The Chemistry Maths Book, Second Edition

392 Chapter 14Partial differential equations

14.2 General solutions

We have seen that the general solution of an ordinary differential equation contains

a number of arbitrary constants, usually nfor an nth-order equation, and that,

in an application, the values of these constants are obtained by the imposition

of appropriate initial or boundary conditions. The general solution of a partial

differential equation, on the other hand, contains a number of arbitrary functions,

often nsuch functions for an nth-order equation. For example, the 1-dimensional

wave equation, equation 1 in the list given in Section 14.1,

(14.1)

has solution

f(x, 1 t) 1 = 1 F(x 1 + 1 vt) (14.2)

where Fis an arbitrary function of the variableu 1 = 1 x 1 + 1 vt. Thus,

and equation (14.1) is satisfied. The general solution of the equation is

f(x, 1 t) 1 = 1 F(x 1 + 1 vt) 1 + 1 G(x 1 − 1 vt) (14.3)

where Fand Gare both arbitrary functions.

1

The particular functions in any

application are determined by appropriate initial and boundary conditions, and one

important example is that discussed in Section 14.7.

EXAMPLE 14.1Verify that the function

f(x, 1 t) 1 = 13 x

2

1 − 12 xvt 1 + 13 v

2

t

2

is a solution of the wave equation (14.1), and has the general form (14.3).

∂

∂

=

∂

∂

=

∂

∂

=

f

t

dF

du

u

t

dF

du

f

t

dF

du

vv,

2

2

2

2

2

∂

∂

=

∂

∂

=,

∂

∂

=

f

x

dF

du

u

x

dF

du

f

x

dF

du

2

2

2

2

∂

∂

=

∂

∂

2

22

2

2

f 1

x

f

v t

1

This form of the general solution was first given by d’Alembert in 1747. The origins of the wave equation lie

in the discussion De motu nervi tensi(On the motion of a tense string) by Brook Taylor in 1713. In 1727 Johann

Bernoulli suggested to his son Daniel that he take up Taylor’s problem again: ‘Of a musical string, of given length

and weight, stretched by a given weight, to find its vibrations’. The equation was derived by d’Alembert in

Recherches sur la courbe que forme une corde tendue mise en vibration, 1747, by considering the string to be

composed of infinitesimal masses and applying Newton’s force law to each element of mass. Euler published his

own solution in Sur la vibration des cordesin 1750, and Daniel Bernoulli explored the idea of the superposition of

normal modes in his Réflexions et éclaircissementsin 1755. The debate over the kinds of functions acceptable

as solutions of the wave equation was continued for about thirty years by these three, without any one of them

being convinced by the others. The problem was resolved through the work of Fourier, Dirichlet, Riemann, and

Weierstrass over the next hundred years, and involved a reconsideration of the meaning of function, continuity,

and convergence.