Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 512

Solutions of the homogeneous sixth-order spheroidal system can be represented by
the propagator matrix Y ( r ,ρ) (3.299), which is the solution of the homogeneous
di ff erential system

d Y ( r ,ρ )
dr =

A ( r ) · Y ( r ,ρ), (9.98)

with the initial condition

Y (ρ,ρ) = I , (9.99)

where I is the unit matrix that propagates the fundamental solutions from radius ρ
to radius r . Each column vector of Y is a linearly independent solution of
d y ( r )
dr =

A ( r ) · y ( r ). (9.100)

ff

The solution of the non-homogeneous di erential system

d y ( r )
dr =

A ( r ) · y ( r ) + g ( r ) (9.101)

is then given by

r

y ( r ) = Y ( r ,ρ)g(ρ) d ρ + Y ( r , b ) y ( b ). (9.102)

b

If g ( r ) = G δ( r − r 0 ), where G is a constant vector, the solution is

⎩

Y ( r , r 0 ) G + Y ( r , b ) y ( b ), r > r 0

y ( r ) = (9.103)

Y ( r , b ) y ( b ), r < r 0 .

G δ ( r If g ( r ) = − r 0 ), the solution is

⎩

d Y ( r , r 0 )
dr 0

− G + y ( r , b ) y ( b ), r > r 0

y ( r ) = (9.104)

Y ( r , b ) y ( b ),

r < r 0 .

Integration from r 0 to r and then back to r 0 gives

Y ( r , r 0 ) Y ( r 0 , r ) = I , (9.105)

while integration from r to r 0 and then back to r gives

Y ( r 0 , r ) Y ( r , r 0 ) = I . (9.106)

Di ff erentiating (9.105) with respect to r 0 yields

d Y ( r , r 0 )
dr 0

=− Y ( r , r 0 ) d Y ( r 0 , r ) Y ( r 0 , r )

dr 0 .

(9.107)