Introduction to PDEs

What is a PDE?

A partial differential equation involves partial derivatives of an unknown function of several variables:

$F\left(x, y, \ldots, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \ldots, \frac{\partial^2 u}{\partial x^2}, \frac{\partial^2 u}{\partial x\partial y}, \frac{\partial^2 u}{\partial y^2}, \ldots\right) = 0$

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Order and Linearity

Order: Highest derivative appearing.

Linear PDE: Linear in $u$ and its derivatives. Can be written:

$\sum_{|\alpha| \leq n} a_\alpha(\mathbf{x})\frac{\partial^{|\alpha|}u}{\partial x_1^{\alpha_1}\cdots\partial x_m^{\alpha_m}} = f(\mathbf{x})$

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Classification of Second Order Linear PDEs (2D)

Consider:

$a\frac{\partial^2 u}{\partial x^2} + 2b\frac{\partial^2 u}{\partial x\partial y} + c\frac{\partial^2 u}{\partial y^2} + \text{lower order} = 0$

Define discriminant: $\Delta = b^2 - ac$.

Type	Condition	Discriminant	Example
Elliptic	$b^2 < ac$	$\Delta < 0$	Laplace: $u_{xx} + u_{yy} = 0$
Hyperbolic	$b^2 > ac$	$\Delta > 0$	Wave: $u_{tt} = c^2 u_{xx}$
Parabolic	$b^2 = ac$	$\Delta = 0$	Heat: $u_t = Du_{xx}$

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Boundary Conditions

Dirichlet

Specify the value of the function on the boundary:

$u(\mathbf{x}) = f(\mathbf{x}) \text{ on } \partial\Omega$

Neumann

Specify the normal derivative on the boundary:

$\frac{\partial u}{\partial n} = \mathbf{n} \cdot \nabla u = g(\mathbf{x}) \text{ on } \partial\Omega$

Mixed (Robin)

Linear combination:

$\alpha u + \beta\frac{\partial u}{\partial n} = h(\mathbf{x})$

Initial Conditions

For time-dependent problems, specify initial state:

$u(\mathbf{x}, 0) = u_0(\mathbf{x})$

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Well-Posed Problems

A problem is well-posed (Hadamard) if:

1. A solution exists
2. The solution is unique
3. The solution depends continuously on data

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Physical Interpretation

PDE	Physical setting
Laplace	Steady-state heat distribution, electrostatics
Heat equation	Temperature evolution in a conductor
Wave equation	Vibrating string, electromagnetic waves
Diffusion	Particle transport, heat flow

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Superposition Principle

For linear homogeneous PDEs: if $u_1$ and $u_2$ are solutions, so is $u_1 + u_2$.

This enables building complex solutions from simple ones.

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Why PDEs Matter

• Model physical phenomena with spatial and temporal variation
• Describe systems from quantum mechanics to fluid dynamics
• Foundation for understanding continuum physics
• Essential for finite element methods and numerical simulation