`pdepe`

Solves one-dimensional parabolic and elliptic partial differential equations.

Category: Mathematics > Numerical Integration and Differential Equations

Syntax

sol = pdepe(m, pdefun, icfun, bcfun, xmesh, tspan)

Solves the PDE system on the mesh xmesh at the times tspan.

sol = pdepe(m, pdefun, icfun, bcfun, xmesh, tspan, options)

Also applies options from odeset.

Description

The spatial derivatives are discretized on xmesh and the resulting system of ODEs is advanced in time. The heat equation u_t = u_xx corresponds to c = 1, f = dudx and s = 0.

Use pdeval to evaluate the solution between mesh points. Choose the mesh fine enough in regions where the solution changes quickly.

Input arguments

m: Symmetry: 0 slab, 1 cylindrical, 2 spherical.
pdefun: Handle [c, f, s] = pdefun(x, t, u, dudx) defining c*u_t = x^(-m) * (x^m * f)_x + s.
icfun: Handle u0 = icfun(x) for the initial condition.
bcfun: Handle [pl, ql, pr, qr] = bcfun(xl, ul, xr, ur, t) for the conditions p + q*f = 0 at both ends.
xmesh, tspan: Vectors of spatial points and output times, each with at least 3 and 2 entries respectively.

Output arguments

sol: Array of size numel(tspan)-by-numel(xmesh)-by-npde: sol(i,j,k) is component k at time tspan(i) and point xmesh(j).

Example

Heat equation with fixed ends

m = 0;
pdefun = @(x, t, u, dudx) deal(1, dudx, 0);
icfun = @(x) sin(pi*x);
bcfun = @(xl, ul, xr, ur, t) deal(ul, 0, ur, 0);
x = linspace(0, 1, 11);
t = linspace(0, 0.1, 5);
sol = pdepe(m, pdefun, icfun, bcfun, x, t);
fprintf('%.3f %.3f\n', sol(end, 6, 1), exp(-pi^2*0.1))

Output

0.376 0.373

pdepe