where  is the unknown and the  and  are arbitrary functions of . The biggest subclass of Abel equations known to be solvable was discovered by our research team and is the AIR 4-parameter class. New in Maple 15, two additional 1-parameter classes of Abel equations, beyond the AIR class, are now also solvable.

Examples

This equation, depending on one parameter , is now solved in terms of Bessel  functions

The related class of Abel equations that is now entirely solvable consists of the set of equations that can be obtained from  by changing variables

where the four  are arbitrary rational functions of ; this is the most general transformation that preserves the form of Abel equations and thus generates Abel ODE classes.

Another Abel equation, also depending on one parameter, is now, surprisingly, solvable entirely in terms of powers, which will be integer whenever  is integer

As in the case of equation , the entire set of equations obtained by changing variables in  using the transformation that generates Abel ODE classes are now also solvable.

By using new algorithms, the dsolve  command can now solve 11 additional nonlinear 2nd order ODE families that have no point symmetries and admit no integrating factors depending only on two of . Each of these families of equations is parametrized by arbitrary functions of the independent and dependent variables, and the equations are solved by dsolve  in Maple 15 for any values of these parametrizing functions. For 3 of these 11 families of equations, a general solution is computable directly, skipping the reduction of order step. For the other 8 families of equations, the original 2nd order ODE is systematically reduced to one of 1st order and dsolve  will return a general solution when the 1st order equation is solvable with the existing algorithms. In what follows, an illustration of some of these newly solvable families of equations is presented.

Example

This ODE has no point symmetries, the determining PDE  for them only admits both infinitesimals  equal to zero:

A solution for any of the members of this ODE family , that is regardless of the form of the mapping , can now be computed

Three new commands were added to PDEtools .

FunctionFieldSolutions  is a new command that involves an innovative approach to computing exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. In these cases, due to the presence of nonpolynomial objects, the original Maple approach by decoupling the system using differential polynomial extensions, although powerful, sometimes fails in solving the problem. The new approach in FunctionFieldSolutions  skips entirely that step and instead searches for solutions that can be written as a power series (with some upper bound degree ) in the mathematical functions and its derivatives (up to some upper bound differential order ), having for coefficients multivariable polynomials (with some upper bound degree ). In turn these polynomials have undetermined coefficients that get adjusted, resulting in the solution. To compute the key values of  to construct these function field solutions, the new command maps the problem into one that can be tackled with the existing PDEtools:-Library:-UpperBounds (see PDEtools[Library] ).

SymmetryCommutator  is a new command to compute the commutator between two symmetries, given either as lists of infinitesimals  or as infinitesimal generator  procedures. This command is useful when studying the properties of a group of symmetries or when deriving the group constants or relations to make the group complete.

SymmetryGauge  is a new command to gauge PDE symmetries. It is well known that an ODE symmetry can be rewritten in different ways (see Xgauge ). It is not so well known, but the same happens with PDE symmetries. The ability to rewrite a symmetry is relevant for a number of purposes. First, that permits identifying that two apparently different symmetries are actually the same by rewriting them in evolutionary form (gauge ). Second, symmetries that appear as dynamical ones can frequently be rewritten as pointlike, transforming the usability of the symmetry from perhaps very difficult into straightforward. Finally, depending on the form of the symmetry, by rewriting them it is frequently possible to simplify its form considerably so that invariants of canonical coordinates become computable and so the symmetry can be used to reduce the number of independent variables of PDE systems.

Examples 

with(PDEtools):

For FunctionFieldSolutions , consider the following nonlinear coupled system in  that involves the mathematical function

declare((F, G)(x, y));

sys[1] := {-F(x,y)*diff(F(x,y),y)+diff(F(x,y),x) = -F(x,y)*(exp(x)-1/y)+exp(x)*y+1/x, -F(x,y)*diff(G(x,y),y)+diff(G(x,y),x) = -F(x,y)*(exp(x)+1/y)+exp(x)*y+1/x};

Typically, systems of this type do not admit polynomial solutions

PolynomialSolutions(sys[1]);   # returns NULL

On the other hand, these systems frequently admit function field solutions:

sol[1] := FunctionFieldSolutions(sys[1]);

Verify this solution

pdetest(sol[1], sys[1]);

The following nonlinear system is also not solvable by pdsolve

sys[2] := {-F(x,y)*diff(F(x,y),y)+diff(F(x,y),x) = -F(x,y)*(exp(x)-1/y*exp(x/y))+exp(x)*y+exp(x/y)/x, -F(x,y)*diff(G(x,y),y)+diff(G(x,y),x) = -F(x,y)*(exp(x)+cos(x*y)/y)+exp(x)*y+cos(x*y)/x};

Its solution via FunctionFieldSolutions takes time to be computed; in cases like this, the use of optional arguments to restrict the degrees in which the solution would depend on mathematical functions, can diminish the computational time

sol[2] := FunctionFieldSolutions(sys[2], mathfunctiondegree = 1);

pdetest(sol[2], sys[2]);

Examples

For SymmetryCommutator , consider two list of infinitesimals  corresponding to a symmetry transformation  where there are two independent () and one dependent variable,

S[1], S[2] := [_xi[x] = x, _xi[t] = 1, _eta[u] = t], [_xi[x] = 1, _xi[t] = 1/t, _eta[u] = x^2];

For illustration purposes compute also the corresponding infinitesimal generators as differential operators

G[1] := InfinitesimalGenerator(S[1], u(x,t), expanded);

G[2] := InfinitesimalGenerator(S[2], u(x,t), expanded);

The symmetry commutator is ; when the first symmetry () is a differential operator procedure, the output is also a differential operator

SymmetryCommutator(G[1], G[2], u(x,t));

The output can be requested as a list or a procedure and the input can also be of mixed types, here we pass  and

SymmetryCommutator(G[1], S[2], u(x,t), output = list);

The prolongation order of the commutator is by default the one of the given infinitesimals, and the same happens with the output's jetnotation, but both can be requested to be different using the optional arguments prolongation = n, and notation = <any jet notation>

SymmetryCommutator(G[1], G[2], u(x,t), prolongation = 2);

SymmetryCommutator(G[1], G[2], u(x,t), prolongation = 2, notation = jetnumbers);

Examples

For SymmetryGauge , consider the list of infinitesimals  corresponding to a symmetry transformation  where there are two independent and one dependent variables,

S := [_xi[x] = 1, _xi[t] = t, _eta[u] = u];

For illustration purposes let's construct also the general form of a 2nd order PDE admitting this symmetry; for that we use the InvariantEquation  command

PDE := InvariantEquation(S, u(x,t), order = 2, name = Lambda);

This general form can be particularized in different ways, for instance

pde := isolate(op(-1, PDE) = subsop(-1 = NULL, PDE), diff(u(x,t), x,x));

Now: the general form of the symmetry S admitted by  is not  but

S1 := SymmetryGauge(S, u(x,t));

Verify that S1 is a symmetry of

SymmetryTest(S1, pde);

Note the presence of arbitrary functions in ; there are as many as the number of independent variables. The input of SymmetryGauge can also be the infinitesimal generator procedure corresponding to , in which case the output of SymmetryGauge will also be a procedure; the option expanded of SymmetryGauge is the same one of InfinitesimalGenerator

G := InfinitesimalGenerator(S, u(x,t), expanded, jetnotation = jetvariables);

SymmetryGauge(G, u(x,t), expanded);

The prolongation order of the symmetry returned is by default the one of the given symmetry, but can also be requested to be different using the optional argument prolongation = n, where n is a positive integer. To rewrite the symmetry in evolutionary form use the _xi = 0 optional argument

S0_xi := SymmetryGauge(S, u(x,t), xi = 0);

Verify that S0_xi is a symmetry of pde

SymmetryTest(S0_xi, pde);

Compare with the gauge where _eta = 0

S0_eta := SymmetryGauge(S, u(x,t), eta = 0);

SymmetryTest(S0_eta, pde);

You can also indicate the value of each of the infinitesimals appearing in the output, see SymmetryGauge