
211From Frictional Instabilities to Friction-Induced Self-Organization
∫
′
ψτ=γτ
′′
τ=
XXXfXdX(,)(,; ,0)( )
1
(8.16)
Comparing Equation (8.16) with Equation (8.15), one can construct the
Green’s function by replacing τ by τ – τ′:
∑
γτ
′′
τ=
′
=
∞
−πτ−
′
τ
XX em
X(,;,02 sin( )sin
m
()()
2
(8.17)
where X′ and τ′
are integration parameters. Then, the solution of the non-
homogeneous problem is given in terms of Green’s function (Mortazavi et
al. 2012):
∫∫
ψτ=
′
ψτ+
′
τγτ−
′
τ
τ=
τ
=
∈τ
XXdG
dX(,)(,) exp(,,)
G
1
(8.18)
Substituting Equation (8.17) into Equation (8.18) yields the solution of
Equation (8.14):
∑
∫
∑
ψτ=ππ
′′′
+
π−∫
π−
<
()
=
∞
−πτ
′
=
=
∞
−∫τ
XemX mX fX ...