kdv_exact, a MATLAB code which evaluates an exact solution of the Korteweg-deVries (KdV) partial differential equation (PDE) that represents the motion of a soliton.
The equation for u(x,t), the height of the wave, has the form
ut - 6 u ux + uxxx = 0for which an exact solution is
u(x,t) = - 1/2 v ( sech ( 1/2 * sqrt ( v ) * ( x - v * t - a ) )^2where parameter "a" is an arbitrary phase, and "v" represents the wave velocity.
The computer code and data files described and made available on this web page are distributed under the MIT license
kdv_exact is available in a MATLAB version and an Octave version and a Python version.
kdv_etdrk4, a MATLAB code which uses the exponential time differencing (ETD) RK4 method to solve the Korteweg-deVries (KdV) partial differential equation (PDE), by Aly-Khan Kassam, Lloyd Trefethen.
kdv_ift, a MATLAB code which uses the Inverse Fourier Transform (IFT) method to solve the Korteweg-deVries (KdV) partial differential equation (PDE), by Aly-Khan Kassam, Lloyd Trefethen.
pendulum_nonlinear_exact, a MATLAB code which evaluates an exact formula for the solution of the the ordinary differential equations (ODE) that represent the behavior of a nonlinear pendulum of length L under a gravitational force of strength G.
navier_stokes_2d_exact, a MATLAB code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 2D.
navier_stokes_3d_exact, a MATLAB code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 3D.
stokes_2d_exact, a MATLAB code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.