Extrapolation methods are a class of arbitrary-order methods with automatic order and step-size control. The error estimate comes from computing a solution over an interval ...

Around 1870, Marius Sophus Lie realized that many of the methods for solving differential equations could be unified using group theory. Lie symmetry methods are central to ...

Following are two examples of nonlinear systems of ODEs that can be solved symbolically using DSolve. The first three equations in this system of four nonlinear ODEs can be ...

Mathematica incorporates the latest algorithms for solving industrial-scale linear systems, automatically switching between optimal dense and sparse algorithms—and handling ...

The general form of a linear second-order PDE is Here uu(x,y), and a, b, c, d, e, f, and g are functions of x and y only—they do not depend on u. If g0, the equation is ...

All the functionality of NumericalMath`Butcher`is now available in the newly created Numerical Differential Equation Analysis Package.

Convolution and correlation are central to many kinds of operations on lists of data. They are used in such areas as signal and image processing, statistical data analysis, ...

ButcherSimplify is an option to RungeKuttaOrderConditions that specifies whether to apply Butcher's row- and column-simplifying conditions.

Wronskian[{y_1, y_2, ...}, x] gives the Wronskian determinant for the functions y_1, y_2, ... depending on x.Wronskian[eqn, y, x] gives the Wronskian determinant for the ...

Complicated algebraic expressions can usually be written in many different ways. Mathematica provides a variety of functions for converting expressions from one form to ...