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Chapter 12: Partial Differential Equations

Chapter 12: Partial Differential Equations

Mathematical Physics, Part 2. Theory, numbered worked examples.

Theory

Basic Concepts of Partial Differential Equations

A partial differential equation (PDE) involves partial derivatives of an unknown function \(u\) of two or more variables, often time \(t\) and one or more space variables. The order of the highest derivative is the order of the PDE. As for ODEs, second-order PDEs are the most important in applications.

The Wave Equation

One-dimensional wave equation

A vibrating elastic string (a violin string, say) is modelled by the one-dimensional wave equation \begin{equation*}\tag{12.1} \pd{^2u}{t^2}=c^2\pd{^2u}{x^2},\qquad c^2=\frac T\rho, \end{equation*} for the deflection \(u(x,t)\), together with the following conditions. The string is fixed at \(x=0\) and \(x=L\), giving the

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