Partial Differential Equations (PDEs)

Partial Differential Equations (PDEs) are the mathematical foundation of modern physics and engineering. From the propagation of heat in a solid to the evolution of quantum mechanical wavefunctions, PDEs describe systems where the rate of change depends on multiple independent variables—usually space $\mathbf{x}$ and time $t$. Unlike Ordinary Differential Equations (ODEs), which track the evolution of a single point or a discrete set of particles, PDEs track the evolution of continuous fields (like temperature, pressure, or probability density).

1. Classification of Second-Order Linear PDEs

In rigorous mathematical analysis, we classify PDEs to determine the qualitative behavior of their solutions and the boundary conditions required for well-posedness. Consider the general second-order linear PDE for a function $u(x,y)$:

$A{u}_{xx}+B{u}_{xy}+C{u}_{yy}+D{u}_x+E{u}_y+Fu=G.$

The classification depends on the sign of the discriminant $B^2-4AC$:

1. Elliptic ($B^2-4AC<0$): These describe steady-state phenomena. The prototypical example is the Laplace Equation ${\nabla }^{2}u=0$. Solutions are “harmonic functions” and are incredibly smooth (analytic) inside their domain. Disturbances at the boundary are felt instantaneously throughout the interior.
2. Parabolic ($B^2-4AC=0$): These describe diffusion and dissipative processes. The Heat Equation $u_t=\alpha {\nabla }^{2}u$ is the central model. They exhibit a “smoothing property”—even if the initial state is jagged or discontinuous, the state at any $t>0$ is smooth.
3. Hyperbolic ($B^2-4AC>0$): These describe wave motion and signal propagation. The Wave Equation $u_{tt}=c^2{\nabla }^{2}u$ is the primary example. Unlike the others, hyperbolic equations preserve singularities (sharp edges) which travel along paths called characteristics at a finite speed $c$.

2. The Big Three: Laplace, Heat, and Wave

2.1 The Laplace and Poisson Equations

The Laplace equation $\Delta u=0$ is the equilibrium state of heat or potential. It is characterized by the Maximum Principle: a harmonic function cannot have a local maximum or minimum in the interior of its domain—the extreme values must occur on the boundary. If a source term exists, we have the Poisson Equation: ${\nabla }^{2}u=f.$ In electrostatics, $u$ would be the electric potential and $f$ the charge density.

2.2 The Heat Equation

The evolution of temperature in a medium with thermal diffusivity $\alpha$: $\frac{\partial u}{\partial t}=\alpha \Delta u.$ The fundamental solution (Green’s function) in free space is a Gaussian that spreads over time: $\Phi (x,t)=\frac{1}{\sqrt{4\pi \alpha t}}{e}^{-\frac{x^2}{4\alpha t}}.$ This formula reveals that a point source of heat at $t=0$ immediately affects every point in space, representing an “infinite speed of propagation,” a characteristic feature of parabolic systems.

2.3 The Wave Equation

The propagation of vibrations and signals: $\frac{{\partial }^{2}u}{\partial t^2}=c^2\Delta u.$ For the 1D case on an infinite line, the general solution is d’Alembert’s solution: $u(x,t)=\frac{f(x-ct)+f(x+ct)}{2}+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds,$ where $f$ is the initial position and $g$ is the initial velocity. This explicitly shows that the value at $(x,t)$ only depends on the initial data within the “domain of dependence” $[x-ct,x+ct]$.

3. Separation of Variables

One of the most powerful analytical techniques is the method of Separation of Variables. Suppose we want to solve $u_t=k{u}_{xx}$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$.

1. Assume a Product Form: Let $u(x,t)=X(x)T(t)$.
2. Separate Variables: $\frac{T^{\prime }}{kT}=\frac{X^{\prime \prime }}{X}=-\lambda$. Since one side depends only on $t$ and the other only on $x$, both must equal a constant $-\lambda$.
3. Solve the Spatial ODE: $X^{\prime \prime }+\lambda X=0$ with $X(0)=X(L)=0$ yields eigenvalues ${\lambda }_n=(\frac{n\pi }{L}{)}^2$ and eigenfunctions $X_n(x)=\sin (\frac{n\pi x}{L})$.
4. Solve the Temporal ODE: $T_n^{\prime }=-k{\lambda }_n{T}_n⟹T_n(t)=e^{-k(n\pi /L{)}^{2}t}$.
5. Superposition: By linearity, the general solution is: $u(x,t)={\sum }_{n=1}^{\infty }{A}_n\sin \left(\frac{n\pi x}{L}\right)e^{-k{\left(\frac{n\pi }{L}\right)}^{2}t}.$ The constants $A_n$ are determined by the Fourier expansion of the initial condition $u(x,0)$.

4. Boundary Conditions and Well-Posedness

A PDE problem is well-posed in the sense of Hadamard if:

1. A solution exists.
2. The solution is unique.
3. The solution depends continuously on the data (stability).

Boundary Conditions (BCs)

• Dirichlet: $u=f$ on the boundary (e.g., specifying temperature).
• Neumann: $\frac{\partial u}{\partial n}=f$ on the boundary (e.g., specifying heat flux).
• Robin: $au+b\frac{\partial u}{\partial n}=f$ (e.g., convective cooling).

Green’s Functions and Fundamental Solutions

For linear operators $\mathcal{L}$, we can define a Green’s function $G(x,\xi )$ such that $\mathcal{L}G=\delta (x-\xi )$. This represents the “impulse response” of the differential operator. The solution for any source $f(x)$ can then be found via the integral: $u(x)={\int }_{\Omega }G(x,\xi )f(\xi )d\xi .$

5. Numerical Methods: FDM vs FEM

When analytical solutions are unavailable (e.g., irregular domains or nonlinearities), we discretize the PDE.

Finite Difference Method (FDM)

FDM approximates derivatives using Taylor series expansions on a grid: $\frac{{\partial }^{2}u}{\partial x^2}\approx \frac{u_{i+1}-2u_i+u_{i-1}}{\Delta x^2}.$ It is straightforward but struggles with complex geometries.

Finite Element Method (FEM)

FEM relies on the Weak Formulation. We multiply the PDE by a “test function” $v$ and integrate over the domain. Using the divergence theorem, we lower the derivative order. For the Laplace equation: ${\int }_{\Omega }\nabla u\cdot \nabla vd\mathbf{x}={\int }_{\Omega }fvd\mathbf{x}.$ The domain is divided into small “elements” (triangles/tetrahedra), and the solution is approximated by simple polynomials on each element. This handles complex boundaries elegantly.

6. Python Implementation: 1D Heat Diffusion

The following code implements the Explicit Finite Difference Method for the heat equation. Note the stability requirement (the CFL condition): $dt\le \frac{d{x}^2}{2\alpha }$.

import numpy as np
import matplotlib.pyplot as plt

def heat_equation_1d(L, T, alpha, nx, nt):
    dx = L / (nx - 1)
    dt = T / nt
    r = alpha * dt / dx**2
    
    if r > 0.5:
        raise ValueError(f"Stability condition failed: r = {r:.4f}")

    # Initial condition: A temperature spike in the middle
    u = np.zeros(nx)
    u[int(0.4*nx):int(0.6*nx)] = 1.0 
    
    # Store results for visualization
    history = [u.copy()]
    
    for _ in range(nt):
        u_next = u.copy()
        for i in range(1, nx - 1):
            u_next[i] = u[i] + r * (u[i+1] - 2*u[i] + u[i-1])
        u = u_next
        history.append(u.copy())
        
    return np.array(history)

# Parameters
history = heat_equation_1d(L=1.0, T=0.1, alpha=0.01, nx=50, nt=1000)
x = np.linspace(0, 1.0, 50)

# Plotting specific time steps
plt.figure(figsize=(10, 5))
for t_idx in [0, 100, 500, 1000]:
    plt.plot(x, history[t_idx], label=f"t = {t_idx/1000:.3f}")
plt.title("Time Evolution of 1D Heat Diffusion")
plt.xlabel("Position (x)"); plt.ylabel("Temperature (u)")
plt.legend(); plt.grid(True); plt.show()