diffalg(deprecated)/power_series_solution - Help

diffalg

 power_series_solution
 expand the non-singular zero of a characterizable differential ideal into integral power series

 Calling Sequence power_series_solution (point, order, J, 'syst', 'params')

Parameters

 point - list or set of names or equations order - non-negative integer J - characterizable differential ideal syst - (optional) name params - (optional) name

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function power_series_solution computes a formal integral power series solution of the differential system equations $J=0$, inequations $J\ne 0$. Such a system is formally integrable. See the last example below.
 • The parameter point furnishes the point of expansion of the formal power series. It is a set or a list of equations $x=v$ where $x$ is one of the derivation variables  and $v$ is its value.
 If point is a singular point of equations (J), then power_series_solution returns FAIL. Nevertheless, this does not mean that no formal power series solution exists at that point.
 • When point is not singular, the series is truncated at the order given by the parameter order. They  could be expanded up to any order, though convergence is not guaranteed.
 The result is presented as a list of equations $\mathrm{ui}=\mathrm{si}\left(\mathrm{x1}\mathrm{...}\mathrm{xm}\right)$, where the $\mathrm{ui}$ are the differential indeterminates and the $\mathrm{si}$ are series in the derivation variables.
 • The series involve parameters corresponding to initial conditions to be given.
 The parameters appear as  $\mathrm{_C}$u, where u is a differential indeterminate if it represents the value of the solution at point, or _Cu_x, where  x is some derivation variable, if it represents the value of the value of the first derivative of $u$ according to x at point.
 The parameters  must satisfy a triangular system of polynomial equations and inequations given by syst in terms of the parameters involved in the power series solution.
 If present, the variable params receives the subset of the parameters involved in the power series solution that can almost be chosen arbitrarily if not for some inequations in syst.
 • If J is a radical differential ideal represented by a list of characterizable differential ideals, the function power_series_solution is mapped on its component.
 • The command with(diffalg,power_series_solution) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x\right],\mathrm{ranking}=\left[u,v\right]\right)$
 ${R}{≔}{\mathrm{ODE_ring}}$ (1)
 > $\mathrm{p1}≔{u}_{x}^{2}-{v}_{[]};$$\mathrm{p2}≔{v}_{x}^{2}-4{v}_{[]}$
 ${\mathrm{p1}}{≔}{{u}}_{{x}}^{{2}}{-}{{v}}_{{[}{]}}$
 ${\mathrm{p2}}{≔}{{v}}_{{x}}^{{2}}{-}{4}{}{{v}}_{{[}{]}}$ (2)
 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2}\right],\left[{v}_{x}\right],R\right)$
 ${J}{≔}\left[{\mathrm{characterizable}}\right]$ (3)
 > $\mathrm{sol}≔\mathrm{power_series_solution}\left(\left[x=0\right],50,{J}_{1},'\mathrm{syst}','\mathrm{params}'\right)$
 ${\mathrm{sol}}{≔}\left[{u}{}\left({x}\right){=}{\mathrm{_Cu}}{+}{x}{}{\mathrm{_Cu_x}}{+}\frac{{1}}{{4}}{}\frac{{{x}}^{{2}}{}{\mathrm{_Cv_x}}}{{\mathrm{_Cu_x}}}{,}{v}{}\left({x}\right){=}{\mathrm{_Cv_x}}{}{x}{+}{{x}}^{{2}}{+}{\mathrm{_Cv}}\right]$ (4)
 > $\mathrm{syst};$$\mathrm{params}$
 $\left[{{\mathrm{_Cu_x}}}^{{2}}{-}{\mathrm{_Cv}}{=}{0}{,}{{\mathrm{_Cv_x}}}^{{2}}{-}{4}{}{\mathrm{_Cv}}{=}{0}{,}{\mathrm{_Cu_x}}{\ne }{0}{,}{\mathrm{_Cv_x}}{\ne }{0}\right]$
 $\left\{{\mathrm{_Cv}}\right\}$ (5)
 > $\mathrm{initial_cond}≔\mathrm{_Cv}={c}^{2},\mathrm{_Cu_x}=c,\mathrm{_Cv_x}=-2c,\mathrm{_Cu}=\mathrm{u0}$
 ${\mathrm{initial_cond}}{≔}{\mathrm{_Cv}}{=}{{c}}^{{2}}{,}{\mathrm{_Cu_x}}{=}{c}{,}{\mathrm{_Cv_x}}{=}{-}{2}{}{c}{,}{\mathrm{_Cu}}{=}{\mathrm{u0}}$ (6)
 > $\mathrm{subs}\left(\mathrm{initial_cond},\mathrm{syst}\right)$
 $\left[{0}{=}{0}{,}{0}{=}{0}{,}{c}{\ne }{0}{,}{-}{2}{}{c}{\ne }{0}\right]$ (7)
 > $\mathrm{sol}≔\mathrm{simplify}\left(\mathrm{subs}\left(\mathrm{initial_cond},\mathrm{sol}\right)\right)$
 ${\mathrm{sol}}{≔}\left[{u}{}\left({x}\right){=}{\mathrm{u0}}{+}{x}{}{c}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}{,}{v}{}\left({x}\right){=}{{c}}^{{2}}{-}{2}{}{c}{}{x}{+}{{x}}^{{2}}\right]$ (8)

Let us explain now why, in general, we have to start from a characterizable differential system instead of any differential system. Consider the  differential system given by these two differential polynomials.

 > $R≔\mathrm{differential_ring}\left(\mathrm{ranking}=\left[u\right],\mathrm{derivations}=\left[x,y\right]\right):$
 > $\mathrm{p1}≔{u}_{x}^{2}-4{u}_{[]};$$\mathrm{p2}≔{u}_{y}-{u}_{[]}$
 ${\mathrm{p1}}{≔}{{u}}_{{x}}^{{2}}{-}{4}{}{{u}}_{{[}{]}}$
 ${\mathrm{p2}}{≔}{{u}}_{{y}}{-}{{u}}_{{[}{]}}$ (9)

We are looking for a solution starting as:

 > $u\left(x,y\right)=u\left(0,0\right)+xu[x]\left(0,0\right)+yu[y]\left(0,0\right)+\frac{{x}^{2}u[x,x]\left(0,0\right)}{2}+xyu[x,y]\left(0,0\right)$
 ${u}{}\left({x}{,}{y}\right){=}{u}{}\left({0}{,}{0}\right){+}{x}{}{u}{[}{x}{]}{}\left({0}{,}{0}\right){+}{y}{}{u}{[}{y}{]}{}\left({0}{,}{0}\right){+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{u}{[}{x}{,}{x}{]}{}\left({0}{,}{0}\right){+}{x}{}{y}{}{u}{[}{x}{,}{y}{]}{}\left({0}{,}{0}\right)$ (10)

It seems that we can choose  an initial condition $u\left(0,0\right)={c}^{2}$ ($c\ne 0$) and that, by differentiating the equations, all the coefficients in the expansion can be  expressed in terms of $c$.

The first terms do not lead to any problem:

 > $\mathrm{solve}\left(\mathrm{subs}\left({u}_{[]}={c}^{2},{u}_{x}=u[x]\left(0,0\right),{u}_{y}=u[y]\left(0,0\right),{u}_{x,x}=u[x,x]\left(0,0\right),\left\{\mathrm{p1},\mathrm{p2},\mathrm{differentiate}\left(\mathrm{p1},x,R\right)\right\}\right),\left\{u[x]\left(0,0\right),u[y]\left(0,0\right),u[x,x]\left(0,0\right)\right\}\right)$
 $\left\{{u}{[}{x}{]}{}\left({0}{,}{0}\right){=}{2}{}{c}{,}{u}{[}{y}{]}{}\left({0}{,}{0}\right){=}{{c}}^{{2}}{,}{u}{[}{x}{,}{x}{]}{}\left({0}{,}{0}\right){=}{2}\right\}{,}\left\{{u}{[}{x}{]}{}\left({0}{,}{0}\right){=}{-}{2}{}{c}{,}{u}{[}{y}{]}{}\left({0}{,}{0}\right){=}{{c}}^{{2}}{,}{u}{[}{x}{,}{x}{]}{}\left({0}{,}{0}\right){=}{2}\right\}$ (11)

To compute the next term we can either differentiate $\mathrm{p1}$ or $\mathrm{p2}$. The problem is that the results obtained are not compatible.

 > $\mathrm{solve}\left(\mathrm{subs}\left({u}_{x,y}=u[x,y]\left(0,0\right),{u}_{x}=2c,{u}_{y}={c}^{2},\mathrm{differentiate}\left(\mathrm{p1},y,R\right)\right),\left\{u[x,y]\left(0,0\right)\right\}\right)$
 $\left\{{u}{[}{x}{,}{y}{]}{}\left({0}{,}{0}\right){=}{c}\right\}$ (12)
 > $\mathrm{solve}\left(\mathrm{subs}\left({u}_{x,y}=u[x,y]\left(0,0\right),{u}_{x}=2c,{u}_{y}={c}^{2},\mathrm{differentiate}\left(\mathrm{p2},x,R\right)\right),\left\{u[x,y]\left(0,0\right)\right\}\right)$
 $\left\{{u}{[}{x}{,}{y}{]}{}\left({0}{,}{0}\right){=}{2}{}{c}\right\}$ (13)

The system $\mathrm{p1}=0,\mathrm{p2}=0$ is not formally integrable as it stands. The only solution of the system is:

 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2}\right],R\right)$
 ${J}{≔}\left[{\mathrm{characterizable}}\right]$ (14)
 > $\mathrm{rewrite_rules}\left(J\right)$
 $\left[\left[{{u}}_{{[}{]}}{=}{0}\right]\right]$ (15)
 > $\mathrm{power_series_solution}\left(\left[x,y\right],1000,{J}_{1}\right)$
 $\left[{u}{}\left({x}{,}{y}\right){=}{0}\right]$ (16)