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PDEtools

 separability
 determine the conditions for sum or product separability of PDEs

 Calling Sequence separability(PDE, F(x,y,...), '*')

Parameters

 PDE - partial differential equation or a set or list of them F(x, y, ...) - unknown function in PDE or a set or list of them '*' - (optional) indicate that the program should test product separability

Description

 • separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE or system of them. In the case of a single PDE, a complete solution is defined to be a solution that depends on $\sum _{i=1}^{n}\left({\mathrm{diff_ord}}_{i}+1\right)$ parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each of the independent variables. In the case of a system of PDEs where the integrability conditions where taken into account, the number of parameters is the sum of the number of parameters involved in the separable solution of each PDE.
 • By default (given only the two first arguments), the program will check separability by sum.
 • separability requires that the PDE be polynomial in the derivatives of the unknown function. If, in the process of checking separability, an independent variable is found to be of differentiation order zero, then an error message indicating that this constitutes a degenerate case will be displayed; the only exception is when the variable appears in the PDE only through derivatives of the function F(x,y,...).
 • If there is a complete solution of the PDE through separation of variables, separability returns 0, indicating that the separability conditions were identically satisfied. Otherwise, the program returns one or more expressions that must vanish for the PDE to become separable. In this way, separability can also be used to set up the equations that must be satisfied for a given PDE to become separable.
 • This function is part of the PDEtools package, and so it can be used in the form separability(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[separability](..).

Examples

Separate a Hamilton-Jacobi PDE by sum.

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $\mathrm{pde1}≔\frac{1{\left(\frac{\partial }{\partial r}S\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)}^{2}}{2m}+a\left(r\right)+\frac{1\left({\left(\frac{\partial }{\partial \mathrm{θ}}S\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)}^{2}+2mb\left(\mathrm{θ}\right)\right)}{2m{r}^{2}}+\frac{1{\left(\frac{\partial }{\partial \mathrm{φ}}S\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)}^{2}}{2m{r}^{2}{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}}+\frac{c\left(\mathrm{φ}\right)}{{r}^{2}{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}}-\left(\frac{\partial }{\partial t}S\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)=0$
 ${\mathrm{pde1}}{≔}\frac{{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{S}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right)}^{{2}}}{{2}{}{m}}{+}{a}{}\left({r}\right){+}\frac{{\left(\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{S}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right)}^{{2}}{+}{2}{}{m}{}{b}{}\left({\mathrm{\theta }}\right)}{{2}{}{m}{}{{r}}^{{2}}}{+}\frac{{\left(\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{S}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right)}^{{2}}}{{2}{}{m}{}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{+}\frac{{c}{}\left({\mathrm{\phi }}\right)}{{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{S}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){=}{0}$ (1)
 > $\mathrm{separability}\left(\mathrm{pde1},S\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)$
 ${0}$ (2)

The following is an example of a PDE that does not separate by sum (unless E=0).

 > $\mathrm{pde2}≔\left(y-z\right)\left(\frac{{\partial }^{2}}{\partial {x}^{2}}u\left(x,y,z\right)\right)+\left(z-x\right)\left(\frac{{\partial }^{2}}{\partial {y}^{2}}u\left(x,y,z\right)\right)+\left(x-y\right)\left(\frac{{\partial }^{2}}{\partial {z}^{2}}u\left(x,y,z\right)\right)=E$
 ${\mathrm{pde2}}{≔}\left({y}{-}{z}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right){+}\left({z}{-}{x}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right){+}\left({x}{-}{y}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right){=}{E}$ (3)
 > $\mathrm{separability}\left(\mathrm{pde2},u\left(x,y,z\right)\right)$
 $\left({y}{-}{z}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right){+}\left({z}{-}{x}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right){+}\left({x}{-}{y}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}\right)\right)$ (4)

The Helmholtz equation in two variables separates by sum and by product.

 > $\mathrm{pde3}≔\frac{{\partial }^{2}}{\partial {x}^{2}}u\left(x,y\right)+\frac{{\partial }^{2}}{\partial {y}^{2}}u\left(x,y\right)+{w}^{2}u\left(x,y\right)=0$
 ${\mathrm{pde3}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){+}{{w}}^{{2}}{}{u}{}\left({x}{,}{y}\right){=}{0}$ (5)
 > $\mathrm{separability}\left(\mathrm{pde3},u\left(x,y\right)\right)$
 ${0}$ (6)
 > $\mathrm{separability}\left(\mathrm{pde3},u\left(x,y\right),\mathrm{*}\right)$
 ${0}$ (7)

A nonlinear PDE system with two unknowns u, v, of three independent variables x, y, z - invoke the declare facility to avoid redundancy in the display of the output (derivatives are displayed as indexed objects):

 > $\mathrm{PDEtools}:-\mathrm{declare}\left(\left(u,v\right)\left(x,y,z\right)\right)$
 ${u}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{u}$
 ${v}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{v}$ (8)
 > $\mathrm{sys}≔\left[\left(\frac{{\partial }^{3}}{\partial z\partial y\partial x}u\left(x,y,z\right)\right)v\left(x,y,z\right)+\left(\frac{{\partial }^{2}}{\partial y\partial x}u\left(x,y,z\right)\right)\left(\frac{\partial }{\partial z}v\left(x,y,z\right)\right)+\left(\frac{{\partial }^{2}}{\partial z\partial x}u\left(x,y,z\right)\right)\left(\frac{\partial }{\partial y}v\left(x,y,z\right)\right)+\left(\frac{\partial }{\partial x}u\left(x,y,z\right)\right)\left(\frac{{\partial }^{2}}{\partial z\partial y}v\left(x,y,z\right)\right)+\left(\frac{{\partial }^{2}}{\partial z\partial y}u\left(x,y,z\right)\right)\left(\frac{\partial }{\partial x}v\left(x,y,z\right)\right)+\left(\frac{\partial }{\partial y}u\left(x,y,z\right)\right)\left(\frac{{\partial }^{2}}{\partial z\partial x}v\left(x,y,z\right)\right)+\left(\frac{\partial }{\partial z}u\left(x,y,z\right)\right)\left(\frac{{\partial }^{2}}{\partial y\partial x}v\left(x,y,z\right)\right)+u\left(x,y,z\right)\left(\frac{{\partial }^{3}}{\partial z\partial y\partial x}v\left(x,y,z\right)\right)=0,\frac{{\partial }^{3}}{\partial z\partial y\partial x}u\left(x,y,z\right)+\frac{{\partial }^{3}}{\partial z\partial y\partial x}v\left(x,y,z\right)=0\right]$
 ${\mathrm{sys}}{≔}\left[{u}{}{{v}}_{{x}{,}{y}{,}{z}}{+}{{u}}_{{x}{,}{y}{,}{z}}{}{v}{+}{{u}}_{{x}}{}{{v}}_{{y}{,}{z}}{+}{{u}}_{{y}}{}{{v}}_{{x}{,}{z}}{+}{{u}}_{{z}}{}{{v}}_{{x}{,}{y}}{+}{{u}}_{{x}{,}{y}}{}{{v}}_{{z}}{+}{{u}}_{{x}{,}{z}}{}{{v}}_{{y}}{+}{{u}}_{{y}{,}{z}}{}{{v}}_{{x}}{=}{0}{,}{{u}}_{{x}{,}{y}{,}{z}}{+}{{v}}_{{x}{,}{y}{,}{z}}{=}{0}\right]$ (9)

As in the case of a single PDE, for PDE systems, unless indicated otherwise, the test is regarding separability by sum

 > $\mathrm{separability}\left(\mathrm{sys}\right)$
 ${0}$ (10)

Note that these separability test results are obtained only considering related integrability conditions, not actually separating the variables. Regarding separation by product,

 > $\mathrm{remain}≔\left[\mathrm{separability}\left(\mathrm{sys},\mathrm{*}\right)\right]$
 ${\mathrm{remain}}{≔}\left[\left({{v}}_{{x}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{x}}^{{2}}\right){}\left({{v}}_{{y}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{y}}^{{2}}\right){}\left({{v}}_{{z}}{}{{v}}_{{y}}{}{{v}}_{{x}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{x}}{}{{u}}_{{y}}{}{{u}}_{{z}}\right){}{{u}}_{{z}}{}{{v}}_{{z}}{,}\left({{v}}_{{x}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{x}}^{{2}}\right){}\left({{v}}_{{z}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{z}}^{{2}}\right){}\left({{v}}_{{z}}{}{{v}}_{{y}}{}{{v}}_{{x}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{x}}{}{{u}}_{{y}}{}{{u}}_{{z}}\right){}{{u}}_{{y}}{}{{v}}_{{y}}{,}\left({{v}}_{{y}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{y}}^{{2}}\right){}\left({{v}}_{{z}}^{{2}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{z}}^{{2}}\right){}\left({{v}}_{{z}}{}{{v}}_{{y}}{}{{v}}_{{x}}{}{{u}}^{{2}}{+}{{v}}^{{2}}{}{{u}}_{{x}}{}{{u}}_{{y}}{}{{u}}_{{z}}\right){}{{v}}_{{x}}{}{{u}}_{{x}}\right]$ (11)

This result means there exists no separable solution unless the three equations above are zero, that is: there is no solution really involving the product of three functions respectively depending on x, y, z. That doesn't mean there exists no separable solution where one or both of the unknowns {u(x, y, z) v(x, y, z)} depend on less variables (i.e. is a constant with respect to one or more of x, y, z). For example, consider the restriction

 > $\mathrm{restriction}≔\left[u\left(x,y,z\right)=u\left(x,z\right),v\left(x,y,z\right)=v\left(y,z\right)\right]$
 ${\mathrm{restriction}}{≔}\left[{u}{=}{u}{}\left({x}{,}{z}\right){,}{v}{=}{v}{}\left({y}{,}{z}\right)\right]$ (12)

Evaluate remain at this restriction

 > $\mathrm{eval}\left(\mathrm{remain},\mathrm{restriction}\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (13)

So with this restriction the separability conditions are satisfied. In any case the solutions separable by + and by * can be computed  with pdsolve using the HINT option; for this particular system only the solutions separable by + have u and v depending on the three variables

 > $\mathrm{pdsolve}\left(\mathrm{sys},\mathrm{HINT}=\mathrm{+}\right)$
 $\left\{{u}{=}{\mathrm{_F1}}{}\left({x}\right){+}{\mathrm{_F2}}{}\left({y}\right){+}{\mathrm{_F3}}{}\left({z}\right){,}{v}{=}{\mathrm{_F4}}{}\left({x}\right){+}{\mathrm{_F5}}{}\left({y}\right){+}{\mathrm{_F6}}{}\left({z}\right)\right\}$ (14)
 > $\mathrm{pdetest}\left(,\mathrm{sys}\right)$
 $\left[{0}{,}{0}\right]$ (15)
 > $\mathrm{pdsolve}\left(\mathrm{sys},\mathrm{HINT}=\mathrm{*}\right)$
 $\left\{{u}{=}{\mathrm{_F1}}{}\left({x}\right){}{\mathrm{_F2}}{}\left({y}\right){}{\mathrm{_C1}}{,}{v}{=}{\mathrm{_F4}}{}\left({x}\right){}{\mathrm{_F5}}{}\left({y}\right){}{\mathrm{_C2}}\right\}$ (16)
 > $\mathrm{pdetest}\left(,\mathrm{sys}\right)$
 $\left[{0}{,}{0}\right]$ (17)