Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 249

where

k 1 = hf ( r i ,y i ),

k 2 = hf r i +

h
2 ,y i +

k 1
2

,

hf r i

(3.282)

h
2 ,y i

k 2
2

k 3 = + + ,

k 4 = hf ( r i + h ,y i + k 3 ).

Since k 1 = hf i ,Taylorexpansionintwovariablesgives

hf r i +

h
2 ,y i +

k 1
2

k 2 =

48 D 3 f r = r i + O h 5 .

hf +

h 2
2 Df +

h 3 h 4

8 D 2 f +

= (3.283)

This expression for k 2 can be used to develop the expansion,
f r i

⎩

h 2

h
2 ,y i

k 2
2

h

2 ∂ f

k 2

2 ∂ f

1
2! 2 ∂

k 2

2 ∂

+ + = f ( r ,y) +

∂ r + ∂y + ∂ r +

f

∂y

⎭ r = r i ,

h 3

1
3! 2 ∂

k 2

2 ∂

+

∂ r +

+···

f (3.284)

∂y

required for the calculation of k 3 . Thus,

h 2 h h

1
2! 2 ∂

k 2

2 ∂

1
2 2 ∂

k 2

2 ∂ 2 ∂ f

k 2

2 ∂ f ∂ r +

f =

∂ r + ∂ r +

∂y ∂y ∂y

h 2 ∂

2 f
∂ r 2 +

2 f
∂ r ∂y +

2 f
∂y

1
8

2 hk 2 ∂ k 2 ∂ =

2

1
8

h 2 ∂

2 h hf

2 Df ∂

2 f
∂ r 2 +

h 2

2 f
∂ r ∂y

= +

r = r i +

h 2 f 2

h 3 fDf ∂

2 f
∂y

O h 4 . + + (3.285)

2

Then,

r = r i + h 2 h 2

8 Df ∂

h 3

2 f
∂ r ∂y +

2 f
∂y

O h 4

1
2! 2 ∂

k 2

2 ∂

f ∂

8 D 2 f ∂ r +

f = +

2

∂y

r = r i + h 2

h 3
8 DfDf y

O h 4 ,

8 D 2 f

= + (3.286)