
976 HIGHER-ORDER PAR TIAL DIFFERENTIAL EQUATIONS
Initial conditions are prescribed:
w = f
0
(x) at t = 0,
∂
t
w = f
1
(x) at t = 0.
Solution:
w(x, t) =
Z
t
0
Z
∞
−∞
E (x − y, t − τ )Φ(y, τ) dy dτ
+
∂
∂t
Z
∞
−∞
E (x − y, t)[f
0
(y) − f
′′
0
(y)] dy +
Z
∞
−∞
E (x − y, t)[f
1
(y) −f
′′
1
(y)] dy.
⊙ Literature: S. A. Gabov and A. G. Sveshnikov (1990).
4. a(x)
∂
2
w
∂t
2
−
∂
∂x
hh
b(x)
∂w
∂x
+ c(x)
∂
3
w
∂
2
t∂x
ii
= Φ(x, t).
This equation describes one-dimensional longitudinal vibrations of a rigid Rayleigh bar of
variable cross-section and takes into account the inertial effects of the transverse motion.
The coefficients of the equation are positive functions and can be expressed as follows via
geometric and