BSplineKit.jl
Tools for Bspline based Galerkin and collocation methods in Julia.
Features
This package provides:

Bspline bases of arbitrary order on uniform and nonuniform grids;

evaluation of splines and their derivatives and integrals;

spline interpolations and function approximation;

basis recombination, for generating bases satisfying homogeneous boundary conditions using linear combinations of Bsplines. Supported boundary conditions include Dirichlet, Neumann, Robin, and generalisations of these;

banded Galerkin and collocation matrices for solving differential equations, using Bspline and recombined bases;

efficient "banded" 3D arrays as an extension of banded matrices. These can store 3D tensors associated to quadratic terms in Galerkin methods.
Example usage
The following is a very brief overview of some of the functionality provided by this package.

Interpolate discrete data using cubic splines (Bspline order
k = 4
):xdata = (0:10).^2 # points don't need to be uniformly distributed ydata = rand(length(xdata)) itp = interpolate(xdata, ydata, BSplineOrder(4)) itp(12.3) # interpolation can be evaluated at any intermediate point

Create Bspline basis of order
k = 6
(polynomial degree 5) from a given set of breakpoints:breaks = log2.(1:16) # breakpoints don't need to be uniformly distributed either B = BSplineBasis(BSplineOrder(6), breaks)

Approximate known function by a spline in a previously constructed basis:
f(x) = exp(x) * sin(x) fapprox = approximate(f, B) f(2.3), fapprox(2.3) # (0.07476354233090601, 0.0747642348243861)

Create derived basis satisfying homogeneous Robin boundary conditions on the two boundaries:
bc = Derivative(0) + 3Derivative(1) R = RecombinedBSplineBasis(B, bc) # satisfies u ∓ 3u' = 0 on the left/right boundary

Construct mass matrix and stiffness matrix for the Galerkin method in the recombined basis:
# By default, M and L are Hermitian banded matrices M = galerkin_matrix(R) L = galerkin_matrix(R, (Derivative(1), Derivative(1)))

Construct banded 3D tensor associated to nonlinear term of the Burgers equation:
T = galerkin_tensor(R, (Derivative(0), Derivative(1), Derivative(0)))
See the heat equation example in the docs for the use of these tools to solve partial differential equations.
References

C. de Boor, A Practical Guide to Splines. New York: SpringerVerlag, 1978.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition. Mineola, N.Y: Dover Publications, 2001.

O. Botella and K. Shariff, Bspline Methods in Fluid Dynamics, Int. J. Comput. Fluid Dyn. 17, 133 (2003).