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This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for applying infinitesimal symmetries of differential equations, the generalization of special solutions and the calculation of symmetry and similarity variables.
In this paper the programs APPLYSYM
, QUASILINPDE
and DETRAFO
are described
which aim at the utilization of infinitesimal symmetries of differential equations. The
purpose of QUASILINPDE
is the general solution of quasilinear PDEs. This
procedure is used by APPLYSYM
for the application of point symmetries for either
calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables in a PDE(-system) or for
generalizing given special solutions of ODEs / PDEs with new constant parameters.
The program DETRAFO
performs arbitrary point- and contact transformations of ODEs /
PDEs and is applied if similarity and symmetry variables have been found.
The program APPLYSYM
is used in connection with the program LIEPDE
for formulating and solving the conditions for point- and contact symmetries
which is described in [Wol93]. The actual problem solving is done in all these
programs through a call to the package CRACK
for solving overdetermined
PDE-systems.
The investigation of infinitesimal symmetries of differential equations (DEs) with computer algebra programs attrackted considerable attention over the last years. Corresponding programs are available in all major computer algebra systems. In a review article by W. Hereman [Her95] about 200 references are given, many of them describing related software.
One reason for the popularity of the symmetry method is the fact that Sophus
Lie’s method [Lie75, Lie67] is the most widely used method for computing
exact solutions of non-linear DEs. Another reason is that the first step in this
method, the formulation of the determining equation for the generators of the
symmetries, can already be very cumbersome, especially in the case of PDEs
of higher order and/or in case of many dependent and independent variables.
Also, the formulation of the conditions is a straight forward task involving only
differentiations and basic algebra - an ideal task for computer algebra systems. Less
straight forward is the automatic solution of the symmetry conditions which is the
strength of the program LIEPDE
(for a comparison with another program see
[Wol93]).
The novelty described in this paper are programs aiming at the final third step: Applying symmetries for
calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables of a PDE(-system) or for
generalizing given special solutions of ODEs/PDEs with new constant parameters.
Programs which run on their own but also allow interactive user control are indispensible for these calculations. On one hand the calculations can become quite lengthy, like variable transformations of PDEs (of higher order, with many variables). On the other hand the freedom of choosing the right linear combination of symmetries and choosing the optimal new symmetry- and similarity variables makes it necessary to ‘play’ with the problem interactively.
The focus in this paper is directed on questions of implementation and efficiency, no principally new mathematics is presented.
In the following subsections a review of the first two steps of the symmetry method is given as well as the third, i.e. the application step is outlined. Each of the remaining sections is devoted to one procedure.
To obey classical Lie-symmetries, differential equations
where \(D/Dx^k\) means total differentiation w.r.t. \(x^k\) and from now on lower latin indices of functions \(y^\alpha ,\) (and later \(u^\alpha \)) denote partial differentiation w.r.t. the independent variables \(x^i,\) (and later \(v^i\)). The complete symmetry condition then takes the form
where mod \(H_A = 0\) means that the original PDE-system is used to replace some partial derivatives of \(y^\alpha \) to reduce the number of independent variables, because the symmetry condition (\ref {sbed1}) must be fulfilled identically in \(x^i, y^\alpha \) and all partial derivatives of \(y^\alpha .\)
For point symmetries, \(\xi ^i, \eta ^\alpha \) are functions of \(x^j, y^\beta \) and for contact symmetries they depend on \(x^j, y^\beta \) and \(y^\beta _k.\)
We restrict ourself to point symmetries as those are the only ones that can be applied by
the current version of the program APPLYSYM
(see below). For literature about
generalized symmetries see [Her95].
Though the formulation of the symmetry conditions (\ref {sbed1}), (\ref {sbed2}), (\ref {recur}) is straightforward and handled in principle by all related programs [Her95], the computational effort to formulate the conditions (\ref {sbed1}) may cause problems if the number of \(x^i\) and \(y^\alpha \) is high. This can partially be avoided if at first only a few conditions are formulated and solved such that the remaining ones are much shorter and quicker to formulate.
A first step in this direction is to investigate one PDE \(H_A = 0\) after another, as done in [CHW91]. Two methods to partition the conditions for a single PDE are described by Bocharov/Bronstein [BB89] and Stephani [Ste89].
In the first method only those terms of the symmetry condition \(X H_A = 0\) are calculated which contain at least a derivative of \(y^\alpha \) of a minimal order \(m.\) Setting coefficients of these \(u\)-derivatives to zero provides symmetry conditions. Lowering the minimal order \(m\) successively then gradually provides all symmetry conditions.
The second method is even more selective. If \(H_A\) is of order \(n\) then only terms of the symmetry condition \(X H_A = 0\) are generated which contain \(n'\)th order derivatives of \(y^\alpha .\) Furthermore these derivatives must not occur in \(H_A\) itself. They can therefore occur in the symmetry condition (\ref {sbed1}) only in \(\eta ^\alpha _{j_1\ldots j_n},\) i.e. in the terms
The second method is applied in LIEPDE
. Already the formulation of the remaining
conditions is speeded up considerably through this iteration process. These methods can
be applied if systems of DEs or single PDEs of at least second order are investigated
concerning symmetries.
The second step in applying the whole method consists in solving the determining conditions (\ref {sbed1}), (\ref {sbed2}), (\ref {recur}) which are linear homogeneous PDEs for \(\xi ^i, \eta ^\alpha \). The complete solution of this system is not algorithmic any more because the solution of a general linear PDE-system is as difficult as the solution of its non-linear characteristic ODE-system which is not covered by algorithms so far.
Still algorithms are used successfully to simplify the PDE-system by calculating its standard normal form and by integrating exact PDEs if they turn up in this simplification process [Wol93]. One problem in this respect, for example, concerns the optimization of the symbiosis of both algorithms. By that we mean the ranking of priorities between integrating, adding integrability conditions and doing simplifications by substitutions - all depending on the length of expressions and the overall structure of the PDE-system. Also the extension of the class of PDEs which can be integrated exactly is a problem to be pursuit further.
The program LIEPDE
which formulates the symmetry conditions calls the
program CRACK
to solve them. This is done in a number of successive calls
in order to formulate and solve some first order PDEs of the overdetermined
system first and use their solution to formulate and solve the next subset of
conditions as described in the previous subsection. Also, LIEPDE
can work on
DEs that contain parametric constants and parametric functions. An ansatz for
the symmetry generators can be formulated. For more details see [Wol93] or
[BW92].
The procedure LIEPDE
is called through LIEPDE(problem,symtype,flist,inequ);
All parameters are lists.
The first parameter specifies the DEs to be investigated:
problem has the form {equations, ulist, xlist} where
is a list of equations, each has the form df(ui,..)=...
where the LHS (left
hand side) df(ui,..)
is selected such that
The RHS (right h.s.) of an equations must not include the derivative on the LHS nor a derivative of it.
Neither the LHS nor any derivative of it of any equation may occur in any other equation.
Each of the unknown functions occurs on the LHS of exactly one equation.
is a list of function names, which can be chosen freely.
is a list of variable names, which can be chosen freely.
Equations can be given as a list of single differential expressions and then the program
will try to bring them into the ‘solved form’ df(ui,..)=...
automatically. If
equations are given in the solved form then the above conditions are checked and
execution is stopped it they are not satisfied. An easy way to get the equations in the
desired form is to use FIRST SOLVE({
eq1,eq2,...},{
one highest derivative for each function u})
(see the example of the Karpman equations in LIEPDE.TST
). The example of the
Burgers equation in LIEPDE.TST
demonstrates that the number of symmetries for a
given maximal order of the infinitesimal generators depends on the derivative chosen for
the LHS.
The second parameter symtype of LIEPDE
is a list \(\{\;\}\) that specifies the symmetry to be
calculated. symtype can have the following values and meanings:
{"point"}
Point symmetries with \(\xi ^i=\xi ^i(x^j,u^{\beta }),\; \eta ^{\alpha }=\eta ^{\alpha }(x^j,u^{\beta })\) are determined.
{"contact"}
Contact symmetries with \(\xi ^i=0,\;\eta =\eta (x^j,u,u_k)\) are determined \((u_k = \partial u/\partial x^k)\), which is only applicable if a
single equation (\ref {PDEs}) with an order \(>1\) for a single function \(u\) is to be investigated.
(The symtype {"contact"}
is equivalent to {"general", 1}
(see
below) apart from the additional checks done for {"contact"}
.)
{"general"
, order}
where order is an integer \(>0\). Generalized symmetries \(\xi ^i=0,\) \(\eta ^{\alpha }=\eta ^{\alpha }(x^j,u^{\beta },\ldots ,u^{\beta }_K)\) of a specified order are
determined (where \(_K\) is a multiple index representing order many indices.)
NOTE: Characteristic functions of generalized symmetries (\(= \eta ^{\alpha }\) if \(\xi ^i=0\)) are equivalent if
they are equal on the solution manifold. Therefore, all dependences of
characteristic functions on the substituted derivatives and their derivatives are
dropped. For example, if the heat equation is given as \(u_t=u_{xx}\) (i.e. \(u_t\) is substituted by \(u_{xx}\)) then
{"general", 2}
would not include characteristic functions depending on \(u_{tx}\) or \(u_{xxx}\).
THEREFORE:
If you want to find all symmetries up to a given order then either
avoid using \(H_A=0\) to substitute lower order derivatives by expressions involving higher derivatives, or
increase the order specified in symtype.
For an illustration of this effect see the two symmetry determinations of the
Burgers equation in the file LIEPDE.TST
.
{xi!_
x1 =...,..., eta!_
u1 =...,...}
It is possible to specify an ansatz for the symmetry. Such an ansatz must specify all
\(\xi ^i\) for all independent variables and all \(\eta ^{\alpha }\) for all dependent variables in terms of
differential expressions which may involve unknown functions/constants. The
dependences of the unknown functions have to be declared in advance by using the
DEPEND
command. For example, DEPEND f, t, x, u$
specifies \(f\) to be a function of \(t,x,u\). If one wants to have \(f\) as a function of derivatives of \(u(t,x)\),
say \(f\) depending on \(u_{txx}\), then one cannot write DEPEND f, df(u,t,x,2)$
but instead must write DEPEND f, u!‘1!‘2!‘2$
assuming xlist has been specified as {t,x}
. Because \(t\) is the first variable and \(x\) is
the second variable in xlist and \(u\) is differentiated oncs wrt. \(t\) and twice wrt. \(x\) we
therefore use u!‘1!‘2!‘2
. The character !
is the escape character to allow
special characters like ‘ to occur in an identifier.
For generalized symmetries one usually sets all \(\xi ^i=0\). Then the \(\eta ^{\alpha }\) are equal to the characteristic functions.
The third parameter flist of LIEPDE
is a list \(\{\;\}\) that includes
all parameters and functions in the equations which are to be determined
such that symmetries exist (if any such parameters/functions are specified in
flist then the symmetry conditions formulated in LIEPDE
become non-linear
conditions which may be much harder for CRACK
to solve with many cases
and subcases to be considered.)
all unknown functions and constants in the ansatz xi!_..
and eta!_..
if that has been specified in symtype.
The fourth parameter inequ of LIEPDE
is a list \(\{\;\}\) that includes all non-vanishing
expressions which represent inequalities for the functions in flist.
The result of LIEPDE
is a list with 3 elements, each of which is a list:
If infinitesimal symmetries have been found then the program APPLYSYM
can use them
for the following purposes:
Both methods are described in the following section.
APPLYSYM
In the following we assume that a symmetry generator \(X\), given in (\ref {sbed2}), is known such that ODE(s)/PDE(s) \(H_A=0\) satisfy the symmetry condition (\ref {sbed1}). The aim is to find new dependent functions \(u^\alpha = u^\alpha (x^j,y^\beta )\) and new independent variables \(v^i = v^i(x^j,y^\beta ),\;\; 1\leq \alpha ,\beta \leq p,\;1\leq i,j \leq q\) such that the symmetry generator \(X = \xi ^i(x^j,y^\beta )\partial _{x^i} + \eta ^\alpha (x^j,y^\beta )\partial _{y^\alpha }\) transforms to
Inverting the above transformation to \(x^i=x^i(v^j,u^\beta ), y^\alpha =y^\alpha (v^j,u^\beta )\) and setting
\(H_A(x^i(v^j,u^\beta ), y^\alpha (v^j,u^\beta ),\ldots ) = h_A(v^j, u^\beta ,\ldots )\) this means that
Consequently, the variable \(v^1\) does not occur explicitly in \(h_A\). In the case of an ODE(-system) \((v^1=v)\) the new equations \(0=h_A(v,u^\alpha ,du^\beta /dv,\ldots )\) are then of lower total order after the transformation \(z = z(u^1) = du^1/dv\) with now \(z, u^2,\ldots u^p\) as unknown functions and \(u^1\) as independent variable.
The new form (\ref {sbed3}) of \(X\) leads directly to conditions for the symmetry variable \(v^1\) and the similarity variables \(v^i|_{i\neq 1}, u^\alpha \) (all functions of \(x^k,y^\gamma \)):
The general solutions of (\ref {ql1}), (\ref {ql2}) involve free functions of \(p+q-1\) arguments. From the general solution of equation (\ref {ql2}), \(p+q-1\) functionally independent special solutions have to be selected (\(v^2,\ldots ,v^p\) and \(u^1,\ldots ,u^q\)), whereas from (\ref {ql1}) only one solution \(v^1\) is needed. Together, the expressions for the symmetry and similarity variables must define a non-singular transformation \(x,y \rightarrow u,v\).
Different special solutions selected at this stage will result in different resulting DEs which are equivalent under point transformations but may look quite differently. A transformation that is more difficult than another one will in general only complicate the new DE(s) compared with the simpler transformation. We therefore seek the simplest possible special solutions of (\ref {ql1}), (\ref {ql2}). They also have to be simple because the transformation has to be inverted to solve for the old variables in order to do the transformations.
The following steps are performed in the corresponding mode of the program
APPLYSYM
:
The user is asked to specify a symmetry by selecting one symmetry from all the known symmetries or by specifying a linear combination of them.
Through a call of the procedure QUASILINPDE
(described in a later
section) the two linear first order PDEs (\ref {ql1}), (\ref {ql2}) are investigated and, if possible,
solved.
From the general solution of (\ref {ql1}) 1 special solution is selected and from (\ref {ql2}) \(p+q-1\) special solutions are selected which should be as simple as possible.
The user is asked whether the symmetry variable should be one of the independent variables (as it has been assumed so far) or one of the new functions (then only derivatives of this function and not the function itself turn up in the new DE(s)).
Through a call of the procedure DETRAFO
the transformation \(x^i,y^\alpha \rightarrow v^j,u^\beta \) of the DE(s) \(H_A=0\)
is finally done.
The program returns to the starting menu.
A second application of infinitesimal symmetries is the generalization of a known special solution given in implicit form through \(0 = F(x^i,y^\alpha )\). If one knows a symmetry variable \(v^1\) and similarity variables \(v^r, u^\alpha ,\;\;2\leq r\leq p\) then \(v^1\) can be shifted by a constant \(c\) because of \(\partial _{v^1}H_A = 0\) and therefore the DEs \(0 = H_A(v^r,u^\alpha ,u^\beta _j,\ldots )\) are unaffected by the shift. Hence from
This generalization works only if \(\partial _{v^1}\bar {F} \neq 0\) and if \(\bar {F}\) does not already have a constant additive to \(v^1\).
The method above needs to know \(x^i=x^i(u^\beta ,v^j),\; y^\alpha =y^\alpha (u^\beta ,v^j)\) and \(u^\alpha = u^\alpha (x^j,y^\beta ), v^\alpha = v^\alpha (x^j,y^\beta )\) which may be practically impossible. Better is, to integrate \(x^i,y^\alpha \) along \(X\):
Knowing only the finite transformations
The special solution \(0 = F(x^i,y^\alpha )\) is generalized by the new constant \(\varepsilon \) through
The steps performed in the corresponding mode of the program APPLYSYM
show
features of both techniques:
The user is asked to specify a symmetry by selecting one symmetry from all the known symmetries or by specifying a linear combination of them.
The special solution to be generalized and the name of the new constant have to be put in.
Through a call of the procedure QUASILINPDE
, the PDE (\ref {ql1}) is solved which
amounts to a solution of its characteristic ODE system (\ref {ODEsys}) where \(v^1=\varepsilon \).
QUASILINPDE
returns a list of constant expressions
The new solution is availabe for further generalizations w.r.t. other symmetries.
If one would like to generalize a given special solution with \(m\) new constants because \(m\) symmetries are known, then one could run the whole program \(m\) times, each time with a different symmetry or one could run the program once with a linear combination of \(m\) symmetry generators which again is a symmetry generator. Running the program once adds one constant but we have in addition \(m-1\) arbitrary constants in the linear combination of the symmetries, so \(m\) new constants are added. Usually one will generalize the solution gradually to make solving (\ref {ODEsys}) gradually more difficult.
The call of APPLYSYM
is APPLYSYM
({de, fun, var}, {sym, cons});
de is a single DE or a list of DEs in the form of a vanishing expression or in the form \(\ldots =\ldots \;\;\).
fun is the single function or the list of functions occuring in de.
var is the single variable or the list of variables in de.
sym is a linear combination of all symmetries, each with a different constant coefficient, in form of a list of the \(\xi ^i\) and \(\eta ^\alpha \): {xi_…=…,…,eta_…=…,…}, where the indices after ‘xi_’ are the variable names and after ‘eta_’ the function names.
cons is the list of constants in sym, one constant for each symmetry.
The list that is the first argument of APPLYSYM
is the same as the first argument of
LIEPDE
and the second argument is the list that LIEPDE
returns without its first
element (the unsolved conditions). An example is given below.
What APPLYSYM
returns depends on the last performed modus. After modus 1 the
return is
{{newde, newfun, newvar}, trafo}
where
newde lists the transformed equation(s)
newfun lists the new function name(s)
newvar lists the new variable name(s)
trafo lists the transformations \(x^i=x^i(v^j,u^\beta ), y^\alpha =y^\alpha (v^j,u^\beta )\)
After modus 2, APPLYSYM
returns the generalized special solution.
Weyl’s class of solutions of Einsteins field equations consists of axialsymmetric time independent metrics of the form
LIEPDE
through
depend h,r; prob:={{-20*h**4+16*h**6+3*r**2*h*df(h,r,2)+5*r*h*df(h,r) -20*h**3*r*df(h,r)+4*h**2-5*r**2*df(h,r)**2}, {h}, {r}}; sym:=liepde(prob, {"point"},{},{}); end;
gives
sym := {{}, 3 2 {xi_r= - c10*r - c11*r, eta_h=c10*h*r }, {c10,c11}}.
All conditions have been solved because the first element of sym
is \(\{\}\). The two existing
symmetries are therefore
APPLYSYM
through
newde:=applysym(prob,rest sym);
The interactive session is given below with the user input following the prompt ‘:
’ or
following ‘?’. (Empty lines have been deleted.)
Do you want to find similarity and symmetry variables (1) or generalize a special solution with new parameters (2) or exit the program (3) Input:3: 1;
We enter ‘1’ because we want to reduce dependencies by finding similarity variables and one symmetry variable and then doing the transformation such that the symmetry variable does not explicitly occur in the DE.
---------------------- The 1. symmetry is: 3 xi_r= - r 2 eta_h=h*r ---------------------- The 2. symmetry is: xi_r= - r ---------------------- Which single symmetry or linear combination of symmetries do you want to apply? Enter an expression with ‘sy_(i)’ for the i’th symmetry. Terminate input with ‘$’ or ‘;’. sy_(1);
We could have entered ‘sy_(2);’ or a combination of both as well with the calculation running then differently.
The symmetry to be applied in the following is 3 2 {xi_r = - r ,eta_h = h*r } Terminate the following input with $ or ; . Enter the name of the new dependent variable (which will get an index attached): u; Enter the name of the new independent variables: (which will get an index attached): v;
This was the input part, now the real calculation starts.
The ODE/PDE (-system) under investigation is : 2 2 2 3 0 = 3*df(h,r,2)*h*r - 5*df(h,r) *r - 20*df(h,r)*h *r 6 4 2 + 5*df(h,r)*h*r + 16*h - 20*h + 4*h for the function(s) : h. It will be looked for a new dependent variable u and an independent variable v such that the transformed de(-system) does not depend on u or v. 1. Determination of the similarity variable 2 The quasilinear PDE: 0 = r *(df(u_,h)*h - df(u_,r)*r). The equivalent characteristic system: 3 0= - df(u_,r)*r 2 0= - r *(df(h,r)*r + h) for the functions: h(r) u_(r).
The PDE is equation (\ref {ql2}).
The general solution of the PDE is given through 0 = ff(u_,h*r) with arbitrary function ff(..). A suggestion for this function ff provides: 0 = - h*r + u_ Do you like this choice? (Y or N) y
For the following calculation only a single special solution of the PDE is necessary and
this has to be specified from the general solution by choosing a special function ff
.
(This function is called ff
to prevent a clash with names of user variables/functions.) In
principle any choice of ff
would work, if it defines a non-singular coordinate
transformation, i.e. here \(r\) must be a function of \(u\_\). If we have \(q\) independent variables and \(p\)
functions of them then ff
has \(p+q\) arguments. Because of the condition \(0 = \)ff
one has
essentially the freedom of choosing a function of \(p+q-1\) arguments freely. This freedom is also
necessary to select \(p+q-1\) different functions ff
and to find as many functionally
independent solutions \(u\_\) which all become the new similarity variables. \(q\) of them become
the new functions \(u^\alpha \) and \(p-1\) of them the new variables \(v^2,\ldots ,v^p\). Here we have \(p=q=1\) (one single
ODE).
Though the program could have done that alone, once the general solution ff(..)
is
known, the user can interfere here to enter a simpler solution, if possible.
2. Determination of the symmetry variable 2 3 The quasilinear PDE: 0 = df(u_,h)*h*r - df(u_,r)*r - 1. The equivalent characteristic system: 3 0=df(r,u_) + r 2 0=df(h,u_) - h*r for the functions: r(u_) h(u_) . New attempt with a different independent variable The equivalent characteristic system: 2 0=df(u_,h)*h*r - 1 2 0=r *(df(r,h)*h + r) for the functions: r(h) u_(h) . The general solution of the PDE is given through 2 2 2 - 2*h *r *u_ + h 0 = ff(h*r,--------------------) 2 with arbitrary function ff(..). A suggestion for this function ff(..) yields: 2 2 h *( - 2*r *u_ + 1) 0 = --------------------- 2 Do you like this choice? (Y or N) y
Similar to above.
The suggested solution of the algebraic system which will do the transformation is: sqrt(v)*sqrt(2) {h=sqrt(v)*sqrt(2)*u,r=-----------------} 2*v Is the solution ok? (Y or N) y In the intended transformation shown above the dependent variable is u and the independent variable is v. The symmetry variable is v, i.e. the transformed expression will be free of v. Is this selection of dependent and independent variables ok? (Y or N) n
We so far assumed that the symmetry variable is one of the new variables, but, of course we also could choose it to be one of the new functions. If it is one of the functions then only derivatives of this function occur in the new DE, not the function itself. If it is one of the variables then this variable will not occur explicitly.
In our case we prefer (without strong reason) to have the function as symmetry variable. We therefore answered with ‘no’. As a consequence, \(u\) and \(v\) will exchange names such that still all new functions have the name \(u\) and the new variables have name \(v\):
Please enter a list of substitutions. For example, to make the variable, which is so far call u1, to an independent variable v2 and the variable, which is so far called v2, to an dependent variable u1, enter: ‘{u1=v2, v2=u1};’{u=v,v=u}; The transformed equation which should be free of u: 3 6 2 3 0=3*u *v - 16*u *v - 20*u *v + 5*u 2v v v v Do you want to find similarity and symmetry variables (1) or generalize a special solution with new parameters (2) or exit the program (3) :
We stop here. The following is returned from our APPLYSYM
call:
3 6 {{{3*df(u,v,2)*v - 16*df(u,v) *v 2 3 - 20*df(u,v) *v + 5*df(u,v)}, {u}, {v}}, 1 2*u*v {r=-----------------,h=----------------- sqrt(u)*sqrt(2) sqrt(u)*sqrt(2) }}
The use of APPLYSYM
effectively provided us the finite transformation
where \(c_1, c_2 = const.\) and \(c_1=\tilde {c}^{1/4}.\) Finally, the metric function \(U(p)\) is obtained as an integral from (\ref {g1dgl}),(\ref {g2dgl}).
APPLYSYM
Restrictions of the applicability of the program APPLYSYM
result from limitations of the
program QUASILINPDE
described in a section below. Essentially this means that
symmetry generators may only be polynomially non-linear in \(x^i, y^\alpha \). Though even then the
solvability can not be guaranteed, the generators of Lie-symmetries are mostly very
simple such that the resulting PDE (\ref {PDE}) and the corresponding characteristic ODE-system
have good chances to be solvable.
Apart from these limitations implied through the solution of differential equations with
CRACK
and algebraic equations with SOLVE
the program APPLYSYM
itself is free of
restrictions, i.e. if once new versions of CRACK, SOLVE
would be available then
APPLYSYM
would not have to be changed.
Currently, whenever a computational step could not be performed the user is informed and has the possibility of entering interactively the solution of the unsolved algebraic system or the unsolved linear PDE.
QUASILINPDE
The generalization of special solutions of DEs as well as the computation of similarity
and symmetry variables involve the general solution of single first order linear
PDEs. The procedure QUASILINPDE
is a general procedure aiming at the
general solution of PDEs
for \(\phi , w_i\) regarded now as functions of one variable \(\varepsilon \).
Because the \(a_i\) and \(b\) do not depend explicitly on \(\varepsilon \), one of the equations (\ref {char1}),(\ref {char2}) with non-vanishing right hand side can be used to divide all others through it and by that having a system with one less ODE to solve. If the equation to divide through is one of (\ref {char1}) then the remaining system would be
with the independent variable \(w_k\) instead of \(\varepsilon \). If instead we divide through equation (\ref {char2}) then the remaining system would be
The equation to divide through is chosen by a subroutine with a heuristic to find the “simplest” non-zero right hand side (\(a_k\) or \(b\)), i.e. one which
is constant or
depends only on one variable or
is a product of factors, each of which depends only on one variable.
One purpose of this division is to reduce the number of ODEs by one. Secondly, the general solution of (\ref {char1}), (\ref {char2}) involves an additive constant to \(\varepsilon \) which is not relevant and would have to be set to zero. By dividing through one ODE we eliminate \(\varepsilon \) and lose the problem of identifying this constant in the general solution before we would have to set it to zero.
CRACK
is called. Although being
designed primarily for the solution of overdetermined PDE-systems, CRACK
can also be used to solve simple not overdetermined ODE-systems. This
solution process is not completely algorithmic. Improved versions of
CRACK
could be used, without making any changes of QUASILINPDE
necessary.
If the characteristic ODE-system can not be solved in the form (\ref {char3}), (\ref {char4}) or
(\ref {char3a}) then successively all other ODEs of (\ref {char1}), (\ref {char2}) with non-vanishing right
hand side are used for division until one is found such that the resulting
ODE-system can be solved completely. Otherwise the PDE can not be solved by
QUASILINPDE
.
On either way one ends up with \(n\) equations
The final step is to solve (\ref {charsol2}) for the \(c_i\) to obtain
The call of QUASILINPDE
is QUASILINPDE
(de, fun, varlist);
de is the differential expression which vanishes due to the PDE de\(\; = 0\) or, de may be the differential equation itself in the form \(\;\;\ldots = \ldots \;\;\).
fun is the unknown function.
varlist is the list of variables of fun.
The result of QUASILINPDE
is a list of general solutions
QUASILINPDE
can not
solve the PDE then it returns \(\{\}\). Each solution \(\textit {sol}_i\) is a list of expressions Example 1:
To solve the quasilinear first order PDE
depend u,x,y,z; de:=x*df(u,x)+u*df(u,y)-z*df(u,z) - 1; varlist:={x,y,z}; QUASILINPDE(de,u,varlist);
In this example the procedure returns
QUASILINPDE
returns the result that for an
arbitrary function \(F,\) the equation QUASILINPDE
returns the
result that for an arbitrary function \(F,\) the equation QUASILINPDE
One restriction on the applicability of QUASILINPDE
results from the program CRACK
which tries to solve the characteristic ODE-system of the PDE. So far CRACK
can be
applied only to polynomially non-linear DE’s, i.e. the characteristic ODE-system (\ref {char3}),(\ref {char4}) or
(\ref {char3a}) may only be polynomially non-linear, i.e. in the PDE (\ref {PDE}) the expressions \(a_i\) and \(b\) may only
be rational in \(w_j,\phi \).
The task of CRACK
is simplified as (\ref {charsol1}) does not have to be solved for \(w_j, \phi \). On the other hand
(\ref {charsol1}) has to be solved for the \(c_i\). This gives a second restriction coming from the REDUCE
function SOLVE
. Though SOLVE
can be applied to polynomial and transzendential
equations, again no guarantee for solvability can be given.
DETRAFO
Finally, after having found the finite transformations, the program APPLYSYM
calls the procedure DETRAFO
to perform the transformations. DETRAFO
can
also be used alone to do point- or higher order transformations which involve a
considerable computational effort if the differential order of the expression to
be transformed is high and if many dependent and independent variables are
involved. This might be especially useful if one wants to experiment and try out
different coordinate transformations interactively, using DETRAFO
as standalone
procedure.
To run DETRAFO
, the old functions \(y^{\alpha }\) and old variables \(x^i\) must be known explicitly in terms
of algebraic or differential expressions of the new functions \(u^{\beta }\) and new variables \(v^j\). Then for
point transformations the identity
provides the transformation
DETRAFO
. Non-regular transformations are not performed.
DETRAFO
is not restricted to point transformations. In the case of contact- or higher
order transformations, the total derivatives \(dy^{\alpha }/dv^i\) and \(dx^j/dv^i\) then only include all \(v^i-\) derivatives of \(u^{\beta }\)
which occur in
The call of DETRAFO
is
DETRAFO
({ex\(_1\), ex\(_2\), …, ex\(_m\)},
{ofun\(_1=\)fex\(_1\), ofun\(_2=\)fex\(_2\), …,ofun\(_p=\)fex\(_p\)},
{ovar\(_1=\)vex\(_1\), ovar\(_2=\)vex\(_2\), …, ovar\(_q=\)vex\(_q\)},
{nfun\(_1\), nfun\(_2\), …, nfun\(_p\)},
{nvar\(_1\), nvar\(_2\), …, nvar\(_q\)});
where \(m,p,q\) are arbitrary.
The ex\(_i\) are differential expressions to be transformed.
The second list is the list of old functions ofun expressed as expressions fex in terms of new functions nfun and new independent variables nvar.
Similarly the third list expresses the old independent variables ovar as expressions vex in terms of new functions nfun and new independent variables nvar.
The last two lists include the new functions nfun and new independent variables nvar.
Names for ofun, ovar, nfun and nvar can be arbitrarily chosen.
As the result DETRAFO
returns the first argument of its input, i.e. the list
DETRAFO
The only requirement is that the old independent variables \(x^i\) and old functions \(y^\alpha \) must be given explicitly in terms of new variables \(v^j\) and new functions \(u^\beta \) as indicated in the syntax. Then all calculations involve only differentiations and basic algebra.
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