6.5 Comments and References 389
- a. Solve
∂^2 u
∂x^2=∂u
∂t, 0 <x< 1 , 0 <t,u( 0 ,t)= 0 , u( 1 ,t)= 1 −e−at, 0 <t,
u(x, 0 )= 0 , 0 <x< 1.b.Examine the special case wherea=n^2 π^2 for some integern.6.5 Comments and References
The real development of the Laplace transform began in the late nineteenth
century, when engineer Oliver Heaviside invented a powerful, but unjustified,
symbolic method for studying the ordinary and partial differential equations
of mathematical physics. By the 1920s, Heaviside’s method had been legitima-
tized and recast as the Laplace transform that we now use. Later generalizations
are Schwartz’s theory of distributions (1940s) and Mikusinski’s operational
calculus (1950s). The former seems to be the more general. Both theories give
an interpretation ofF(s)=1, which is not the Laplace transform of any func-
tion, in the sense we use.
There are a number of other transforms, under the names of Fourier, Mellin,
Hänkel, and others, similar in intent to the Laplace transform, in which some
other function replacese−stin the defining integral.Operational Mathematics,
by Churchill, has more information about the applications of transforms. Ex-
tensive tables of transforms will be found inTables of Integral Transformsby
Erdelyi et al. (See the Bibliography.)
Miscellaneous Exercises
1.Solve the heat conduction problem∂^2 u
∂x^2−γ^2 (u−T)=∂u
∂t, 0 <x< 1 , 0 <t,∂u
∂x( 0 ,t)= 0 , ∂u
∂x( 1 ,t)= 0 , 0 <t,u(x, 0 )=T 0 , 0 <x< 1.