Partial Differential 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,… )u(x, y, \dots) of two or more independent variables and its partial derivatives (∂u∂x,∂2u∂y2\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)(x, y) takes the form:
A∂2u∂x2+B∂2u∂x∂y+C∂2u∂y2+D∂u∂x+E∂u∂y+Fu=GA \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

The Discriminant Test

Similar to conic sections in geometry, PDEs are classified based on the discriminant Δ=B2−4AC\Delta = B^2 - 4AC:
  • Elliptic (Δ<0\Delta < 0): Describes steady-state systems with no time dependence (e.g., 2D Laplace's Equation uxx+uyy=0u_{xx} + u_{yy} = 0). Solutions are typically smooth and represent equilibrium states.
  • Parabolic (Δ=0\Delta = 0): Describes time-dependent dissipative processes (e.g., 1D Heat Equation ut=α2uxxu_t = \alpha^2 u_{xx}). Solutions diffuse and smooth out over time.
  • Hyperbolic (Δ>0\Delta > 0): Describes time-dependent wave propagation (e.g., 1D Wave Equation utt=c2uxxu_{tt} = c^2 u_{xx}). Solutions transport information at a finite speed (waves) and preserve discontinuities.

Fourier Series

To solve many linear PDEs on bounded domains, we must represent complex initial or boundary conditions as an infinite sum of simple sine and cosine waves. This decomposition is known as a Fourier Series.

Fourier Series Formula

For a piecewise smooth, periodic function f(x)f(x) with period 2L2L defined on the interval [−L,L][-L, L]:
f(x)=a02+∑n=1∞(ancos⁡(nπxL)+bnsin⁡(nπxL))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)
Where the Euler-Fourier formulas for the coefficients are:
a0=1L∫−LLf(x) dxa_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dxan=1L∫−LLf(x)cos⁡(nπxL) dxa_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dxbn=1L∫−LLf(x)sin⁡(nπxL) dxb_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx

Visualizing Fourier Series

See how adding more terms (harmonics) to the series improves the approximation of square and sawtooth waves. Notice how higher frequencies are required to capture sharp corners (Gibbs Phenomenon).

Fourier Series Visualization

f(t)≈4π∑k=11sin⁡((2k−1)t)2k−1f(t) \approx \frac{4}{\pi} \sum_{k=1}^{1} \frac{\sin((2k-1)t)}{2k-1}
1 Harmonicn = 120 Harmonics

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.

Separation Procedure

  1. Assume Solution Form: Let u(x,t)=X(x)T(t)u(x,t) = X(x)T(t).
  2. 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 xx terms on the left, all tt terms on the right).
  3. Set to Separation Constant: Since a function of xx equals a function of tt for all xx and tt, both sides must equal a constant, typically denoted −λ-\lambda. This converts the single PDE into two separate ordinary differential equations (ODEs).
  4. Solve the Boundary Value Problem (BVP): Solve the spatial ODE (X(x)X(x)) using the given homogeneous boundary conditions (like fixed ends of a string). This restricts λ\lambda to specific discrete values called eigenvalues (λn\lambda_n), which correspond to non-trivial solutions called eigenfunctions (Xn(x)X_n(x)).
  5. Solve the Temporal ODE: Solve the time-dependent ODE (T(t)T(t)) for each found eigenvalue λn\lambda_n. This usually yields exponential decay (Heat) or oscillatory sines/cosines (Wave).
  6. Superposition: Construct the general solution by summing the infinite product solutions: u(x,t)=∑n=1∞Xn(x)Tn(t)u(x,t) = \sum_{n=1}^\infty X_n(x)T_n(t).
  7. Apply Initial Conditions: Substitute t=0t=0 into the series. Use the initial conditions and compute the Fourier coefficients to find the remaining arbitrary constants.

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.
∂u∂t=α2∂2u∂x2\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}
Where u(x,t)u(x,t) is the temperature and α2=k/(cρ)\alpha^2 = k/(c\rho) is the thermal diffusivity (thermal conductivity kk, specific heat cc, density ρ\rho).
Behavior: Applying Separation of Variables yields a spatial ODE X′′+λX=0X'' + \lambda X = 0 (sines/cosines) and a temporal ODE T′+α2λT=0T' + \alpha^2\lambda T = 0 (exponential decay). The solution features terms like e−α2λntsin⁡(λnx)e^{-\alpha^2\lambda_n t}\sin(\sqrt{\lambda_n}x). High-frequency spatial variations (large nn) decay very rapidly, meaning heat quickly smooths out sharp temperature differences.

2. The 1D Wave Equation (Hyperbolic)

Governs the small-amplitude, transverse vibrations of a stretched string, or acoustic waves in a pipe.
∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
Where u(x,t)u(x,t) is the displacement and c=T/ρc = \sqrt{T/\rho} is the wave speed (tension TT, linear density ρ\rho).
Behavior: Both the spatial (XX) and temporal (TT) ODEs are second-order and result in oscillatory sine/cosine solutions. The full series solution represents a superposition of standing waves (normal modes or harmonics) oscillating at discrete frequencies.

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.
∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
Often written as ∇2u=0\nabla^2 u = 0.
Behavior: There is no time derivative. Separation of variables u(x,y)=X(x)Y(y)u(x,y) = X(x)Y(y) leads to X′′/X=−Y′′/Y=λX''/X = -Y''/Y = \lambda. If λ>0\lambda > 0, XX is oscillatory (sines/cosines) and YY is exponential (or hyperbolic sines/cosines sinh⁡,cosh⁡\sinh, \cosh). The choice of the separation constant's sign depends entirely on the specific boundary conditions given on the edges of the rectangular domain.

Wave Equation Visualizer

1D Wave Equation: Vibrating String

Key Takeaways
  • PDE Classification: Second-order PDEs are classified as Elliptic (Δ<0\Delta < 0, steady-state, e.g., Laplace's), Parabolic (Δ=0\Delta = 0, diffusion, e.g., Heat), or Hyperbolic (Δ>0\Delta > 0, propagation, e.g., Wave).
  • 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)u(x,t) = X(x)T(t).
  • Fourier Series: Used to express initial or boundary conditions as an infinite sum of sinusoidal waves, allowing us to perfectly match the eigenfunctions derived from the spatial ODE boundary value problem.
  • Heat Equation (ut=α2uxxu_t = \alpha^2 u_{xx}): Results in exponential decay of temperature differences over time, smoothing the distribution.
  • Wave Equation (utt=c2uxxu_{tt} = c^2 u_{xx}): Results in oscillatory motion in both space and time, supporting traveling and standing waves.
  • Laplace's Equation (uxx+uyy=0u_{xx} + u_{yy} = 0): Results in steady-state solutions built from combinations of trigonometric (oscillating) and hyperbolic (exponential) functions.
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