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Miscellaneous Exercises 211

30.Prove the following identity:

√^1

4 πkt

∫a

b

exp

[

−(ξ−x)

2

4 kt

]

dξ=^12

[

erf

(

b√−x

4 kt

)

−erf

(

a√−x

4 kt

)]

.

31.In Exercise 6 of Section 10, it was shown that the function

w(x,t;ω)=e−pxsin(ωt−px),

wherep=

√

ω/ 2 k, satisfies the heat equation and also the boundary con-

dition

w( 0 ,t;ω)=sin(ωt).

Show how to choose the coefficientB(ω)so that the function

u(x,t)=

∫∞

0

B(ω)e−pxsin(ωt−px)dω

satisfies the boundary condition

u( 0 ,t)=f(t), 0 <t

for a suitable functiont.

32.Use the idea of Exercise 31 to find a solution of

∂^2 u

∂x^2 =

1

k

∂u

∂t,^0 <x,^0 <t,

u( 0 ,t)=h(t), 0 <t,

where

h(t)=

{

1 , 0 <t<T,

0 , T<t.

33.S.E. Serrano and T.E. Unny develop probabilistic mathematical models
for groundwater flow under uncertain conditions [Predicting groundwa-
terflowinaphreaticaquifer,Journal of Hydrology, 95 (1987): 241–268],
and compare the results to measurements. One of the models uses this
nonlinear Boussinesq equation,

S∂∂yt−∂∂x

(

Kh∂∂yx

)

=I+φ, 0 <x<L, 0 <t,

together with the conditions

y( 0 ,t)=y 1 (t), y(L,t)=y 2 (t), 0 <t,

y(x, 0 )=y 0 (x), 0 <x<L.