diffusion_pde, an Octave code which solves the diffusion partial differential equation (PDE) dudt = mu * d2udx2 in one spatial dimension and time, with a constant diffusion coefficient mu, and periodic or zero Neumann boundary conditions, using the forward time centered space (FTCS) solver or ode45().
We solve for u(x,t), the solution of the constant-velocity diffusion equation in 1D,
du/dt - mu d2u/dx2 = 0
over the interval:
0.0 <= x <= 1.0
with constant diffusion coefficient:
mu = 0.5
with periodic boundary conditions:
u(0,t) = u(1,t) for all t
or zero Neumann conditions:
u'(0,t) = u'(1,t) = 0 for all t
and initial condition
u(x,0) = (10x-6)^2 (8-10x)^2 for 0.6 <= x <= 0.8
= 0 elsewhere.
We may use a method known as FTCS (forward time, centered space):
The information on this web page is distributed under the MIT license.
diffusion_pde is available in a MATLAB version and an Octave version and a Python version.
advection_pde, an Octave code which solves the advection PDE dudt + c * dudx = 0 in one spatial dimension, with a constant velocity c, and periodic boundary conditions, using the forward time centered space (FTCS) method.
allen_cahn_pde, an Octave code which sets up and solves the Allen-Cahn reaction-diffusion system of partial differential equations (PDE) in 1 space dimension and time.
artery_pde. an Octave code which solves a partial differential equation (PDE) that models the displacement of arterial walls under pressure.
gray_scott_pde, an Octave code which solves the partial differential equation (PDE) known as the Gray-Scott reaction diffusion equation, displaying a sequence of solutions as time progresses.
schroedinger_linear_pde, an Octave code which solves the complex partial differential equation (PDE) known as Schroedinger's linear equation: dudt = i uxx, in one spatial dimension, with Neumann boundary conditions.
schroedinger_nonlinear_pde, an Octave code which solves the complex partial differential equation (PDE) known as Schroedinger's nonlinear equation: dudt = i uxx + gamma * |u|^2 u, in one spatial dimension, with Neumann boundary conditions.
spiral_pde, an Octave code which solves a pair of reaction-diffusion partial differential equations (PDE) over a rectangular domain with periodic boundary condition, whose solution is known to evolve into a pair of spirals.
string_pde, an Octave code which sets up and solves the partial differential equations (PDE) describing a vibrating string.
wave_pde, an Octave code which uses finite differences in space, and the method of lines in time, to set up and solve the partial differential equations (PDE) known as the wave equations, utt = c uxx.