Powerful Tools for Learning and Doing Mathematics
Differential Equations
Maple 13 applies revolutionary techniques for finding solutions to differential equations that are beyond the scope of standard methods. It also greatly extends the event-handling abilities for numeric solutions and enhances the abilities and performance of the world-class high-index DAE solvers.
Exact Methods for Solving ODEs
Numeric Methods for ODEs and DAEs
Exact Methods for Partial Differential Equations
Exact Methods for Solving ODEs
The theme for exact, symbolic ordinary differential equation solving in Maple 13 is the exploration of alternative methods for finding solutions to DE problems that are beyond the scope of standard methods or for which standard methods give overly complicated results. For example, important nonlinear ODE families, including some which do not have point symmetries, can be solved by constructing and then solving related linear ODEs of higher order. Similarly, by exploring symmetries in non-standard ways, the solution for various ODE initial value problems can be expressed in simple form even when the ODE general solution cannot be computed. These abilities are now built in to the standard DE solver (dsolve), and the techniques are also available separately through a variety of new tools. Highlights include:
- New solutions for nonlinear ODEs, from 1st to 5th order: linearisation by differentiation
- New ODE-IVP solutions from the ODE symmetries
- New solutions for linear ODEs with non-rational coefficients
- New solutions for nonlinear ODEs exploring dynamical symmetries
- New parametric solutions for nonlinear ODEs
- New command for specialising general solutions to satisfy given initial conditions or boundaries
- The tools for finding particular and parametric solutions can now handle a new range of higher order and nonlinear ODE problems by combining standard methods with alternative ways of using symmetries.
Numeric Methods for ODEs and DAEs
Maple 13 greatly extends the event handling capabilities for numeric solutions, as well as supplying enhancements to the algorithms and performance of the numeric solvers.
- Event handling lets you detect when a condition has been met, and then modify the system in a potentially non-differential manner. Event handling is particularly useful for solving DAEs. New capabilities include:
- The ability to control initial triggering of events.
- The addition of new event programming.
- The addition of the side construct for round-off control.
- The ability to restrict a root-finding trigger to an increasing or decreasing region of the trigger expression.
- The ability to interactively query when events have fired.
- Better detection of solution discontinuities.
- Improved performance for DAE problems using an improved projection strategy.
- More effective optimisation, including a compile option which allows certain types of solutions to be compiled using the built-in compiler for increased execution speed.
- The addition of an interactive query option to provide the number of evaluations the ODE performed to compute a solution value for an IVP problem.
Partial Differential Equations
A large number of improvements and additions were made to the PDEtools package, setting a new benchmark for the state-of-the-art in symbolic computation of partial differential equations.
- Most of the symmetry commands in the PDEtools package have been significantly expanded. Solutions can be found to previously intractable problems and there are more tools for finding solutions of a particular form.
- New commands perform fast consistency tests, compute polynomial solutions for PDE systems, transform given solutions into other different solutions exploring the symmetries of the problem, and compute invariants of differential equation systems using a simpler approach by exploring the CharacteristicQ function of the system's symmetries.
- The PDEtools internal library, a collection of over 40 routines for manipulating and programming with differential equations, is now user-accessible.
- The pdetest command can now test if solutions also satisfy given boundary conditions.
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