Primer on Partial Differential Equations

Table of Contents

A partial differential equation (PDE) is an equation that involves partial derivatives of a multivariable function. Typically, they describe how a quantity (like temperature, displacement, pressure) changes over space and time.

If ordinary differential equations (ODEs) describe evolution over one variable (often time), PDEs handle evolution over multiple variables.

The basic form of a PDE is:

F(x1,x2,,xn,u,ux1,ux2,,2uxi2,)=0F\left(x_1, x_2, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2}, \ldots, \frac{\partial^2 u}{\partial x_i^2}, \ldots\right) = 0

where:

Some examples of PDEs include:

The Heat Equation: Describes how heat diffuses through a medium.

ut=α2u\frac{\partial u}{\partial t} = \alpha \nabla^2 u

where α\alpha is the thermal diffusivity and 2\nabla^2 is the Laplacian operator.

The Wave Equation: Describes how waves propagate through a medium.

2ut2=c22u\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u

where cc is the wave speed.

The Laplace Equation: A special case of the heat equation where there is no time dependence.

2u=0\nabla^2 u = 0

This equation describes steady-state solutions.

The Poisson Equation: A generalization of the Laplace equation that includes a source term.

2u=f(x,y,z)\nabla^2 u = f(x, y, z)

where f(x,y,z)f(x, y, z) is a known function.

The Navier-Stokes Equations: Describe the motion of fluid substances.

ut+(u)u=1ρp+ν2u\frac{\partial u}{\partial t} + (u \cdot \nabla) u = -\frac{1}{\rho} \nabla p + \nu \nabla^2 u

where uu is the velocity field, ρ\rho is the fluid density, pp is the pressure, and ν\nu is the kinematic viscosity.


Methods of solving PDEs

There are several methods to solve PDEs, including:


Example: Solving the Heat Equation with Separation of Variables

Consider the one-dimensional heat equation:

ut=α2ux2\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}

with boundary conditions:

To solve this using separation of variables, we assume a solution of the form:

u(x,t)=X(x)T(t)u(x, t) = X(x)T(t)

Substituting this into the heat equation gives:

1αTdTdt=1Xd2Xdx2=λ\frac{1}{\alpha T} \frac{dT}{dt} = \frac{1}{X} \frac{d^2X}{dx^2} = -\lambda

where λ\lambda is a separation constant. This leads to two ODEs:

  1. For T(t)T(t):
dTdt+αλT=0\frac{dT}{dt} + \alpha \lambda T = 0
  1. For X(x)X(x):
d2Xdx2+λX=0\frac{d^2X}{dx^2} + \lambda X = 0

The boundary conditions lead to a Sturm-Liouville problem for X(x)X(x), which has solutions of the form:

Xn(x)=sin(nπxL)X_n(x) = \sin\left(\frac{n\pi x}{L}\right)

with eigenvalues λn=(nπL)2\lambda_n = \left(\frac{n\pi}{L}\right)^2. The solution for T(t)T(t) is:

Tn(t)=eαλntT_n(t) = e^{-\alpha \lambda_n t}

The general solution is a sum of these modes:

u(x,t)=n=1Aneαλntsin(nπxL)u(x, t) = \sum_{n=1}^{\infty} A_n e^{-\alpha \lambda_n t} \sin\left(\frac{n\pi x}{L}\right)

where AnA_n are coefficients determined by the initial condition u(x,0)=f(x)u(x, 0) = f(x) using Fourier series expansion. The coefficients can be found using:

An=2L0Lf(x)sin(nπxL)dxA_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx

This gives the complete solution to the heat equation with the specified boundary and initial conditions.