Partial Differential Equations - Theory & Concepts

Partial Differential Equations

$Learning Objectives$

Define what constitutes a Partial Differential Equation (PDE).
Classify second-order linear PDEs as elliptic, parabolic, or hyperbolic.
Understand the role and structure of Fourier Series in solving PDEs.
Apply the method of Separation of Variables to solve linear PDEs.
Identify and understand the fundamental PDE models: Heat, Wave, and Laplace equations.

While ordinary differential equations (ODEs) involve unknown functions of a single variable, Partial Differential Equations (PDEs) involve unknown functions of multiple independent variables and their partial derivatives. They are essential for describing physical phenomena spread out over space and time, such as heat diffusion, wave propagation, and fluid dynamics.

$Partial Differential Equation (PDE)$

An equation containing an unknown function $u(x, y, \dots)$ of two or more independent variables and its partial derivatives ( $\frac{\partial u}{\partial x}, \frac{\partial^2 u}{\partial y^2}$ , etc.).

Classification of Second-Order Linear PDEs

A general second-order linear PDE in two variables $(x, y)$ takes the form:

General Second-Order Linear PDE

The standard form of a second-order linear PDE in two independent variables.

A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + Fu = G

Variables

Symbol	Description	Unit
$u$	The unknown function dependent on x and y	-
$A, B, C$	Coefficients of the second-order partial derivatives	-
$D, E$	Coefficients of the first-order partial derivatives	-
$F, G$	Functions of x and y	-

$The Discriminant Test$

For the principal part $A u_{xx}+B u_{xy}+C u_{yy}$ , classify the equation using $\Delta = B^2 - 4AC$ at the point under study, provided this principal part is nondegenerate:

Elliptic ( $\Delta < 0$ ): The Laplace equation $u_{xx} + u_{yy} = 0$ is the canonical steady boundary-value example.
Parabolic ( $\Delta = 0$ ): The heat equation $u_t = \alpha^2 u_{xx}$ is the canonical diffusion model; with positive diffusivity it smooths temperature profiles under standard boundary conditions.
Hyperbolic ( $\Delta > 0$ ): The wave equation $u_{tt} = c^2 u_{xx}$ is the canonical propagation model, with finite wave speed $c$ .

For variable coefficients, evaluate the principal part locally; if it degenerates, the classification can change or require separate analysis. Lower-order terms do not enter this discriminant test.

Canonical PDE Classification

Compare the sign of the principal-part discriminant with the canonical model and the kind of problem it represents.

Elliptic: Δ < 0 | Laplace | steady boundary-value model

Parabolic: Δ = 0 | Heat | diffusion in space and time

Hyperbolic: Δ > 0 | Wave | finite-speed propagation

Fourier Series

For a periodic function on $[-L,L]$ , Fourier series represent data using sines and cosines with period $2L$ . For a separated PDE, the appropriate spatial eigenfunctions come from its boundary-value problem and may be sine/cosine functions or another orthogonal family.

Fourier Series Formula

Representation of a piecewise smooth periodic function as a sum of sines and cosines.

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right)

Variables

Symbol	Description	Unit
$f(x)$	The function to represent	-
$L$	Half of the period (period is 2L)	-
$a_0, a_n, b_n$	The Fourier coefficients computed via integrals	-

$Computing Euler-Fourier Coefficients$

STEP-BY-STEP

For $a_0$ : $a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$
For $a_n$ : $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$
For $b_n$ : $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

Projecting Data onto a Spatial Basis

Initial data are represented through inner products with the eigenfunctions selected by the spatial boundary-value problem.

State the spatial domain and boundary conditions

Find the eigenfunctions and their orthogonality weight

Compute each modal coefficient by projection

Check that the resulting series represents the data

Fourier-Series Convergence

For a piecewise-smooth periodic function, Fourier partial sums converge to the function at each continuity point and to the mean of the one-sided limits at a jump. Near a jump, the Gibbs overshoot narrows as more harmonics are included, but its peak relative to the jump does not vanish. The standing-wave simulation below illustrates wave modes rather than plotting Fourier partial sums.

Method of Separation of Variables

This is the primary analytical method for solving linear PDEs with homogeneous boundary conditions. It assumes the multivariable solution can be factored into a product of single-variable functions.

$Method of Separation of Variables$

STEP-BY-STEP

Assume Solution Form: Let $u(x,t) = X(x)T(t)$ .
Substitute into PDE: Plug the assumed form into the PDE. Rearrange algebraically to separate the variables onto opposite sides of the equation (e.g., all $x$ terms on the left, all $t$ terms on the right).
Set a Separation Constant: Since a function of $x$ equals a function of $t$ for all $x$ and $t$ , both sides equal a constant. Choose its sign convention consistently with the spatial boundary-value problem. This converts the PDE into two ordinary differential equations (ODEs).
Solve the Boundary Value Problem (BVP): Solve the spatial ODE ( $X(x)$ ) using the given homogeneous boundary conditions (like fixed ends of a string). This restricts $\lambda$ to specific discrete values called eigenvalues ( $\lambda_n$ ), which correspond to non-trivial solutions called eigenfunctions ( $X_n(x)$ ).
Solve the Temporal ODE: Substitute each eigenvalue $\lambda_n$ into the temporal equation derived from the actual PDE. The heat equation gives a first-order time ODE; the wave equation gives a second-order time ODE.
Superposition: Construct the general solution by summing the infinite product solutions: $u(x,t) = \sum_{n=1}^\infty X_n(x)T_n(t)$ .
Apply Initial Conditions: Project each required initial datum onto the justified spatial eigenfunction basis. For the wave equation this generally means fitting both initial displacement and initial velocity.

Heat and Wave Temporal Modes

The spatial eigenvalue enters different temporal equations: first order for heat and second order for wave.

Heat equation

Tₙ′ + α²λₙTₙ = 0
For λₙ > 0, a decaying exponential mode when α² > 0

Wave equation

Tₙ″ + c²λₙTₙ = 0
For λₙ > 0 and c > 0, an oscillatory mode requiring two initial values

Separation of Variables Workflow

Separation of Variables: Eigenmodes to Fourier Fit

Follow a separable linear PDE from its domain and boundary conditions through the spatial eigenproblem, temporal modes, superposition, and initial-data fit.

This route applies when the linear PDE, domain, and boundary conditions support separation of variables. Homogeneous boundary conditions are assumed. For nonhomogeneous data, use a justified lifting only when the transformed equation still permits the required eigenfunction/modal expansion, including any induced forcing; otherwise choose another method. The spatial boundary-value problem must yield the required eigenfunctions. Derive each temporal equation from the actual PDE: a heat model gives a first-order time equation, while a wave model gives a second-order time equation and generally requires both initial displacement and velocity. Fit initial data only in the eigenfunction basis justified by the spatial problem; for an orthogonal Sturm–Liouville basis use its weight in the projection. Check convergence and compatibility on the stated domain, then verify the PDE and all supplied conditions.

Start: state PDE, domain, BCs, and ICs → Is the PDE linear?; Is the PDE linear? — Yes → Do its operator and domain support separation?; Is the PDE linear? — No → Use a method for nonlinear PDEs; Do its operator and domain support separation? — Yes → Are the boundary conditions homogeneous?; Do its operator and domain support separation? — No → Use another method for this PDE; Are the boundary conditions homogeneous? — Yes → Use the homogeneous boundary conditions; Are the boundary conditions homogeneous? — No → Can a lift preserve a usable modal expansion?; Can a lift preserve a usable modal expansion? — Yes → Lift the data; include induced modal forcing; Can a lift preserve a usable modal expansion? — No → Choose a method for these boundary conditions; Use the homogeneous boundary conditions → Form the spatial boundary-value eigenproblem; Lift the data; include induced modal forcing → Form the spatial boundary-value eigenproblem; Form the spatial boundary-value eigenproblem → Solve for spatial eigenvalues λₙ and eigenfunctions Xₙ; Solve for spatial eigenvalues λₙ and eigenfunctions Xₙ → Do the eigenpairs satisfy the boundary conditions?; Do the eigenpairs satisfy the boundary conditions? — Yes → Derive each forced or unforced temporal ODE; Do the eigenpairs satisfy the boundary conditions? — No → Reformulate or use another spatial method; Derive each forced or unforced temporal ODE → Is the eigenfunction expansion justified?; Is the eigenfunction expansion justified? — Yes → Project initial displacement and, if needed, velocity; Is the eigenfunction expansion justified? — No → Resolve expansion or data compatibility; Project initial displacement and, if needed, velocity → Do the coefficients fit the stated initial data?; Do the coefficients fit the stated initial data? — Yes → Form u(x,t) = Σ Xₙ(x)Tₙ(t); Do the coefficients fit the stated initial data? — No → Resolve expansion or data compatibility; Form u(x,t) = Σ Xₙ(x)Tₙ(t) → PDE, BCs, ICs, and convergence check pass?; PDE, BCs, ICs, and convergence check pass? — Yes → Record solution and domain of validity; PDE, BCs, ICs, and convergence check pass? — No → Recheck model, eigenpairs, projection, and series; Recheck model, eigenpairs, projection, and series → Start: state PDE, domain, BCs, and ICs; Resolve expansion or data compatibility → Use a compatible eigenfunction or other method

Start: state PDE, domain, BCs, and ICs: terminator
Is the PDE linear?: decision
Do its operator and domain support separation?: decision
Are the boundary conditions homogeneous?: decision
Use a method for nonlinear PDEs: terminator
Use another method for this PDE: terminator
Use the homogeneous boundary conditions: process
Can a lift preserve a usable modal expansion?: decision
Choose a method for these boundary conditions: terminator
Lift the data; include induced modal forcing: process
Form the spatial boundary-value eigenproblem: process
Solve for spatial eigenvalues λₙ and eigenfunctions Xₙ: process
Do the eigenpairs satisfy the boundary conditions?: decision
Derive each forced or unforced temporal ODE: process
Is the eigenfunction expansion justified?: decision
Project initial displacement and, if needed, velocity: process
Do the coefficients fit the stated initial data?: decision
Form u(x,t) = Σ Xₙ(x)Tₙ(t): process
PDE, BCs, ICs, and convergence check pass?: decision
Recheck model, eigenpairs, projection, and series: process
Resolve expansion or data compatibility: process
Record solution and domain of validity: terminator
Use a compatible eigenfunction or other method: terminator
Reformulate or use another spatial method: terminator

Fundamental PDE Models

Three classic linear PDEs form the foundation of this field. Each describes a fundamentally different physical process.

$1. The 1D Heat Equation (Parabolic)$

Models the diffusion of thermal energy in a rod over time.

Behavior: For a finite interval with homogeneous Dirichlet boundary conditions, separation gives a positive spatial eigenvalue problem and a temporal ODE $T_n' + \alpha^2\lambda_n T_n = 0$ . With $\alpha^2 > 0$ and $\lambda_n > 0$ , the modal factor decays exponentially; higher eigenvalues decay faster.

1D Heat Equation

Models the parabolic diffusion of heat over space and time.

\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}

Variables

Symbol	Description	Unit
$u(x,t)$	The temperature at position x and time t	-
$\alpha^2$	Thermal diffusivity k/(c\rho)	-

$2. The 1D Wave Equation (Hyperbolic)$

Governs the small-amplitude, transverse vibrations of a stretched string, or acoustic waves in a pipe.

Behavior: For a fixed-end string, the spatial problem has positive eigenvalues and the temporal ODE is $T_n'' + c^2\lambda_n T_n = 0$ . Each mode oscillates at angular frequency $c\sqrt{\lambda_n}$ and requires two initial values; the full solution is a superposition of these modes.

1D Wave Equation

Models hyperbolic wave propagation over space and time.

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Variables

Symbol	Description	Unit
$u(x,t)$	The wave displacement at position x and time t	-
$c$	The wave speed, \sqrt{T/\rho}	-

$3. The 2D Laplace Equation (Elliptic)$

Describes steady-state temperature distribution in a 2D plate (after a long time has passed), or electrostatic potential in a region devoid of charge.

Behavior: There is no time derivative. With the convention $X''/X = -Y''/Y = \lambda$ , a positive $\lambda$ makes $X$ exponential/hyperbolic and $Y$ trigonometric; reversing which separated ratio is assigned $+\lambda$ swaps the coordinates. Boundary conditions determine the useful convention and admissible modes.

2D Laplace Equation

Models elliptic steady-state systems.

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0

Variables

Symbol	Description	Unit
$u(x,y)$	The steady-state quantity, such as temperature or potential	-

$Interactive Simulation$

Interact with the simulation below to visualize standing waves and normal modes in the 1D wave equation.

1D Wave Equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Vibrating string fixed at both ends ( $x=0$ , $x=L$ ). The solution is a superposition of normal modes (standing waves).

Normal Mode (

n

Determines the spatial frequency $\sin(n\pi x/L)$ .

Wave Speed (

c

)1.0

u(x,t)

u_n(x,t) = \sin\left(\frac{1\pi x}{L}\right) \cos\left(\frac{1\pi (1.0) t}{L}\right)

Key Takeaways

PDE Classification: For a nondegenerate principal part $A u_{xx}+B u_{xy}+C u_{yy}$ , the sign of $B^2-4AC$ gives the local elliptic, parabolic, or hyperbolic type.
Separation of Variables: The core analytical technique to convert a single linear PDE into multiple ODEs by assuming the solution factors as $u(x,t) = X(x)T(t)$ .
Fourier Series and Eigenfunction Expansions: Represent compatible data in a basis justified by the spatial boundary-value problem; projections use the associated inner product and weight.
Heat Equation ( $u_t = \alpha^2 u_{xx}$ ): Results in exponential decay of temperature differences over time, smoothing the distribution.
Wave Equation ( $u_{tt} = c^2 u_{xx}$ ): Results in oscillatory motion in both space and time, supporting traveling and standing waves.
Laplace's Equation ( $u_{xx} + u_{yy} = 0$ ): Results in steady-state solutions built from combinations of trigonometric (oscillating) and hyperbolic (exponential) functions.

$References$

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