The nikolaevskiy model for nonlinear seismic waves

Mathl. Camput. Mod&kg Vol. 22, No. 3, pp. 81-82, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/95-$9.50 + 0.00

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The Nikolaevskiy Model for Nonlinear Seismic Waves G. ADOMIAN General Analytics Corporation 155 Clyde Road Athens, GA 30605, U.S.A. (Received December 1994; accepted January Abstract-The seismic waves.

decomposition method [l] is applied to the Nikolaevskiy model [2,3] for nonlinear

Keywords-Nonlinear

The model equation

where

u is particle

nonlinearity,

seismic waves, Decomposition method, Adomian polynomials.

for nonlinear

velocity,

seismic waves [2,3] proposed

x is a moving

space coordinate,

by V.N. Nikolaevskiy

n is a dimensionless

is given as

coefficient

of

and o, fi, r, 6, E are constants dv 8% a?J -~__n~-_~--~~_~--_-~6----~~~ ax2 dt &r

Define Lt = & and L,r(.)

v = v(0) - Lr’nv

Decompose

-

a

ax

3%

d%

d5v

8%

dzs

3x4

8x5

3x6

= s,“(.) dt. The nonlinear

the A, are Adomian’s polynomials generated we have v - v(0). Rearranging, we have

are given

1995)

term ‘u &V is written

for this specific nonlinearity.

a3

a2

as Cr?e Applying

a4

as v(0).

Then

components

of v for n = 1,2,3,.

by 12= -L,%zA,_~

21

- L,‘a

&

v,_~

V,_..l - L,%

a6 a5 G ?I,_1 - L,l& V,_l. 8x6

The m-term

approximant

to v will be

81

to Lp

a6

v - L;l a~v-L;la1173v-L~~7aZ41U-L;ls~21-L;~E~v.

v into Cc=‘=, 21, with vo identified

A, where Ltl

..

G. ADOMIAN

82

The A, are [I]

A2 =vo -

a

3X

A3 = vo 2

a

212 + 01 T& Vl + v2 z ~3 + VI g

a

v2 + u2 $

110,

v1 + v3 ;

vo,

Thus

vo = V(O), VI = -L;‘nAo

- L;‘a

a2

ds2 vo - L;l@ -

63

623

v.

a” 86 a4 - L;lr d24 vo - L,% - vo - LL% vo, dX6

dX5

v2

=

-L,'nAl

We can now form the approximant

a3

a2

- L&x a22 v1 - L,‘j3 m

pm which converges

quite rapidly

VI

and uniformly

to v as shown

in [I].

REFERENCES of Physics: The ~ecom~o~t~on method, Kluwer Acad. Publ., (1994). 1. G. Adomiau, Soiting Elmtier P~o~l~ 2. LA. Beresnev and V.N. Nikolaevskiy, A model for nonlinear seismic waves, Physic D. 66, 1-6 (1993). 3. V.N. Nikotsevskiy, Dynamics of viscoelastic media with internal oscillators, Lecture Notes in Eragineering, Vol. 39, Springer-Verlag, Berlin, (1989).