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Perturbation method for a generalized Burgers' equation
c
CIl:&7462.92 $5 00 + .@I 1992 Pcrgamon RN plc
PERTURBATION METHOD FOR A GENERALIZED BURGERS’ EQUATION C. Department
CURRY
and A.
of Mathematics,
DONATO
University
of Messina, Italy
and A.
YA.
Institute of Physics of the Earth, Academy
POVZNER of Sciences of the U.S.S.R.,
Moscow,
Russia
(Receiced I2 &lurch 1991) Abstract--A perturbation method based on the linear theory of operators [S] for solving non-linear problems is used to find an approximate solution of a generalized Burgers’ equation governing the shock structure in a viscous thermally conducting fluid.
I.
INTRODUCTION
Non-linear wave phenomena in continuous media are usually governed by quasilinear firstorder hyperbolic system whose solution will generally break down in a finite time and a shock may occur. Taking into account dissipative mechanisms like viscosity or thermal conductivity the governing equations modify as they contain, in addition to the hyperbolic part, higher-order space derivatives of the unknown field multiplied by coefficients. If these coefficients are small, that is, a small dissipation occurs, then the higher-order derivatives play a fundamental role balancing the effects of the non-linearity and preventing the formation of shocks. This can be seen by means of a reductive perturbation method which enables us to reduce the general governing system to a single equation describing a far field of the system. For dissipative systems where the higher-order derivatives are of second order, this equation is usually the so-called generalized Burgers’ equation of the form
24 dr+q$+BG(z)$=O
(1.1)
where q(z,q) is a kind of amplitude of the asymptotic wave related to the first term in the asymptotic development, r] is related to the phase function, z is an independent variable related to the time along the characteristic rays, /I is the small parameter characterizing the dissipation and G(z) a suitable function of z depending on the geometry of the problem as well as on the dissipative mechanism. Unfortunately equation (1.1) in general is not integrable but only a similarity solution is available for a special form of G(z) [l]. However a theorem proved by Scott in [2] gives information on the qualitative behaviour of the solution when q(a,O) tends to different constants as q --) f co. In [3], the axi-symmetric motion of a viscous, thermally conducting fluid has been studied. Under the assumption that the coefficients of viscosity and thermal conductivity are small and of the same order of magnitude, an asymptotic expansion around a known similarity solution of the associated hyperbolic system has been considered and the governing system reduced to a generalized Burgers’ equation like equation (1.1) with G(z) = z’ (z = constant). Moreover, by requiring the invariance of the equation with respect to an infinitesimal group of transformations, it is shown that a similarity solution exists in the form q=
zQ(xh
XC--
,“z
(1.2)
Q being a solution of /!IQ” - 2xQ’ + QQ’ + Q = 0, Contributed
by W. F. Ames.
’ = 2.
(1.3)
150
c. cURR6
et al
The last equation is, for small values of /I, a typical non-linear problem with a small parameter and an attempt may be made to give an asymptotic solution by using the procedure outlined in [4] within the framework of the perturbation theory of linear operators used for solving non-linear problems. In order to interpret this asymptotic solution in terms of physical variables we recall the following expressions obtained in [ 11,
WQI
(1.4) giving the leading order terms in the asymptotic expansion around a known similarity solution. In equation (1.4) p is the density of the fluid, u the particle velocity and p the pressure, t and r are the time and space variables while the other constants are defined by h = 1 - I-(v + l)w, w0 -- 1
2 ‘vo = I-(v + 1) + (1 - v)
’
7
K=
wo(l - No) h ’
(kid),
Qd-)l;2
= y
(1.5) in terms of the index of the fluid I and of the similarity exponent 7. Moreover v = 1 corresponds to cylindrical symmetry and v = 2 to spherical symmetry. For the case of the variable x appearing in equation (1.2), it is given by _y =
#h * 2)/Z
h2A20
w-l’2
lfj(V4;
(1.6)
tAo+(h+2):r’
Thus x > 0 for t 2 t,, t, # 0 and r > 0. Moreover, -’ =
(1.7)
With these limitations the first- and the second-order approximations obtained for the solutions of equation (1.3) are defined for every x > 0 and they characterize the asymptotic expansion (1.4). 2. BASIC EQUATIONS:
Equation equations:
TRANSFORMATION
(1.3) may be conveniently
OF VARIABLES
written as a first-order
autonomous
system of
EQ’ = v EC”= 2x~ - QC - EQ EX’= E
(2.1)
where E = p. In matrix form equation (2.1) can be written: EU’= Au,
(2.1’)
u = (Q,v,x)’
where A is the linear operator given by A=A,+&A1=
L.~+((Z.YU-QV)~
?Q
h!]
+E
_Qd+d
[
.
dv a,]
(2.2'
i5t
Perturbation method for a generalized Burgers’ equation
A function e(x) is an invariant of A,, if &e(x) = 0 [S], so an invariant behaves like a constant with respect to AO. Moreover a function 4(x) is said to be an eigenfunction belonging to the eigenvalue i.(x) of A, if [A, - i.(x)]Qt(x) = 0, &n(x) = 0. Now the operator A, has the invariants e,, e, and eige~unction d, where e, = x, e, = 2xQ - *Q2 - c; and 4 the solution of the problem
t&=j.4
C&Q - !JQ'- e,) 8Q is the eigenfunction
of A,
corresponding
to
(2.3)
the eigenvalue
g2 - St L = -, 2
where
g1,2 = Ze, + (4e: - 2e,)“’ are also invariants of AO, so that (b=Q_
(2.4)
Q - gz
Consequently
in terms of the coordinates et, e2, #
In the case we are considering we have x > 0, consequently we must have g2 - g1 # 0 and 1. is not permitted to be zero and in general we avoid considering this singular case. By choosing as new variabfes ii = (g1,g2,#)T, the system (2.1’) may be written as &ii’ = (A, f cA,)ii
(2.6)
with A, given by equation (2.5) and
1
c? ___-
2_2Q*(#J+3-92u+w)
+
[
(92
- Sl)(d, - 1)
29242+5(92-9,M-91
292
(92
- 91Y
J
zj’
(2.7)
Consider S to be a linear operator expressed by the formal series S = &St + &%S,+ . . . . . By performing the transformation may be written as
(2.8)
of variables ii = eeSy, y = (yr,~~,y,)~ &y’= (eSAevS)y
= A*y
the system (2.6) (2.9)
where A *, by the Hausdorff femma, is A*=A+(A,S)++((A,S),S)+
. . ..
(A,S)
= AS - fA
(2.10)
and, taking into account equation (2.8), we easily get A* = A, + &MI + c2M,
+ ...
(2.11)
where M, =(&St)
+ A,
M,
=
t&S,)
+
x,
=
iC(A,A)
A,
-t
(M,,S,)I.
In the following we want to solve the so-called perturbation
problem for the operator A
CU Definition. An operator A* is called the normal form of A if there exists a formal series S such that in the development (2.11) it is possible to find for each Mi a number k so that lvi” = 0 where 1&I!*’ ( + 1) = (A,, Mu”). I = M.L, ,v.’1
To solve a perturbation problem for A is equivalent to finding its normal form A* [SJ, where S, , S,, . . . are smooth, first-order operators. As in the present case A, is diagonally
152
C. CL’RR~ et al.
reducible, the equations (A,,S,)
+ A, = M,
(A,,S,)
+ 2, = M, etc.
with Mi2’ = (A,, MI”) = (A,,( A,, M,)) = 0 may be solved and the solution is MiSt
=
Mi4
Gki(91*92)1
=
1,2.
k=
Fi(S17S2)4*
(2.12)
Once es transforming A to the normal form has been found, then y = $6 is a solution of the system: dy, cdx = cM,y, dy2 E dx =
+ &?M,y,
+ ... =
&G,,(.Y~,J~)
sM,y2 + &‘M,y, + . . . = ~G21(~1,).2)
dy, cdx = A,y,
+ &Mly3 + E’M,~~
= C4Yl.Y2)
+
. . .
+ E~G,,(Y,,Y,)
+ . . .
+ ...
+ E2F2(Y,,Y,)
+ &F,(YlVY,)
+ ~~G,,(y,,y,)
(2.13)
+ . . .lY,.
After equations (2.13) have been solved, the solution 6 to the original system is given by ti=e-Sy=y-sS,y+.s2[-SZy++SI(Sly)]+ 3.
FIRST-ORDER
. . ..
(2.14)
APPROXIXfATlOK
In order to get the first-order approximation we have to characterize the operators M, and S, such that (&S,) + A, = M,. (3.1) By a standard procedure [S] we obtain
M,g, = ~
2Y,
92 -91
M,g, =
w92
- &J,)
92 -91
104
M,4=
-___ g2 -
(3.2)
91
121nl4 - 11 S,g, = S,4 =
-s,g2
=
-
tg2 _
WI, + g2dJ2) (92 -
9d3
dx
dg, = M,g,
dx
d4 -= dx
= ~
=
244 +(s2-9d2
The differential equations corresponding -dg, = M,g,
gl)
“In]lC/ - 11 I//
I0
dlL.
to this normal form are:
%I
92 -
91
w92
-
39,)
92 -91
i.(g,,g,) +M,+=
!!?+L
E
92 -
91
The solution of the first two equations is (1 + wox)“2 - 1 91=2
W0
(3.3)
g2 = 4x - 91,
w. = constant.
(3.4)
Perturbation
153
method for a generalized Burgers’ equation
By choosing wg = 1 and further integrating gives g1 = 2[(1 +
xp2 - I]
g2 = 2[2x + 1 - (I + xp2-J 2(x + 1)
4 = &exp
I
E
x+ 1 (x $ I)“2 2 ;
[
1-
51n 1(x + 1)1/2 - 1I
I
(3.5)
so that the solution Q of equation (2.1) is
= Q2# - Qt
Q
4.
r&--l
SECOND-ORDER
In the second-order approximation
(3.6)
.
APPROXIMATION
the normal form of A to look for is the following:
A* = A, + EMU + E’M,.
So we have to determine M, and S, since M,, S1 have been already evaluated in the firstorder approximation. By using the same procedure as before, from the equation (A,,%) we
+ 2, =
M2
obtain 24Ql
M,g, = --M,g, = M,#=
-
(g2
_
g1)3
16QlfQ2 + 4Ql) (92 - Qd5
so that the differential equations corresponding
-dy, =
M,y,
dY2 =
M, y, + EM,~~ =
dx
dx
dy, dx
+ cM,y,
= ~
31
Yt -YyI
(b
(4.1)
to this normal form are:
24Y, +?Y2
-Y,13
2(2Y* - 3YI) 24~~ Y2 -Y1 - &(Y, - YI)3
i
Yz-YY,
-=;y3+M1y3+cM2y3=
2E---
10 Y2 - Yl
1
16Y,(Y2 + 4YI) (Yz - Y1)5 y3‘
(4.2)
From the first two equations in (4.2) we easily obtain Y2 = 4x - Yl
dy, dx- - z&
3&Y, + (2x - y,)3’
(4.3)
As we are interested in the approximate solution we consider y, = y(: + EJ1 = 2[(1 + x)1/2 - l] + &Y, yz = 2[2x f 1 - (I + x)r’*-J - Eyl where yy is the first-order approximation (4.4) into (4.3), we find 1 =
C(1f
G
already found (3.9,. By substitution of equation
xP2 - 11*+ 3 [(l + x)l’2 - 112 In
1 (1 + x)1/2
2I
(1 + x)“2
(1 + x)l’2 [(l + x)1/2 - l]
1 1 - (1 -5 x)X’2 - 1 (1 + x)1/2 + 2(1 + x)i’2 - 3[1 + x - (1 + x)“ZJ NLM 27:2-B
(4.4)
(G = constant).
c. CURR6 et al.
154
Further integration of (4.2), gives y,=&exp
x.+ 1 x + 1 - (l +I;)l’z] ( & [
- $G [(l + x)‘!2 - 113
- (I(1 + x)‘!’ - 11 + l}ln ol(~:)~~~ - 5Inl(l + x)112 - 11 -E
1, + (1 +
X)
-
7(1 +
39 64(1 +x)‘:* + 15 8(1 + x)t’* (1 + x)1’*
x)‘/~
In[(l
+
3
(1 + x)li* + x)1’* - I]
13 49 5 + [(l + x)“* - l] - 8[(1 + x)l’* - l]* + 8[(1 + x)“* - l-J3 17 5 + 4(1 + x) + 4(1 + x)3’* 1) . Now, by using equation (2.14) we can solve the original problem in the second-order approximation: g1 =e -cslyt = y, - &Sly, + 0(&Z) g2 = e -cSly* = y* - &S,Y, + 0(&2) 4 = e-rslyJ
= y, - &S,y, + o(E*).
So we are able to evaluate the approximate solution of the original problem by means of the relation
We conclude the paper by plotting the solution Q(x) in the first-order approximation for different values of E (see Figs l-3). A first oscillation occurs for x z 0.7 with the amplitude decreasing for small E.
100
0
,
-100
Fig. 1. E = .OOI.
Perturbation
I
method
for a generalized
Burgers’ equation
155
I
1 Fig. 2. E = .Ol.
I
100
0
-100
1 Fig. 3. E = .I. REFERENCES 1. D. G. Crighton. Basic theoretical nonlinear acoustics, in Frontiers in Physical Acou.s~ics (Edited by Societa Italionc di Fisica), pp. l-52. S. 1. F., Bologna, Italy (1986). 2. J. F. Scott Proc. R. Sot. Lord. A373.443 (1981). 3. C. Cun6 and A. Donato, Similarity solutions and asymptotic waves, in Nonlinear Wave Morion (Edited by A. Jefrrcy), pp. 41-53. Longman, New York (1989). 4. A. Ya. Povzncr, Linear methods in problems of nonlinear differential equations with a small parameter. Inr J. Non-linear Mech. 9, 279-323 (1974). 5. V. N. Bogaevsky and A. Ya. Povzner. Linear methods in nonlinear problems with a small parameter. Lecture Notes in Mothemalics 985, 431-448 (1983).
CIl:&7462.92 $5 00 + .@I 1992 Pcrgamon RN plc
PERTURBATION METHOD FOR A GENERALIZED BURGERS’ EQUATION C. Department
CURRY
and A.
of Mathematics,
DONATO
University
of Messina, Italy
and A.
YA.
Institute of Physics of the Earth, Academy
POVZNER of Sciences of the U.S.S.R.,
Moscow,
Russia
(Receiced I2 &lurch 1991) Abstract--A perturbation method based on the linear theory of operators [S] for solving non-linear problems is used to find an approximate solution of a generalized Burgers’ equation governing the shock structure in a viscous thermally conducting fluid.
I.
INTRODUCTION
Non-linear wave phenomena in continuous media are usually governed by quasilinear firstorder hyperbolic system whose solution will generally break down in a finite time and a shock may occur. Taking into account dissipative mechanisms like viscosity or thermal conductivity the governing equations modify as they contain, in addition to the hyperbolic part, higher-order space derivatives of the unknown field multiplied by coefficients. If these coefficients are small, that is, a small dissipation occurs, then the higher-order derivatives play a fundamental role balancing the effects of the non-linearity and preventing the formation of shocks. This can be seen by means of a reductive perturbation method which enables us to reduce the general governing system to a single equation describing a far field of the system. For dissipative systems where the higher-order derivatives are of second order, this equation is usually the so-called generalized Burgers’ equation of the form
24 dr+q$+BG(z)$=O
(1.1)
where q(z,q) is a kind of amplitude of the asymptotic wave related to the first term in the asymptotic development, r] is related to the phase function, z is an independent variable related to the time along the characteristic rays, /I is the small parameter characterizing the dissipation and G(z) a suitable function of z depending on the geometry of the problem as well as on the dissipative mechanism. Unfortunately equation (1.1) in general is not integrable but only a similarity solution is available for a special form of G(z) [l]. However a theorem proved by Scott in [2] gives information on the qualitative behaviour of the solution when q(a,O) tends to different constants as q --) f co. In [3], the axi-symmetric motion of a viscous, thermally conducting fluid has been studied. Under the assumption that the coefficients of viscosity and thermal conductivity are small and of the same order of magnitude, an asymptotic expansion around a known similarity solution of the associated hyperbolic system has been considered and the governing system reduced to a generalized Burgers’ equation like equation (1.1) with G(z) = z’ (z = constant). Moreover, by requiring the invariance of the equation with respect to an infinitesimal group of transformations, it is shown that a similarity solution exists in the form q=
zQ(xh
XC--
,“z
(1.2)
Q being a solution of /!IQ” - 2xQ’ + QQ’ + Q = 0, Contributed
by W. F. Ames.
’ = 2.
(1.3)
150
c. cURR6
et al
The last equation is, for small values of /I, a typical non-linear problem with a small parameter and an attempt may be made to give an asymptotic solution by using the procedure outlined in [4] within the framework of the perturbation theory of linear operators used for solving non-linear problems. In order to interpret this asymptotic solution in terms of physical variables we recall the following expressions obtained in [ 11,
WQI
(1.4) giving the leading order terms in the asymptotic expansion around a known similarity solution. In equation (1.4) p is the density of the fluid, u the particle velocity and p the pressure, t and r are the time and space variables while the other constants are defined by h = 1 - I-(v + l)w, w0 -- 1
2 ‘vo = I-(v + 1) + (1 - v)
’
7
K=
wo(l - No) h ’
(kid),
Qd-)l;2
= y
(1.5) in terms of the index of the fluid I and of the similarity exponent 7. Moreover v = 1 corresponds to cylindrical symmetry and v = 2 to spherical symmetry. For the case of the variable x appearing in equation (1.2), it is given by _y =
#h * 2)/Z
h2A20
w-l’2
lfj(V4;
(1.6)
tAo+(h+2):r’
Thus x > 0 for t 2 t,, t, # 0 and r > 0. Moreover, -’ =
(1.7)
With these limitations the first- and the second-order approximations obtained for the solutions of equation (1.3) are defined for every x > 0 and they characterize the asymptotic expansion (1.4). 2. BASIC EQUATIONS:
Equation equations:
TRANSFORMATION
(1.3) may be conveniently
OF VARIABLES
written as a first-order
autonomous
system of
EQ’ = v EC”= 2x~ - QC - EQ EX’= E
(2.1)
where E = p. In matrix form equation (2.1) can be written: EU’= Au,
(2.1’)
u = (Q,v,x)’
where A is the linear operator given by A=A,+&A1=
L.~+((Z.YU-QV)~
?Q
h!]
+E
_Qd+d
[
.
dv a,]
(2.2'
i5t
Perturbation method for a generalized Burgers’ equation
A function e(x) is an invariant of A,, if &e(x) = 0 [S], so an invariant behaves like a constant with respect to AO. Moreover a function 4(x) is said to be an eigenfunction belonging to the eigenvalue i.(x) of A, if [A, - i.(x)]Qt(x) = 0, &n(x) = 0. Now the operator A, has the invariants e,, e, and eige~unction d, where e, = x, e, = 2xQ - *Q2 - c; and 4 the solution of the problem
t&=j.4
C&Q - !JQ'- e,) 8Q is the eigenfunction
of A,
corresponding
to
(2.3)
the eigenvalue
g2 - St L = -, 2
where
g1,2 = Ze, + (4e: - 2e,)“’ are also invariants of AO, so that (b=Q_
(2.4)
Q - gz
Consequently
in terms of the coordinates et, e2, #
In the case we are considering we have x > 0, consequently we must have g2 - g1 # 0 and 1. is not permitted to be zero and in general we avoid considering this singular case. By choosing as new variabfes ii = (g1,g2,#)T, the system (2.1’) may be written as &ii’ = (A, f cA,)ii
(2.6)
with A, given by equation (2.5) and
1
c? ___-
2_2Q*(#J+3-92u+w)
+
[
(92
- Sl)(d, - 1)
29242+5(92-9,M-91
292
(92
- 91Y
J
zj’
(2.7)
Consider S to be a linear operator expressed by the formal series S = &St + &%S,+ . . . . . By performing the transformation may be written as
(2.8)
of variables ii = eeSy, y = (yr,~~,y,)~ &y’= (eSAevS)y
= A*y
the system (2.6) (2.9)
where A *, by the Hausdorff femma, is A*=A+(A,S)++((A,S),S)+
. . ..
(A,S)
= AS - fA
(2.10)
and, taking into account equation (2.8), we easily get A* = A, + &MI + c2M,
+ ...
(2.11)
where M, =(&St)
+ A,
M,
=
t&S,)
+
x,
=
iC(A,A)
A,
-t
(M,,S,)I.
In the following we want to solve the so-called perturbation
problem for the operator A
CU Definition. An operator A* is called the normal form of A if there exists a formal series S such that in the development (2.11) it is possible to find for each Mi a number k so that lvi” = 0 where 1&I!*’ ( + 1) = (A,, Mu”). I = M.L, ,v.’1
To solve a perturbation problem for A is equivalent to finding its normal form A* [SJ, where S, , S,, . . . are smooth, first-order operators. As in the present case A, is diagonally
152
C. CL’RR~ et al.
reducible, the equations (A,,S,)
+ A, = M,
(A,,S,)
+ 2, = M, etc.
with Mi2’ = (A,, MI”) = (A,,( A,, M,)) = 0 may be solved and the solution is MiSt
=
Mi4
Gki(91*92)1
=
1,2.
k=
Fi(S17S2)4*
(2.12)
Once es transforming A to the normal form has been found, then y = $6 is a solution of the system: dy, cdx = cM,y, dy2 E dx =
+ &?M,y,
+ ... =
&G,,(.Y~,J~)
sM,y2 + &‘M,y, + . . . = ~G21(~1,).2)
dy, cdx = A,y,
+ &Mly3 + E’M,~~
= C4Yl.Y2)
+
. . .
+ E~G,,(Y,,Y,)
+ . . .
+ ...
+ E2F2(Y,,Y,)
+ &F,(YlVY,)
+ ~~G,,(y,,y,)
(2.13)
+ . . .lY,.
After equations (2.13) have been solved, the solution 6 to the original system is given by ti=e-Sy=y-sS,y+.s2[-SZy++SI(Sly)]+ 3.
FIRST-ORDER
. . ..
(2.14)
APPROXIXfATlOK
In order to get the first-order approximation we have to characterize the operators M, and S, such that (&S,) + A, = M,. (3.1) By a standard procedure [S] we obtain
M,g, = ~
2Y,
92 -91
M,g, =
w92
- &J,)
92 -91
104
M,4=
-___ g2 -
(3.2)
91
121nl4 - 11 S,g, = S,4 =
-s,g2
=
-
tg2 _
WI, + g2dJ2) (92 -
9d3
dx
dg, = M,g,
dx
d4 -= dx
= ~
=
244 +(s2-9d2
The differential equations corresponding -dg, = M,g,
gl)
“In]lC/ - 11 I//
I0
dlL.
to this normal form are:
%I
92 -
91
w92
-
39,)
92 -91
i.(g,,g,) +M,+=
!!?+L
E
92 -
91
The solution of the first two equations is (1 + wox)“2 - 1 91=2
W0
(3.3)
g2 = 4x - 91,
w. = constant.
(3.4)
Perturbation
153
method for a generalized Burgers’ equation
By choosing wg = 1 and further integrating gives g1 = 2[(1 +
xp2 - I]
g2 = 2[2x + 1 - (I + xp2-J 2(x + 1)
4 = &exp
I
E
x+ 1 (x $ I)“2 2 ;
[
1-
51n 1(x + 1)1/2 - 1I
I
(3.5)
so that the solution Q of equation (2.1) is
= Q2# - Qt
Q
4.
r&--l
SECOND-ORDER
In the second-order approximation
(3.6)
.
APPROXIMATION
the normal form of A to look for is the following:
A* = A, + EMU + E’M,.
So we have to determine M, and S, since M,, S1 have been already evaluated in the firstorder approximation. By using the same procedure as before, from the equation (A,,%) we
+ 2, =
M2
obtain 24Ql
M,g, = --M,g, = M,#=
-
(g2
_
g1)3
16QlfQ2 + 4Ql) (92 - Qd5
so that the differential equations corresponding
-dy, =
M,y,
dY2 =
M, y, + EM,~~ =
dx
dx
dy, dx
+ cM,y,
= ~
31
Yt -YyI
(b
(4.1)
to this normal form are:
24Y, +?Y2
-Y,13
2(2Y* - 3YI) 24~~ Y2 -Y1 - &(Y, - YI)3
i
Yz-YY,
-=;y3+M1y3+cM2y3=
2E---
10 Y2 - Yl
1
16Y,(Y2 + 4YI) (Yz - Y1)5 y3‘
(4.2)
From the first two equations in (4.2) we easily obtain Y2 = 4x - Yl
dy, dx- - z&
3&Y, + (2x - y,)3’
(4.3)
As we are interested in the approximate solution we consider y, = y(: + EJ1 = 2[(1 + x)1/2 - l] + &Y, yz = 2[2x f 1 - (I + x)r’*-J - Eyl where yy is the first-order approximation (4.4) into (4.3), we find 1 =
C(1f
G
already found (3.9,. By substitution of equation
xP2 - 11*+ 3 [(l + x)l’2 - 112 In
1 (1 + x)1/2
2I
(1 + x)“2
(1 + x)l’2 [(l + x)1/2 - l]
1 1 - (1 -5 x)X’2 - 1 (1 + x)1/2 + 2(1 + x)i’2 - 3[1 + x - (1 + x)“ZJ NLM 27:2-B
(4.4)
(G = constant).
c. CURR6 et al.
154
Further integration of (4.2), gives y,=&exp
x.+ 1 x + 1 - (l +I;)l’z] ( & [
- $G [(l + x)‘!2 - 113
- (I(1 + x)‘!’ - 11 + l}ln ol(~:)~~~ - 5Inl(l + x)112 - 11 -E
1, + (1 +
X)
-
7(1 +
39 64(1 +x)‘:* + 15 8(1 + x)t’* (1 + x)1’*
x)‘/~
In[(l
+
3
(1 + x)li* + x)1’* - I]
13 49 5 + [(l + x)“* - l] - 8[(1 + x)l’* - l]* + 8[(1 + x)“* - l-J3 17 5 + 4(1 + x) + 4(1 + x)3’* 1) . Now, by using equation (2.14) we can solve the original problem in the second-order approximation: g1 =e -cslyt = y, - &Sly, + 0(&Z) g2 = e -cSly* = y* - &S,Y, + 0(&2) 4 = e-rslyJ
= y, - &S,y, + o(E*).
So we are able to evaluate the approximate solution of the original problem by means of the relation
We conclude the paper by plotting the solution Q(x) in the first-order approximation for different values of E (see Figs l-3). A first oscillation occurs for x z 0.7 with the amplitude decreasing for small E.
100
0
,
-100
Fig. 1. E = .OOI.
Perturbation
I
method
for a generalized
Burgers’ equation
155
I
1 Fig. 2. E = .Ol.
I
100
0
-100
1 Fig. 3. E = .I. REFERENCES 1. D. G. Crighton. Basic theoretical nonlinear acoustics, in Frontiers in Physical Acou.s~ics (Edited by Societa Italionc di Fisica), pp. l-52. S. 1. F., Bologna, Italy (1986). 2. J. F. Scott Proc. R. Sot. Lord. A373.443 (1981). 3. C. Cun6 and A. Donato, Similarity solutions and asymptotic waves, in Nonlinear Wave Morion (Edited by A. Jefrrcy), pp. 41-53. Longman, New York (1989). 4. A. Ya. Povzncr, Linear methods in problems of nonlinear differential equations with a small parameter. Inr J. Non-linear Mech. 9, 279-323 (1974). 5. V. N. Bogaevsky and A. Ya. Povzner. Linear methods in nonlinear problems with a small parameter. Lecture Notes in Mothemalics 985, 431-448 (1983).