

 > $\mathrm{PDE_sys}≔\left\{{\mathrm{diff}\left(g\left(x,y\right),x\right)}^{2}=g\left(x,y\right)-f\left(x,y\right),\mathrm{diff}\left(f\left(x,y\right),x\right)=g\left(x,y\right)+f\left(x,y\right),\mathrm{diff}\left(h\left(x,y,z\right),x\right)=zk\left(x,y\right)\right\}$
 ${\mathrm{PDE_sys}}{≔}\left\{{\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}{,}{y}\right)\right)}^{{2}}{=}{g}{}\left({x}{,}{y}\right){-}{f}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right){=}{g}{}\left({x}{,}{y}\right){+}{f}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({x}{,}{y}{,}{z}\right){=}{z}{}{k}{}\left({x}{,}{y}\right)\right\}$ (6)
 > $\mathrm{splitsys}\left(\mathrm{PDE_sys},\left\{f\left(x,y\right),g\left(x,y\right),h\left(x,y,z\right),k\left(x,y\right)\right\}\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({x}{,}{y}{,}{z}\right){=}{z}{}{k}{}\left({x}{,}{y}\right)\right\}{,}\left\{{\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}{,}{y}\right)\right)}^{{2}}{=}{g}{}\left({x}{,}{y}\right){-}{f}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right){=}{g}{}\left({x}{,}{y}\right){+}{f}{}\left({x}{,}{y}\right)\right\}$ (7)