Partial differential equations (PDEs)

Trends

A PDE is a differential equation with two or more independent variables (or coordinates).
- Typically $t$, $x$, $y$, $z$.
- But often more general coordinates, including state space coordinates
  - Composition
  - Temperature
  - etc.
The order of a PDE is the order of the highest derivative appearing in the equation.
“A linear PDE is one in which all of the partial derivatives appear in linear form and none of the coefficients depends on the dependent variable.” (Hoffman)
- The coefficients can be functions of the independent variables.
- This is analogous to linear ODEs.

Laplace equation:
$$\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = 0.$$
Diffusion equation:
$$\frac{\partial f}{\partial t} = \alpha \frac{\partial^2f}{\partial x^2}$$
Wave equation:
$$\frac{\partial f}{\partial t} = c \frac{\partial f}{\partial x}$$

The above examples are all homogeneous. If we add a function like $F(x,y,t)$ to the above equations, they become nonhomogeneous.
- Note that $F$ does not involve $f$.
Nonhomogeneous terms do not change the basic behavior of the PDE, or complicate the numerical solution method.

Often we have a coupled system of ODEs.
Examples
- Combined heat and mass transfer
- Navier-Stokes equations
- Equations of motion
- Maxwell equations

Use subscripts to denote derivatives $$f_t \equiv \frac{\partial f}{\partial t}.$$ $$f_{xx} \equiv \frac{\partial^2f}{\partial x^2},$$ $$f_{xy} \equiv \frac{\partial^2f}{\partial x\partial y}.$$

Gradient
$$ \vec{\nabla} f = \vec{i}\frac{\partial f}{\partial x} + \vec{j}\frac{\partial f}{\partial y}$$
Divergence
$$ \vec{\nabla} \cdot \vec{f} = \frac{\partial f_0}{\partial x} + \frac{\partial f_1}{\partial y}$$
Laplacian $\nabla^2 \equiv \vec{\nabla}\cdot\vec{\nabla}$
$$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$$
(Curl)

PDE classification is important as it determines the character of the PDE, hence the types of solution methods we apply to it.
Quasi-linear PDE (linear in highest derivative): $$ Af_{xx} + Bf_{xy} + Cf_{yy} + Df_x + Ef_y + Ff = G.$$
- Coefficients $A$, $B$, and $C$ may depend on $x$, $y$, $f_x$, and $f_y$.
- Coefficients $D$, $E$, and $F$ may depend on $x$, $y$, and $f$.
- $G$ may depend on $x$, $y$.
- Note, $x$, $y$, are generic (could be time).
Discriminant: $B^2 - 4AC$.
- Elliptic: $B^2 - 4AC$ < 0.
  - Elliptic problems are diffusive
- Parabolic: $B^2 - 4AC$ = 0.
  - Parabolic problems are unsteady diffusive
- Hyperbolic: $B^2 - 4AC$ > 0.
  - Hyperbolic problems are convective (wave character).
Technically, the whole PDE is of one form or another, but often we refer to the character of individual terms in a PDE.
- For example $$\nabla^2\phi + \nabla\cdot(u\phi) = S$$ We would say the first term elliptic, and the second term is hyperbolic.

Elliptic
- Each point depends on every other point in the domain.
- There is no direction of information propagation.
- (Diffusion problems).

Parabolic
- A given point in the domain depends on every other point spatially, but only depends on past times, not future times.
- Future points depend on past points, but not vice-versa.
- (Marching problems, including unsteady diffusive problems).
- In the figure above, the lower part of the plot is the domain of dependence. The upper part of the plot is the range of influence.

Hyperbolic
- A given point depends limited regions of the spatial domain.
- This region is defined by characteristic lines (or “characteristics”).
  - Information propagates along characteristics.
- (Wave propagation, convection).
Example: 1-D wave propagation: