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:

$$F\left(x_1, x_2, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2}, \ldots, \frac{\partial^2 u}{\partial x_i^2}, \ldots\right) = 0$$

where:

• $u$ is the unknown function of the variables $x_1, x_2, \ldots, x_n$.
• $F$ is a given function that relates the variables and their derivatives.
• The partial derivatives of $u$ with respect to the variables are denoted as $\frac{\partial u}{\partial x_i}$, and higher-order derivatives can be represented similarly.

Some examples of PDEs include:

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

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

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

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

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

where $c$ is the wave speed.

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

$$\nabla^2 u = 0$$

This equation describes steady-state solutions.

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

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

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

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

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

where $u$ is the velocity field, $\rho$ is the fluid density, $p$ is the pressure, and $\nu$ is the kinematic viscosity.

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Methods of solving PDEs

There are several methods to solve PDEs, including:

• Separation of Variables: This method involves assuming that the solution can be written as a product of functions, each depending on a single variable. This reduces the PDE to a set of ODEs.
• Method of Characteristics: This method is used for first-order PDEs and involves transforming the PDE into a set of ODEs along certain curves called characteristics.
• Fourier Series and Transforms: These methods are used to solve linear PDEs by transforming them into algebraic equations in the frequency domain.
• Finite Difference and Finite Element Methods: These are numerical methods used to approximate solutions to PDEs by discretizing the domain and solving the resulting algebraic equations.

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Example: Solving the Heat Equation with Separation of Variables

Consider the one-dimensional heat equation:

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

with boundary conditions:

• $u(0, t) = 0$
• $u(L, t) = 0$
• $u(x, 0) = f(x)$ where $L$ is the length of the rod and $f(x)$ is the initial temperature distribution.

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

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

Substituting this into the heat equation gives:

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

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

1. For $T(t)$:

$$\frac{dT}{dt} + \alpha \lambda T = 0$$

2. For $X(x)$:

$$\frac{d^2X}{dx^2} + \lambda X = 0$$

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

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

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

$$T_n(t) = e^{-\alpha \lambda_n t}$$

The general solution is a sum of these modes:

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

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

$$A_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx$$

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

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