MATLAB Mathematics - SERC - Index of

Partial Differential Equations
equation can vanish at isolated values of x if they are mesh points.
Discontinuities in c and/or s due to material interfaces are permitted provided
that a mesh point is placed at each interface.
At the initial time t = t 0 , for all x the solution components satisfy initial
conditions of the form
uxt ( , 0 ) = u 0 ( x)
(5-4)
At the boundary x = a or x = b, for all t the solution components satisfy a
boundary condition of the form
pxtu ( , , ) + qxt ( , )f ⎛x, t, u,
∂ ------
u ⎞ = 0
⎝ x⎠
∂
(5-5)
qxt ( , ) is a diagonal matrix with elements that are either identically zero or
never zero. Note that the boundary conditions are expressed in terms of the
flux f rather than ∂u ⁄ ∂x . Also, of the two coefficients, only p can depend on u .
MATLAB Partial Differential Equation Solver
This section describes:
• The PDE solver, pdepe
• PDE solver basic syntax
• Additional PDE solver arguments
The PDE Solver
The MATLAB PDE solver, pdepe, solves initial-boundary value problems for
systems of parabolic and elliptic PDEs in the one space variable x and time t.
There must be at least one parabolic equation in the system.
The pdepe solver converts the PDEs to ODEs using a second-order accurate
spatial discretization based on a fixed set of nodes specified by the user. The
discretization method is described in [9]. The time integration is done with
ode15s. The pdepe solver exploits the capabilities of ode15s for solving the
differential-algebraic equations that arise when Equation 5-3 contains elliptic
equations, and for handling Jacobians with a specified sparsity pattern. ode15s
changes both the time step and the formula dynamically.
5-91