Chebfun has extensive capabilities for dealing with linear and nonlinear differential and integral operators, and includes continuous analogues of linear algebra notions like QR and singular value decomposition.
The mathematical basis of Chebfun is piecewise polynomial interpolation implemented with “Chebyshev technology”. The foundations are described, with Chebfun examples, in Approximation Theory and Approximation Practice. The Chebfun2 extension works with functions of two variables defined on a rectangle in the x-y plane. To get a sense of the breadth and power of Chebfun, a great place to start is by looking at some examples. Chebfun’s code is open source and is available on Github.