Energy Integral and Uniqueness of Solutions

MATH 467 Partial Differential Equations

J. Robert Buchanan

Department of Mathematics

Fall 2017

Objectives

In this lesson we will learn:

▶ how to express the total energy in a vibrating string as an

integral, ▶ how to use the energy integral to establish the uniqueness

of solutions to the wave equation.

Initial Boundary Value Problem

Consider the initial boundary value problem for a string of length L > 0.

u_{tt} = c^{2}u_{xx} for 0 < x < L and t > 0
u(0, t) = u(L, t) = 0
u(x, 0) = f (x)
u_{t}(x, 0) = g(x)

During the derivation of the wave equation we set c^{2} = T_{0}/ρ where ρ is the mass density of the string and T_{0} is the tension in the string.

Kinetic Energy

Kinetic energy for a point mass is defined as mv^{2}/2 where v is velocity. For the distributed mass of the string, kinetic energy is

K (t) = \frac{1}{2} ∫_{0}^{L}ρ(u_{t}(x, t))^{2} dx.