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Solve 1-D parabolic and elliptic PDEs

solves a system of parabolic and elliptic PDEs with one spatial variable
`sol`

= pdepe(`m`

,`pdefun`

,`icfun`

,`bcfun`

,`xmesh`

,`tspan`

)*x* and time *t*. At least one equation must be
parabolic. The scalar `m`

represents the symmetry of the problem (slab,
cylindrical, or spherical). The equations being solved are coded in
`pdefun`

, the initial value is coded in `icfun`

, and the
boundary conditions are coded in `bcfun`

. The ordinary differential
equations (ODEs) resulting from discretization in space are integrated to obtain approximate
solutions at the times specified in `tspan`

. The `pdepe`

function returns values of the solution on a mesh provided in
`xmesh`

.

`[`

also finds where functions of (`sol`

,`tsol`

,`sole`

,`te`

,`ie`

] = pdepe(`m`

,`pdefun`

,`icfun`

,`bcfun`

,`xmesh`

,`tspan`

,`options`

)*t*,*u*(*x*,*t*)), called event functions, are zero. In the output, `te`

is
the time of the event, `sole`

is the solution at the time of the event, and
`ie`

is the index of the triggered event. `tsol`

is a
column vector of times specified in `tspan`

, prior to the first terminal
event.

For each event function, specify whether the integration is to terminate at a zero and
whether the direction of the zero-crossing matters. Do this by setting the
`'Events'`

option of `odeset`

to a function, such as
`@myEventFcn`

, and creating a corresponding function:
[`value`

,`isterminal`

,`direction`

] =
`myEventFcn`

(`m`

,`t`

,`xmesh`

,`umesh`

).
The `xmesh`

input contains the spatial mesh and `umesh`

is
the solution at the mesh points.

If

`uji = sol(j,:,i)`

approximates component`i`

of the solution at time`tspan(j)`

and mesh points`xmesh`

, then`pdeval`

evaluates the approximation and its partial derivative ∂*u*/∂_{i}*x*at the array of points`xout`

and returns them in`uout`

and`duoutdx`

:`[uout,duoutdx] = pdeval(m,xmesh,uji,xout)`

. The`pdeval`

function evaluates the partial derivative ∂*u*/∂_{i}*x*rather than the flux. The flux is continuous, but at a material interface the partial derivative may have a jump.

The time integration is done with the `ode15s`

solver. `pdepe`

exploits the capabilities of
`ode15s`

for solving the differential-algebraic equations that arise when
the PDE contains elliptic equations, and for handling Jacobians with a specified sparsity
pattern.

After discretization, elliptic equations give rise to algebraic equations. If the elements
of the initial-conditions vector that correspond to elliptic equations are not consistent with
the discretization, `pdepe`

tries to adjust them before beginning the time
integration. For this reason, the solution returned for the initial time may have a
discretization error comparable to that at any other time. If the mesh is sufficiently fine,
`pdepe`

can find consistent initial conditions close to the given ones.
If `pdepe`

displays a message that it has difficulty finding consistent
initial conditions, try refining the mesh. No adjustment is necessary for elements of the
initial conditions vector that correspond to parabolic equations.

[1] Skeel, R. D. and M. Berzins, "A
Method for the Spatial Discretization of Parabolic Equations in One Space Variable,"
*SIAM Journal on Scientific and Statistical Computing*, Vol. 11, 1990,
pp.1–32.