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Partial Differential Equations: learn PDEs by working through them
This course covers the standard first pass at partial differential equations: what makes an equation a PDE, how to classify second-order linear equations as elliptic, parabolic, or hyperbolic, and how to derive and solve the heat, wave, and Laplace equations. You'll work through the method of characteristics for first-order equations, separation of variables with Fourier series for the classical equations, Sturm-Liouville theory as the general framework behind eigenfunction expansions, and forced/nonhomogeneous problems including the forced wave equation. The course ends with Fourier and Laplace transform methods for problems on infinite domains, and energy methods for proving uniqueness.
Each session, the tutor asks you to derive a step, compute a coefficient, or explain why a condition is needed, rather than showing you a finished solution. If you separate variables but mishandle a boundary condition, the tutor asks what that condition means physically before pointing you to the fix. Over the course it tracks whether you can both explain the reasoning behind a method (why separation of variables requires homogeneous boundary conditions, say) and actually carry out the computation (finding the coefficients in a Fourier sine series), and it returns to earlier equations once you've seen new tools, so you reinterpret d'Alembert's solution after learning transforms, for instance.
Who it is for
- Students in an undergraduate PDE course who want to talk through derivations and solution steps
- Engineering and physics majors using PDEs to model heat flow, vibration, or potential problems
- Students who have finished multivariable calculus and ODEs and want a structured next step
- Anyone reviewing separation of variables, Fourier series, or characteristics before an exam
- Instructors or tutors looking for a Socratic way to supplement lecture-based PDE courses
What you will be able to do
- Classify a second-order linear PDE as elliptic, parabolic, or hyperbolic and name its canonical form
- Solve first-order linear PDEs by the method of characteristics
- Derive the heat and wave equations from physical conservation laws
- Solve the heat, wave, and Laplace equations by separation of variables with the correct boundary and initial conditions
- Solve forced and nonhomogeneous problems using eigenfunction expansions
- Apply Fourier and Laplace transform methods to PDEs on infinite or semi-infinite domains
A taste of a lesson on Separation of Variables: Heat Equation
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 17 topics in Partial Differential Equations
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
What is a PDE?
Defines a partial differential equation as an equation involving partial derivatives of an unknown function of several variables. Introduces order, linearity vs nonlinearity, homogeneous vs nonhomogeneous equations, and the idea of a solution and a well-posed problem (existence, uniqueness, stability). Contrasts with ODEs the student already knows.
First-Order Linear PDEs and the Method of Characteristics
Solves first-order linear (and quasilinear) PDEs such as transport equations using the method of characteristics: reducing the PDE to a family of ODEs along characteristic curves. Covers the transport/advection equation, initial conditions, and interpreting solutions as traveling profiles.
Builds on What is a PDE?
Classification of Second-Order Linear PDEs
Classifies second-order linear PDEs in two variables as elliptic, parabolic, or hyperbolic using the discriminant, and connects each type to a canonical example: Laplace's equation, the heat equation, and the wave equation. Explains why the classification predicts qualitative behavior (smoothing, propagation, equilibrium).
Builds on What is a PDE?
Deriving the Heat and Wave Equations
Derives the one-dimensional heat equation from conservation of energy and Fourier's law, and the one-dimensional wave equation from Newton's second law applied to a vibrating string under tension. Identifies physical meaning of coefficients (diffusivity, wave speed) and the role of boundary and initial conditions.
Builds on Classification of Second-Order Linear PDEs
Boundary and Initial Conditions
Explains the types of boundary conditions (Dirichlet, Neumann, Robin, periodic) and initial conditions needed for heat, wave, and Laplace problems, and why the number and type of conditions required depends on the equation's order and classification. Discusses physical interpretations (fixed temperature, insulated end, fixed/free string end).
Builds on Deriving the Heat and Wave Equations
Fourier Series
Introduces Fourier series representation of periodic functions, computing sine and cosine coefficients, and the notions of even/odd extensions leading to Fourier sine and cosine series on a finite interval. Covers convergence (pointwise, at jump discontinuities) at an intuitive level.
Builds on What is a PDE?
Separation of Variables: Heat Equation
Solves the homogeneous 1D heat equation with homogeneous boundary conditions by assuming a product solution, deriving an ODE eigenvalue problem in space and an exponential decay in time, and assembling the full solution as a Fourier series using initial data. Discusses why only certain eigenvalues/eigenfunctions arise from the boundary conditions.
Builds on Boundary and Initial Conditions, Fourier Series
Sturm-Liouville Theory and Eigenfunction Expansions
Generalizes the eigenvalue problems seen in separation of variables to the Sturm-Liouville framework: self-adjoint operators, real eigenvalues, orthogonality of eigenfunctions, and expansion of arbitrary functions in eigenfunction series. Explains why this justifies the separation-of-variables method beyond simple sine/cosine cases.
Builds on Separation of Variables: Heat Equation
Separation of Variables: Homogeneous Wave Equation
Solves the homogeneous 1D wave equation with fixed or free boundary conditions by separation of variables, obtaining standing wave (normal mode) solutions, and superposes modes using Fourier series to match initial position and velocity. Interprets solutions physically as vibrating string modes and frequencies.
Builds on Separation of Variables: Heat Equation
D'Alembert's Solution and Characteristics for the Wave Equation
Derives the general solution of the 1D homogeneous wave equation on the infinite line via the method of characteristics, obtaining d'Alembert's formula in terms of left- and right-traveling waves. Uses it to solve initial value problems and discusses domains of dependence and influence, and reflection for finite/semi-infinite strings.
Builds on First-Order Linear PDEs and the Method of Characteristics, Separation of Variables: Homogeneous Wave Equation
Nonhomogeneous Heat Equation with Source Terms
Solves the 1D heat equation with a forcing/source term and/or nonhomogeneous boundary conditions, using techniques such as shifting to homogenize boundary data and eigenfunction expansion (Duhamel-type) to handle the source term. Builds the general strategy of superposing a particular solution and a homogeneous solution.
Builds on Sturm-Liouville Theory and Eigenfunction Expansions
The One-Dimensional Forced Wave Equation
Solves the 1D wave equation with an external forcing term (e.g., f(x,t) representing applied force per unit mass) and given boundary/initial conditions. Uses eigenfunction expansion: expands both the forcing and the unknown solution in the spatial eigenfunctions from the homogeneous problem, reduces the PDE to an infinite system of forced ODEs in time (solved via variation of parameters or Duhamel's principle), and reassembles the series solution. Discusses resonance when forcing frequency matches a natural mode frequency.
Builds on Separation of Variables: Homogeneous Wave Equation, Nonhomogeneous Heat Equation with Source Terms
Laplace's Equation and Separation of Variables in 2D
Solves Laplace's equation on rectangular and circular domains via separation of variables, deriving harmonic function solutions and using Fourier series to match boundary data (Dirichlet problem). Introduces the mean value property and maximum principle as qualitative features of harmonic functions.
Builds on Sturm-Liouville Theory and Eigenfunction Expansions
Heat and Wave Equations in Higher Dimensions
Extends separation of variables to 2D/3D heat and wave equations on rectangular and circular/spherical domains, introducing double Fourier series and Bessel functions for circular domains. Discusses vibrating membranes and nodal patterns as physical interpretation.
Builds on Separation of Variables: Homogeneous Wave Equation, Laplace's Equation and Separation of Variables in 2D
Fourier Transform Methods for PDEs on Infinite Domains
Applies the Fourier transform in the spatial variable to solve heat and wave equations on the whole real line, converting the PDE into an ODE in time for the transformed variable, then inverting (e.g., to derive the heat kernel/Gaussian solution). Discusses convolution and the physical meaning of the heat kernel as a smoothing/averaging operator.
Builds on Fourier Series, D'Alembert's Solution and Characteristics for the Wave Equation
Laplace Transform Methods for PDEs
Applies the Laplace transform in the time variable to solve initial-boundary value problems for the heat and wave equations, particularly useful for problems with time-dependent boundary data or signal/forcing functions, reducing the PDE to an ODE in the spatial variable. Compares this approach to eigenfunction expansion for handling forcing terms.
Builds on The One-Dimensional Forced Wave Equation
Energy Methods and Uniqueness
Uses energy (integral) estimates to prove uniqueness and well-posedness for the heat and wave equations, showing how multiplying the PDE by the solution (or its derivative) and integrating yields conserved or decaying quantities. Connects this to the physical notions of energy conservation (wave) and energy dissipation (heat).
Builds on Deriving the Heat and Wave Equations, Boundary and Initial Conditions
Practice questions in Partial Differential Equations
A few of the kinds of questions you will work through, with answers.
Is the equation u_t + u u_x = 0 linear or nonlinear? Explain.
See the answer
Nonlinear
The term u u_x involves a product of the unknown function with its own derivative, so the equation is not linear in u.
Solve u_t + c u_x = 0, u(x,0) = f(x), using characteristics.
See the answer
u(x,t) = f(x - ct)
Along lines x - ct = constant, u is constant, so the initial profile f just translates to the right at speed c.
Classify u_tt - 4 u_xx = 0 as elliptic, parabolic, or hyperbolic.
See the answer
Hyperbolic
Writing it as A u_xx + B u_xt + C u_tt = 0 with A = -4, B = 0, C = 1, the discriminant B² - 4AC = 16 > 0, which makes it hyperbolic, matching its wave-equation form.
Find the Fourier sine series coefficient b_1 for f(x) = 1 on (0, π), where f(x) = sum b_n sin(nx).
See the answer
b_1 = 4/π
b_n = (2/π)∫₀^π 1·sin(nx) dx = (2/π)(1-cos(nπ))/n; for n=1 this gives (2/π)(2)/1 = 4/π.
Write d'Alembert's solution for u_tt = c² u_xx with u(x,0) = f(x), u_t(x,0) = g(x).
See the answer
u(x,t) = [f(x+ct) + f(x-ct)]/2 + (1/2c)∫ from x-ct to x+ct of g(s) ds
This combines a right- and left-traveling wave built from the initial displacement with an integral of the initial velocity over the domain of dependence.
For u_tt - c² u_xx = f(x,t) with u = 0 at x = 0, L, why do we expand both u and f in the same sine eigenfunctions sin(nπx/L)?
See the answer
Because those eigenfunctions satisfy the boundary conditions and diagonalize the spatial operator
Expanding in the eigenfunctions of the associated Sturm-Liouville problem turns the PDE into a decoupled set of ordinary differential equations, one for each mode's time-dependent coefficient, with the forcing term's coefficients acting as the forcing for each ODE.
Questions about this subject
Do I need to have seen PDEs before this course?
No. It assumes multivariable calculus, linear constant-coefficient ODEs, and basic eigenvalues/eigenvectors, but no prior PDE exposure.
Does this cover numerical methods like finite differences?
No. The course stays with analytic methods: characteristics, separation of variables, Fourier and Laplace transforms, and Sturm-Liouville theory. It stops short of numerical schemes and nonlinear systems.
How does the forced wave equation fit in if most of the course is homogeneous problems?
It comes after you've built up separation of variables for the homogeneous wave equation and eigenfunction expansion for the nonhomogeneous heat equation, so the forced wave equation combines both ideas: expanding the solution and the forcing term in the same eigenfunctions.
What's the difference between this and a course on Fourier analysis?
Fourier series and transforms show up here as tools for solving specific PDEs, not as a subject on their own, so the emphasis is on connecting coefficients and eigenfunctions to boundary conditions and physical solutions rather than on Fourier theory in general.
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