Problem Solving Guide

PDEs · Axiom Academy

A systematic approach to solving any partial differential equation problem. Read Carefully: Identify the PDE, domain, boundary conditions, and initial conditions Classify the PDE: Determine order, linearity, homogeneity, and type (parabolic/hyperbolic/elliptic) Choose Method: Select separation of variables, transform methods, or characteristic methods based on classification Apply Technique: Execute the solution method systematically, checking each step Verify Solution: Check that your solution satisfies the PDE and all conditions Order: Highest derivative present (1st order: , 2nd order: ) Linearity: Linear if no products/powers of u or its derivatives Homogeneity: Homogeneous if all terms contain u or its derivatives Type: For 2nd order, compute where Common Problem Types and Strategies Heat Equation on Finite Domain: Separation of variables yields exponentially decaying Fourier series Wave Equation with Fixed Ends: Separation gives standing waves; use d'Alembert for infinite domains Laplace on Rectangle: Separate variables in Cartesian coordinates with homogeneous BCs on opposite sides Laplace in Circle/Disk: Use polar coordinates; solutions involve Bessel functions or simple harmonics Non-homogeneous PDEs: Solve homogeneous problem first, then add particular solution Non-homogeneous BCs: Transform to homogeneous BCs using appropriate substitution Example Walkthrough: Heat Equation on Rod

This is the written version of the interactive lesson above. See the full PDEs course.