New Features in Maple 15: Differential Equations	Optimisation Package

Maple is the uncontested leader for computing symbolic solutions to differential equations. Numerous improvements in Maple 15 further expand the classes of problems that can be handled.

Kamke Benchmark
Ordinary Differential Equations
Partial Differential Equations
PDE Tools



Kamke Benchmark
Maple 15 now computes symbolic solutions to 97% of the 1390 linear and nonlinear ODEs from the famous text, Differentialgleichungen by Kamke. Mathematica® 8 only handles 79%. Maple also solves these ODEs almost 10 times faster than Mathematica.



Looking at the details, Maple is the only system to solve any of the systems in some classes of ODEs:

Ordinary Differential Equations (ODEs)
By using new algorithms, the dsolve command can now solve two additional 1st order and eleven 2nd order nonlinear ODE families, all of them parametrized by arbitrary functions of the independent and dependent variables and out of reach of the previously existing solving methods.

Partial Differential Equations (PDEs)
A significant number of improvements and additions were made to the PDEtools package, setting a new benchmark for the state-of-the-art in symbolic computation and partial differential equation solving and symmetry analysis.

New commands: FunctionFieldSolutions, SymmetryCommutator and SymmetryGauge

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 was to decouple the system using differential polynomial extensions. This approach, although powerful, sometimes failed to solve the problem. The new approach in FunctionFieldSolutions skips that step entirely and instead searches for solutions that can be written as a power series (with some upper bound degree n) in the mathematical functions and its derivatives (up to some upper bound differential order m), having for coefficients multivariable polynomials (with some upper bound degree r). In turn these polynomials have undetermined coefficients that get adjusted, resulting in the solution. To compute the key values of n, m, r 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 ξ = 0). 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.

• The following commands were significantly enhanced in their functionality: ConservedCurrents, D_Dx, DeterminingPDE, Eta_k, InfinitesimalGenerator, Infinitesimals, IntegratingFactors, InvariantEquation and Solve.
• In Maple 15, the entire set of symmetry commands of PDEtools:
• Automatically use textbook mathematical jet notation on input and output;
• Handle list of infinitesimals or the corresponding symmetry generator operators in equal footing;
• Understand different jet notations as representing the same mathematical objects;
• Return list of infinitesimals with infinitesimal labels in mathematical notation, unless your input contains infinitesimals without labels;
• Return results in the jet notation of the input you pass to them;
• Optionally return in any jet notation you request.

Optimisation Package