SYSTEMATIC MATHEMATICS

Partial Differential Equations

How does temperature spread through a plate, how does a stretched string move, and how can a shape settle into equilibrium? A partial differential equation connects one unknown field to changes in several directions. Before solving anything, this course identifies the field, region, time interval, initial state, boundary information, and the meaning of every symbol and derivative. It then develops one- and multidimensional transport, diffusion, waves, equilibrium, conservation laws, Fourier series and transforms, energy estimates, fundamental solutions, Green functions, weak solutions, and a first numerical bridge. Every proposed formula is checked against both the equation and all data, rather than presenting three famous equations as the whole subject.

Before this course: Completed Vector Calculus, Differential Equations & Dynamical Systems, Linear Algebra, and Real Analysis Units 1–6. Students must be able to use gradients, divergence, integration by parts, line and surface integral theorems, ODE existence and systems, eigenvalues and orthogonality, Fourier convergence ideas, multivariable derivatives, and multiple integrals.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

How to read a PDE line, one symbol at a time

Objective: What does Δu mean when x and y are the spatial variables?

A PDE formula is compressed language. Before manipulating it, expand every symbol into a noun or an action: what is unknown, where it lives, which direction changes, what is supplied, and what must be found.

Read the objects before manipulating the equation: In “L[u]=f in Ω, u=g on ∂Ω”, u is the unknown field; L is a rule that differentiates or combines u; square brackets mean “apply L to u”; f is the interior source; Ω (capital omega) is the region; ∂Ω is the edge of that region; and g supplies values on that edge. The result to establish is: For u_t=k(u_xx+u_yy)+q, the subscript t means one time derivative, xx and yy mean two derivatives in the named spatial direction, k is diffusivity, and q is added heat per unit time. The equation says “time change = diffusion caused by curvature + local supply.” Keep the domain, time interval, forcing, initial data, boundary data, and solution class attached to every step.

The worked question is: On the square 0<x<π, 0<y<π, read and verify u(x,y,t)=e^{-2t}sin x sin y for u_t=u_xx+u_yy with zero boundary values. Begin with “Differentiate in time: u_t=−2e^{-2t}sin x sin y.” and finish with “On x=0, x=π, y=0, or y=π, one sine factor is zero; hence all four edges have value zero.” Then check the interior PDE and every relevant data surface separately. A plot can illustrate the result but cannot replace these checks.

For u_t=k(u_xx+u_yy)+q, the subscript t means one time derivative, xx and yy mean two derivatives in the named spatial direction, k is diffusivity, and q is added heat per unit time. The equation says “time change = diffusion caused by curvature + local supply.”

Read the equality sign as a balance between the left and right sides, not as an instruction to solve immediately.

Translate each derivative: u_t asks how the field changes while position is fixed; u_xx and u_yy measure how its spatial slopes themselves change.

Then attach the region, initial state, and boundary data; without them the displayed differential law still describes many possible fields.

On the square 0<x<π, 0<y<π, read and verify u(x,y,t)=e^{-2t}sin x sin y for u_t=u_xx+u_yy with zero boundary values.

  1. Differentiate in time: u_t=−2e^{-2t}sin x sin y.
  2. Differentiate twice in each spatial direction: u_xx=−e^{-2t}sin x sin y and u_yy=−e^{-2t}sin x sin y, so their sum equals u_t.
  3. On x=0, x=π, y=0, or y=π, one sine factor is zero; hence all four edges have value zero.

Result: The formula satisfies both the interior heat equation and all four boundary edges; at t=0 its initial field is sin x sin y.