**python_exact**,
a Python code which
evaluates exact solutions to a few selected examples of
ordinary differential equations (ODE) and partial differential
equations (PDE).

These exact solutions can be used to test out the correctness of a solution algorithm.

The information on this web page is distributed under the MIT license.

**python_exact** is available in
a C version and
a C++ version and
a Fortran77 version and
a Fortran90 version and
a MATLAB version and
an Octave version and
a Python version.

biharmonic_exact, a Python code which evaluates exact solutions w(x,y) to the biharmonic equation del^2 w = 0 or wxxxx + 2 wxxyy + wyyyy = 0

burgers_exact, a Python code which evaluates exact solutions of time-dependent 1D viscous Burgers equation.

fisher_exact, a Python code which returns an exact solution of the Kolmogorov Petrovsky Piskonov Fisher partial differential equation (PDE) ut=uxx+u*(1-u).

flame_exact, a Python code which returns the exact solution of an ordinary differential equation (ODE) which models the growth of a ball of flame in a combustion process. The exact solution is defined in terms of the Lambert W function.

helmholtz_exact, a Python code which evaluates an exact formula for the solution of the Helmholtz equation, a sort of eigenvalue equation that can arise after separation of variables is a applied to the wave equation. This example considers a problem in which Z(x,y) represents the vertical deflection of the surface of a vibrating circular membrane such as a drumhead.

kdv_exact, a Python code which evaluates an exact solution of the Korteweg-deVries (KdV) partial differential equation (PDE).

lindberg_exact, a Python code which evaluates the exact solution of the Lindberg ordinary differential equations (ODE), a system that is extremely difficult to solve accurately.

logistic_exact, a Python code which evaluates an exact solution of the logistic equation, an ordinary differential equation (ODE) which models population growth in the face of a limited carrying capacity.

navier_stokes_2d_exact, a Python 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 Python code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 3D.

porous_medium_exact, a Python code which returns an exact solution of the porous medium equation (PME), dudt=Del^2(u^m), a partial differential equation (PDE) related to the diffusion equation, based on the Barenblatt solution.

sine_gordon_exact, a Python code which returns an exact solution of the Sine-Gordon equation, a partial differential equation (PDE) of the form uxy=sin(u).

spiral_exact, a Python code which computes a 2D velocity vector field that is an exact solution of the continuity equation.

stokes_2d_exact, a Python code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.