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In a *partial differential equation* (PDE), the function being
solved for depends on several variables, and the differential equation can include
partial derivatives taken with respect to each of the variables. Partial differential
equations are useful for modelling waves, heat flow, fluid dispersion, and other
phenomena with spatial behavior that changes over time.

The MATLAB^{®} PDE solver `pdepe`

solves initial-boundary
value problems for systems of PDEs in one spatial variable *x* and
time *t*. You can think of these as ODEs of one variable that
also change with respect to time.

`pdepe`

uses an informal classification for the 1-D equations
it solves:

Equations with a time derivative are

*parabolic*. An example is the heat equation $$\frac{\partial u}{\partial t}=\frac{{\partial}^{2}u}{\partial {x}^{2}}$$.Equations without a time derivative are

*elliptic*. An example is the Laplace equation $$\frac{{\partial}^{2}u}{\partial {x}^{2}}=0$$.

`pdepe`

requires at least one parabolic equation
in the system. In other words, at least one equation in the system must include a
time derivative.

`pdepe`

also solves certain 2-D and 3-D problems that reduce to
1-D problems due to angular symmetry (see the argument description for the symmetry
constant `m`

for more information).

Partial Differential Equation Toolbox™ extends this functionality to generalized problems in 2-D and 3-D with Dirichlet and Neumann boundary conditions.

A 1-D PDE includes a function *u*(*x*,*t*) that depends on time *t* and one spatial variable
*x*. The MATLAB PDE solver `pdepe`

solves systems of 1-D parabolic
and elliptic PDEs of the form

$$c\left(x,t,u,\frac{\partial u}{\partial x}\right)\frac{\partial u}{\partial t}={x}^{-m}\frac{\partial}{\partial x}\left({x}^{m}f\left(x,t,u,\frac{\partial u}{\partial x}\right)\right)+s\left(x,t,u,\frac{\partial u}{\partial x}\right).$$

The equation has the properties:

The PDEs hold for

*t*_{0}≤*t*≤*t*_{f}and*a*≤*x*≤*b*.The spatial interval [

*a*,*b*] must be finite.`m`

can be 0, 1, or 2, corresponding to*slab*,*cylindrical*, or*spherical*symmetry, respectively. If*m*> 0, then*a*≥ 0 must also hold.The coefficient $$f\left(x,t,u,\frac{\partial u}{\partial x}\right)$$ is a flux term and $$s\left(x,t,u,\frac{\partial u}{\partial x}\right)$$ is a source term.

The flux term must depend on the partial derivative ∂

*u*/∂*x*.

The coupling of the partial derivatives with respect to time is restricted to
multiplication by a diagonal matrix $$c\left(x,t,u,\frac{\partial u}{\partial x}\right)$$. The diagonal elements of this matrix are either zero or positive.
An element that is zero corresponds to an elliptic equation, and any other element
corresponds to a parabolic equation. There must be at least one parabolic equation.
An element of *c* that corresponds to a parabolic equation can
vanish at isolated values of *x* if they are mesh points (points
where the solution is evaluated). Discontinuities in *c* and
*s* due to material interfaces are permitted provided that a
mesh point is placed at each interface.

To solve PDEs with `pdepe`

, you must define the equation
coefficients for *c*, *f*, and
*s*, the initial conditions, the behavior of the solution
at the boundaries, and a mesh of points to evaluate the solution on. The
function call `sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan)`

uses this information to calculate a solution on the specified mesh:

Together, the `xmesh`

and
`tspan`

vectors form a 2-D grid that
`pdepe`

evaluates the solution on.

You must express the PDEs in the standard form expected by
`pdepe`

. Written in this form, you can read off the
values of the coefficients *c*, *f*, and
*s*.

In MATLAB you can code the equations with a function of the form

function [c,f,s] = pdefun(x,t,u,dudx) c = 1; f = dudx; s = 0; end

`pdefun`

defines the equation $$\frac{\partial u}{\partial t}=\frac{{\partial}^{2}u}{\partial {x}^{2}}$$. If there are multiple equations, then At the initial time *t* =
*t*_{0}, for all *x*,
the solution components satisfy initial conditions of the form

$$u\left(x,{t}_{0}\right)={u}_{0}\left(x\right).$$

In MATLAB you can code the initial conditions with a function of the form

function u0 = icfun(x) u0 = 1; end

`u0 = 1`

defines an initial condition of `u0`

is a vector with each element defining the initial condition of one
equation.At the boundary *x* = *a* or
*x* = *b*, for all *t*,
the solution components satisfy boundary conditions of the form

$$p\left(x,t,u\right)+q\left(x,t\right)f\left(x,t,u,\frac{\partial u}{\partial x}\right)=0.$$

*q*(*x*,*t*) is a diagonal matrix with elements that are either zero or
never zero. Note that the boundary conditions are expressed in terms of the flux
*f*, rather than the partial derivative of
*u* with respect to *x*. Also, of the two
coefficients *p*(*x*,*t*,*u*) and *q*(*x*,*t*), only *p* can depend on
*u*.

In MATLAB you can code the boundary conditions with a function of the form

function [pL,qL,pR,qR] = bcfun(xL,uL,xR,uR,t) pL = uL; qL = 0; pR = uR - 1; qR = 0; end

`pL`

and `qL`

are the coefficients for the left boundary, while
`pR`

and `qR`

are the coefficients for the
right boundary. In this case `bcfun`

defines the boundary
conditions$$\begin{array}{l}{u}_{L}\left({x}_{L},t\right)=0\\ {u}_{R}\left({x}_{R},t\right)=1\end{array}$$

If there are multiple equations, then the outputs `pL`

,
`qL`

, `pR`

, and `qR`

are
vectors with each element defining the boundary condition of one
equation.

The default integration properties in the MATLAB PDE solver are selected to handle common problems. In some cases,
you can improve solver performance by overriding these default values. To do
this, use `odeset`

to create an
`options`

structure. Then, pass the structure to
`pdepe`

as the last input argument:

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

Of the options for the underlying ODE solver `ode15s`

, only
those shown in the following table are available for
`pdepe`

.

Category | Option Name |
---|---|

Error control | |

Step-size | |

Event logging |

After you solve an equation with `pdepe`

, MATLAB returns the solution as a 3-D array `sol`

, where
`sol(i,j,k)`

contains the `k`

th component
of the solution evaluated at `t(i)`

and
`x(j)`

. In general, you can extract the `k`

th
solution component with the command `u = sol(:,:,k)`

.

The time mesh you specify is used purely for output purposes, and does not
affect the internal time steps taken by the solver. However, the spatial mesh
you specify can affect the quality and speed of the solution. After solving an
equation, you can use `pdeval`

to evaluate the
solution structure returned by `pdepe`

with a different
spatial mesh.

An example of a parabolic PDE is the heat equation in one dimension:

$$\frac{\partial \mathit{u}}{\partial \mathit{t}}=\frac{{\partial}^{2}\mathit{u}}{\partial {\mathit{x}}^{2}}.$$

This equation describes the dissipation of heat for $0\le \mathit{x}\le \mathit{L}$ and $\mathit{t}\ge 0$. The goal is to solve for the temperature $\mathit{u}\left(\mathit{x},\mathit{t}\right)$. The temperature is initially a nonzero constant, so the initial condition is

$$\mathit{u}\left(\mathit{x},0\right)={\mathit{T}}_{0}.$$

Also, the temperature is zero at the left boundary, and nonzero at the right boundary, so the boundary conditions are

$$\mathit{u}\left(0,\mathit{t}\right)=0,$$

$$\mathit{u}\left(\mathit{L},\mathit{t}\right)=1.$$

To solve this equation in MATLAB, you need to code the equation, initial conditions, and boundary conditions, then select a suitable solution mesh before calling the solver `pdepe`

. You either can include the required functions as local functions at the end of a file (as in this example), or save them as separate, named files in a directory on the MATLAB path.

**Code Equation**

Before you can code the equation, you need to make sure that it is in the form that the `pdepe`

solver expects:

$$\mathit{c}\left(\mathit{x},\mathit{t},\mathit{u},\frac{\partial \mathit{u}}{\partial \mathit{x}}\right)\frac{\partial \mathit{u}}{\partial \mathit{t}}={\mathit{x}}^{-\mathit{m}}\frac{\partial}{\partial \mathit{x}}\left({\mathit{x}}^{\mathit{m}}\mathit{f}\left(\mathit{x},\mathit{t},\mathit{u},\frac{\partial \mathit{u}}{\partial \mathit{x}}\right)\right)+\mathit{s}\left(\mathit{x},\mathit{t},\mathit{u},\frac{\partial \mathit{u}}{\partial \mathit{x}}\right).$$

In this form, the heat equation is

$$1\cdot \frac{\partial \mathit{u}}{\partial \mathit{t}}={\mathit{x}}^{0}\frac{\partial}{\partial \mathit{x}}\left({\mathit{x}}^{0}\frac{\partial \mathit{u}}{\partial \mathit{x}}\right)+0.$$

So the values of the coefficients are as follows:

$$\mathit{m}=0$$

$$\mathit{c}=1$$

$$\mathit{f}=\frac{\partial \mathit{u}}{\partial \mathit{x}}$$

$$\mathit{s}=0$$

The value of $\mathit{m}$ is passed as an argument to `pdepe`

, while the other coefficients are encoded in a function for the equation, which is

function [c,f,s] = heatpde(x,t,u,dudx) c = 1; f = dudx; s = 0; end

(Note: All functions are included as local functions at the end of the example.)

**Code Initial Condition**

The initial condition function for the heat equation assigns a constant value for ${\mathit{u}}_{0}$. This function must accept an input for $\mathit{x}$, even if it is unused.

function u0 = heatic(x) u0 = 0.5; end

**Code Boundary Conditions**

The standard form for the boundary conditions expected by the `pdepe`

solver is

$$\mathit{p}\left(\mathit{x},\mathit{t},\mathit{u}\right)+\mathit{q}\left(\mathit{x},\mathit{t}\right)\mathit{f}\left(\mathit{x},\mathit{t},\mathit{u},\frac{\partial \mathit{u}}{\partial \mathit{x}}\right)=0.$$

Written in this form, the boundary conditions for this problem are

$$\mathit{u}\left(0,\mathit{t}\right)+\left(0\cdot \mathit{f}\right)=0,$$

$$\left(\mathit{u}\left(\mathit{L},\mathit{t}\right)-1\right)+\left(0\cdot \mathit{f}\right)=0.$$

So the values for $\mathit{p}$ and $\mathit{q}$ are

$${\mathit{p}}_{\mathit{L}}={\mathit{u}}_{\mathit{L}},{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{q}}_{\mathit{L}}=0.$$

${\mathit{p}}_{\mathit{R}}={\mathit{u}}_{\mathit{R}}-1,{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{q}}_{\mathit{R}}=0$.

The corresponding function is then

function [pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = ur - 1; qr = 0; end

**Select Solution Mesh**

Use a spatial mesh of 20 points and a time mesh of 30 points. Since the solution rapidly reaches a steady state, the time points near $\mathit{t}=0$ are more closely spaced together to capture this behavior in the output.

L = 1; x = linspace(0,L,20); t = [linspace(0,0.05,20), linspace(0.5,5,10)];

**Solve Equation**

Finally, solve the equation using the symmetry $\mathit{m}$, the PDE equation, the initial condition, the boundary conditions, and the meshes for $\mathit{x}$ and $\mathit{t}$.

m = 0; sol = pdepe(m,@heatpde,@heatic,@heatbc,x,t);

**Plot Solution**

Use `imagesc`

to visualize the solution matrix.

colormap hot imagesc(x,t,sol) colorbar xlabel('Distance x','interpreter','latex') ylabel('Time t','interpreter','latex') title('Heat Equation for $0 \le x \le 1$ and $0 \le t \le 5$','interpreter','latex')

**Local Functions**

function [c,f,s] = heatpde(x,t,u,dudx) c = 1; f = dudx; s = 0; end function u0 = heatic(x) u0 = 0.5; end function [pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = ur - 1; qr = 0; end

Several available example files serve as excellent starting points for most common 1-D PDE problems. To explore and run examples, use the Differential Equations Examples app. To run this app, type

odeexamples

To open an individual file for editing, type

edit exampleFileName.m

To run an example, type

exampleFileName

This table contains a list of the available PDE example files.

Example File | Description | Example Link |
---|---|---|

| Simple PDE that illustrates the formulation, computation, and plotting of the solution. | |

| Problem that involves discontinuities. | |

| Problem that requires computing values of the partial derivative. | |

| System of two PDEs whose solution has boundary layers at
both ends of the interval and changes rapidly for small
| |

| System of PDEs with step functions as initial conditions. |

[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.

`bvp4c`

| `ode45`

| `odeset`

| `pdepe`

| `pdeval`