Partial Differential Equations

A PDE involves partial derivatives of a dependent variable with respect to 2 or more independent variables.

A general PDE in implicit form is $f(x,y,u,u_x,u_y,u_{xx},u_{xy},\ldots) = 0$.

Here:

• $x,y$ - independent variables
• $u(x,y)$ - dependent variable
• $u_x$ - first partial derivative w.r.t. $x$
• $u_y$ - first partial derivative w.r.t. $y$
• $u_{xx}$ - 2nd partial derivative w.r.t. $x$
• $u_{yy}$ - 2nd partial derivative w.r.t. $y$
• $u_{xy}$ - partial derivative of $u_x$ w.r.t. $y$
• $u_{yx}$ - partial derivative of $u_y$ w.r.t. $x$

Terminology

Order

Highest derivative appearing in a PDE.

Types

Linear

Dependent variable and all derivatives appear linearly. Won’t have: $u_x^2$, $\sin(u)$, $u\cdot u_{xx}$.

Non-linear

When a PDE is not linear.

Quasi-linear

When highest-order derivative terms appear linearly. but coefficients may depend on lower-order derivatives or variables. A subset of non-linear PDEs.

Homogeneous

Every term contains $u$ or its derivatives. There are no constants.

Not discussed for non-linear PDEs.

Non-homogenous

When a PDE is not homogeneous.

Forms

Implicit Form

The PDE is written without solving for any specific derivative. Everything is inside one general function.

Explicit Form

The PDE is solved for the highest-order derivatives.

Normal Form

The PDE is solved for only one highest-order derivative. Basically explicit form for the specific highest derivative you care about.

Used to solve PDEs by characteristic curves.

Classification of Second-Order PDEs

Second-order equations are central in heat flow, vibrations and potential theory. Their classification determines what solving technique is appropriate.

Canonical Form

A second-order linear PDE with 1 dependent variable and 2 independent variables.

Here $A$ to $G$ are functions of $x$ or $y$ or both or constants.

Classification depends on the principal part $L(u) = Au_{xx} + Bu_{xy} + Cu_{yy}$. When $A, B, C$ are functions of $x$ and $y$, the classification may differ across different points.

Discriminant

For a PDE in canonical form, its discriminant is:

Elliptic

When $D \lt 0$.

Examples:

• $u_{xx} + u_{yy} = 0$ (Laplace Equation)

Hyperbolic

When $D \gt 0$.

Examples:

• $u_{xx} - u_{yy} = 1$

Parabolic

When $D = 0$.

Examples:

• $u_t = k u_{xx}$ (Heat Equation)