symmetry analysis of generalized burger's equation
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Symmetry Analysis of GeneralizedBurgers Equation
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Chapter I
Introduction
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1.1 Introduction
The Partial differential equations (PDEs) arising in many physical fields like the
condense matter physics, fluid Mechanics, physics and optics, etc. which exhibit a
rich variety of non linear phenomena. When the inhomogenetics of media and non-
uniformity of boundaries are taken into account in various physical situations, the
variable coefficient PDEs often can provide more powerful and realistic molds than
their constant- coefficient counterparts in describing a large variety of real phenomena.
It is known that to find exact solutions of the PDEs is always one of the central themes
of Mathematics and Physics. In the last few decades, remarkable progress has been
made in understanding the integrability and non-integrability of nonlinear partial
differential equations.
The motivation for the present thesis comes from the work on the GBE with
variable viscosity
ut + uux =(t)
2uxx, (1.1)
by Doyle and Englefield (1990). They arrived at (1.1) by considering an application
of the Burgers equation
ut + uux =
2uxx, (1.2)
in the formation and decay of nonplane shock waves. In (1.2) x is a coordinate moving
with the wave at the speed of sound and u is the velocity fluctuations. The coefficient
of uxx, which allows for viscosity, is approximated by a constant in (1.2), but is
in reality a function of the time t (Lighthill (1956)), say, (t). Doyle and Englefield
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(1990) by using the method for defining an optimal system of group-invariant solutions
(Olver (1986)) showed that (t) satisfies the first order equation
=
c1 + c2c3t2 + (c1 c2)t + c4 , (1.3)
and obtained the following five distinct expressions for (t) by setting certain of the
constants ci equal to zero:
1(t) = et, (1.4)
2(t) = e1/t, (1.5)
3(t) = (t + )r, (1.6)
4(t) =
t +
t +
r, = 0 = , (1.7)
5(t) = exp2tan1(t + )
. (1.8)
Sachdev, Nair and Tikekar (1988) through u = (1+ t)(1n)/2
(1+n)/2(1n)
f(), =
1/2(1 + t)(1n)/2x reduced another GBE
ut + u(1n)/(1+n)ux =
2(1 + t)nuxx, (1.9)
to
f 2f(1n)/(1+n)f + (n + 1)(f + f) = 0, (1.10)
for which an exact solution in terms of an error function is obtained in a manner
described below: Equation (1.10) is integrated once under the conditions f and f
vanish as to give
f (1 + n)f2/(1+n) + (1 + n)f = 0, (1.11)
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which is once again integrated to yield the solution f = e(1+n)2/2[h()](1+n)/(1n),
where h() = [f(0)](1n)/(1+n) 2(1 n)erf[1n2
]. Doyle and Engelfield (1990)
observed that equation (1.11) being the Bernoullis equation can be converted to a
linear equation and therefore solved in terms of an error function. The forms (1.4)-
(1.8) included those of Scott (1981a) and Sachdev, Nair and Tikekar (1988) as special
cases.
Mayil Vaganan (1994) employed the direct method to derive the similarity reduc-
tions of the Burgers equation as well as its generalizations
ut + uux +ju
2t= uxx, (1.12)
ut + u2ux +
ju
2t= uxx, (1.13)
ut + u2ux = uxx, (1.14)
ut + uux + f(x, t) = g(t)uxx, (1.15)
ut + uux + f(t)u
= g(t)uxx. (1.16)
The Burgers equation (1.2) is introduced by Batman (1915). Equation (1.2) is
in fact a special case of mathematical models of turbulance (Burgers (1939, 1940)).
A similarity solution in the form u(x, t) = t1/2S(z), z(x, t) = x/
2t, where S(z)
satisfies a Riccati equation is obtained by Burgers (1950). Hopf (1950) and Cole
(1951) linearized the Burgers equation (1.2) through the Cole-Hopf transformation
u = x/ to the heat equation t = (/2)xx.
Now we describe some important works on the GBEs: Lardner and Arya (1980)
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obtained matched asymptotic solutions of the Burgers equation with linear damping
ut + uux + u =
2
uxx, (1.17)
under the constraint that the shock is thin. They have also considered an extended
form of (1.17) in which the coefficient ofux is p0u + q0u2 + r0c(t), where p0, q0 and r0
are constants. The very presence ofu in (1.17) is due to the fact that the equation of
motion includes a small viscous damping term proportional to the velocity (Crighton
(1979)). The N-wave solutions for (1.17) are obtained by Sachdev and Joseph (1994).
Asymptotic solutions to the modified Burgers equation
ut + u2ux = uxx, (1.18)
with two initial disturbances, namely, N-wave and sinusoid, for small values of the
dissipation coefficient are derived by Lee-Bapty and Crighton (1987).
Lighthill (1956) and Leibovich and Seebass (1974) derived the nonplanar Burgers
equation
ut + uux +ju
2t=
2uxx, (1.19)
where 0 < < 1 are constants and j = 0, 1, 2 to describe the propagation of weakly
nonlinear longitudinal waves in gases or liquids from a non planar source. Sachdev,
Joseph and Nair (1994) have given an exact N-wave solutions for (1.19).
The GBE describing the progressive waves with cylindrical symmetry j = 1 or
spherical symmetry j = 2 is
vx + vv j2x
v = v. (1.20)
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The existence of an exact solution of the form x1/2v(x, ) = (x/2) to (1.20) is first
noted by Chong and Sirovich (1973) and subsequently Rudenko and Soluyan (1977)
provided the closed form expression for . Sinai (1976) found the the profile of the
axisymmetric body which gives rise to a similarity solution in the steady supersonic
flow problem governed by an equation of the form (1.20).
Grundy, Sachdev and Dawson (1994) obtained large time solutions of an initial
value problem (IVP) for a GBE:
ut + (u+1)x +
ju
2t= uxx, u(x, 1) = u1(x). (1.21)
Sachdev and his collobarators (1986, 1987, 1988) by inserting the self-similar form
u = tpf(), = x/
t into the GBEs
ut + uux + u
=
2uxx, (1.22)
ut + uux +
ju
2t=
2uxx, (1.23)
ut + uux =
2g(t)uxx, (1.24)
obtained equations for f() in the form
f a1fqf + a2f + a3fr = 0, (1.25)
which is transformed through H() = fq to an equation of the form
HH + aH2
+ f()HH + g()H2 + bH + c = 0, (1.26)
which they called the Euler-Painleve transcendent (EPT). Here f() and g() are
arbitrary functions and a,b,c are real constants. Equation (1.26) extends the class
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of nonlinear ordinary differential equations (ODEs) studied by Euler and Painleve
(Kamke (1943)) for which b = c = 0. Equation (1.26) with b = c = 0 is exactly
linearizable to
v + f v + (a + 1)gv = 0, (1.27)
through H = v1
a+1 . It has been established that for the Burgers equation and its
generalizations b = 0 and therefore (1.26) is, in general, not linearizable.
Recently Rao, Sachdev and Mythily Ramaswamy (2001, 2002, 2003) obtained the
self-similar reductions of the GBEs (1.22), (1.23) and investigated them in detail
for positive single-hump, monotonic (bounded or unbounded) solutions and also for
solutions with a finite zero by assuming certain asymptotic conditions at .
The following GBE
ut + unux +
j2t +
u +
+ x
un+1 = 2 uxx, (1.28)
has been studied for N-wave solutions by Sachdev, Joseph and Mayil Vaganan (1996)
and for similarity solutions by Mayil Vaganan and Asokan (2003).
We now list below the GBEs which are studied in this project:
ut + f(t)uux + l(t)u + g(t)uxx = 0, (1.29)
ut + unux =
(t)
2uxx, (1.30)
ut + unux(
j
2t+ )u + (+
xun+1) =
(t)
2uxt, (1.31)
where f(t), g(t) and l(t) are variable coefficients, n is a positive integer.
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The structure of the thesis is arranged as follows. In chapter 2, the Painleve
test is extended to equation (1.29) in order to obtain the constraints on the variable
coefficients for it to possess the Painleve property. In chapter 3, auto-Backlund trans-
formation is presented via the truncated Painleve expansion, also analytic solutions
are obtained. In chapter 4 the generalized burgers equation is linearized to the heat
equation under the generalized Cole-Hopf transformation.
References
1. Bianchi, L. Lezioni sulla teoria dei gruppi continui finiti di trasformazioni. Pisa:
Spoerri, 1918.
2. Bluman, G. W. and Cole, J. D. The general similarity solution of the heat
equation, J. Math. Mech. 18: 1025 - 1042 (1969).
3. Bluman, G. W. and Cole, J. D. Similarity methods for Differential Equations.
Applied Mathematical Sciences No. 13, Springer-Verlag, New York, 1974.
4. Bluman, G. W. and Kumei, S. Symmetries and Differential Equations. Springer-
Verlag, New York, 1989.
5. Bluman, G. W. and Anco, S. C. Symmetries and Integration methods for Dif-
ferential Equations, Applied Mathematical Sciences, No. 154, Spriner-Verlag,
New York, 2002.
6. Burgers, J. M. Mathematical examples illustrating relations occuring in the
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theory of turbulent for motion, Trans. Roy. Neth. Acad. Sci. 17: 1 - 53
(1939).
7. Burgers, J. M. Application of a model system to illustrate some points of the
statistical theory of free turbulence, Proc. Roy. Neth. Acad. Sci. 43: 2 - 12
(1940).
8. Burgers, J. M. The formation of vortex sheets in a simplified type of turbulent
motion, Proc. Roy. Neth. Acad. Sci. 53: 122 - 133 (1950).
9. Chong, T. H. and Sirovich, L. Nonlinear effects in steady supersonic dissipative
gasdynamics - II. Three dimensional axisymmetric flow, J. Fluid Mech. 58: 53
- 63 (1973).
10. Chowdhury, A. R. and Naskar, M. J. Phys. A: Math. Gen. 19: 1775 - 1781
(1986).
11. Cole, J. D. On a quasi-linear parabolic equation occurring in aerodynamics,
Quart. Appl. Math. 9: 225 - 236 (1951).
12. Doyle, J. and Englefield, J. Similarity solutions of a generalized Burgers equa-
tion, IMA J. Appl. Math. 44: 145 - 153 (1990).
13. Grundy, R. E., Sachdev, P. L. and Dawson, C. N. Large time solution of an
initial value problem for a generalised Burgers equation, Nonlinear Diffusion
Phenomenon.(Eds: P. L. Sachdev and R. E. Grundy) pp.68-83. Narosa Pub-
lishing House, New Delhi, 1994.
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14. Gungor, F. Symmetries and invariant solutions of the two-dimensional variable
coefficient Burgers equation, J. Phys. A: Math. Gen. 34: 4313 - 4321 (2001).
15. Hopf, E. The partial differential equation ut + uux = uxx, Commun. Pure Appl.
Math. 3: 201 - 230 (1950).
16. Ibragimov, N. H. CRC Hand Book of Analysis of Differential Equations, Exact
Solutions and Conservation Laws, CRC Press, New York, 1994.
17. Kalyani, R. Some solutions of Burgers equation, J. Maths. Phys. Sci. 5: 109 -
120 (1971).
18. Kamke, E. Differential gleichungen: Losungsmethoden and Losungen, Akademis-
che Verlagagesellschaft, Leipzig, 1943.
19. Kuznetsov, V. P. Soviet Phys. Acoust. 16: 467 - 470 (1971).
20. Lee-Bapty, I. P. and Crighton, D. G. Nonlinear wave motion governed by the
modified Burgers equation, Phil. Trans. R. Soc. Lond. A 323: 173 - 209
(1987).
21. Lighthill, M. J. Viscosity effects in sound waves of finite amplitude. Surveys
in Mechanics (Eds. G. K. Batchelor and R. M. Davies) Cambridge University
Press, pp. 250 - 351 (1956). .
22. Mayil Vaganan, B. Exact Analytic Solutions for some classes of partial differ-
ential equations, Ph. D. Thesis, Indian Institute of Science, Banglore, India,
1994.
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23. Mayil Vaganan, B. and Asokan, R. Direct Similarity Analysis of Generalized
Burgers Equations and Perturbation Solutions of Euler-Painleve Transcendents,
Stud. Appl. Math. 111(4): 435 - 451 (2003).
24. Olver, P. J. Symmetry groups and group invariant solutions of partial differential
equations, J. Diff. Geom. 14: 497 - 542 (1979).
25. Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate
Text in Mathematics No. 107, Springer-Verlag, New York, 1986.
26. Rudenko, O. V. and Soluyan, S. I. Theoretical foundations of nonlinear acoustics
(English translation by R. T. Beyer), New York, Consultants Bureau (Plenum),
1977.
27. Schwarz, F. J. Phys. A: Math. Gen. 20: 1613 - 1614 (1987).
28. Sinai, Y. L. Similarity solution of the axisymmetric Burgers equation, Phys.
Fluids, 19: 1059 - 1060 (1976).
29. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Analysis of self
similar solutions of a generalized Burgers equation with nonlinear damping,
Math. Problems Eng. 7: 253 - 282 (2001).
30. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Analysis of the self
similar solutions of the non-planar Burgers equation, Nonlinear Analysis 51:
1447 - 1472 (2002).
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31. Srinivasa Rao, Ch., Sachdev, P. L. and Mythily Ramasamy Self-similar solutions
of a generalized Burgers equation with nonlinear damping, Nonlinear Analysis:
Real World Applications 4: 723 - 741 (2003).
32. Steven Nerney, Edward J. Schmahl and Musielak, Z. E. Analytic solutions of
the vector Burgers equation, Quart. Appl. Math. 1: 63 - 71 (1996).
33. Tajiri, M., Kawamoto, S. and Thushima, K. Reduction of Burgerrs equation to
Riccati equation, Math. Japon. 28: 125 - 133 (1983).
34. Taylor, G. I. The conditions necessary for discontinuous motion in gases, Proc.
R. Soc. Lond. A 84: 371 - 377 (1910).
35. Webb, G. M. and Zank, G. P. Painleve analysis of the two dimensional Burgers
equation, J. Phys. A: Math. Gen. 23: 5465 - 5477 (1990).
36. Zabolotskaya, E. A. and Kholohlov, R. V. Sov. Phys. Acoust. 15: 35 - 40
(1969).
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Chapter II
Painleve Analysis of Generalized Burg-ers Equation
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2.1 Introduction
Nowadays, a wide class of nonlinear evolution equations (NLEEs) has been derived
to describe a variety of nonlinear wave phenomena in different physical contests,
including nonlinear optics, hydrodynamics, condensed matter, plasma physics and
quantum field theory. The study of integrable models continues to attract much
attention from many mathematicians and physicists.
The painleve property for ordinary differential equations is defined as follows.
The solutions of a sytem of ordinary differential equations are regarded as (analytic)
functions of a complex (time) variable. The movable singularities of the solution
(as a function of complex t) whose location depends on the initial conditions and
are, hence movable. Fixed singularities occur at points where the coefficients of the
equation are singular. The system is said to possess the Painleve property when all
the movable singularities are single valued.
Experience has shown that, when a system possess the Painleve property, the
system will be Integrable. Albowitz have proven that when a partial differential
equation is solvable by the inverse scattering transform and a system of ordinary
differential equation is obtained from this pde by an exact similarity reduction then
the solution associated with the Gelfand-Levitan-Marchenko equation will possess
the Painleve property. Furthermore, they conjecture that, when all the odes obtained
by exact similarity transforms from a given pde have the Painleve property, perhaps
after a change of variables,then the pde will be integrable. In another context
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Chudnovsky has observed that a certain pde which arises a compatabality condition
in a generalization of the theory of Isomonodromy. Deformation is meromorphic [in
(x, t)].It is somewhat remarkable that a connection between integrability and the
Painleve property has been noted since the work of Kowalevskaya, and, as of yet, the
precise equivalence of these concepts remains to be determined.
In this chapter we define a Painleve property for partial differential equations
that does not refer to that for ordinary differential equations.Indeed, we believe that
by extending the definition of the Painleve property it is possible to treat the phe-
nomenon of integrable behaviour in a unified manner. For the past few years, several
inhomogeneous NPDEs (INPDEs) have been studied from the soliton point of view.
This chapter is devoted to study the generalized Burgers equation
ut + f(t)uux + g(t)uxxx + l(t)u = 0, (2.1)
where f(t), g(t) and l(t) are time dependent coefficients.
Equation (2.1) arises in various areas of Mathematical Physics and Nonlinear
Dynamics.
2.2 Painleve Property
One major difference between analytic functions of one complex variable and analytic
functions of several complex variables cannot be isolated. If f = f(z1,...,zn) is a
meromorphic function of N complex variable (2N real variables), the singularities of
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f occur along analytic manifolds of dimension 2N-2. These manifolds are determined
by conditions of the form
(z1,...,zn) = 0 (2.2)
where is an analytic function of in a neighbourhood of the manifold. Therefore, we
say that a pde has the Painleve property when the solutions of the pde are single-
valued about the movable, singularity manifold. To be precise, if the singulariy
manifold is determined by (2.1) and u = u(z1,...,zn) is a solution of the pde, then we
assume that
u = u(z1,...,zn) =
j=0
ujj (2.3)
where
= (z1,...,zn)
and
uj = uj(z1,...,zn)
are analytic functions of (z1,...,zn) in a neighbourhood of the manifold (2.1) , and
is an integer.Substituting (2.2) into the pde determines the possible values of and
defines the recursion relations for uj , j = 0, 1, 2,.... The process is analogous to that
for ordinary differential equations.
2.3 Painleve Analysis for (2.1)
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According to the method of Kurskal (1989), the solutions for (2.1) can be expanded
interms of the Laurent series as follows.
u(x, t) = (x, t)j=0
j(x, t)uj , (2.4)
where (x, t) = x + (t), uj(x, t) = uj(t) are analytic functions in a neighbourhood
of (x, t) = 0, the non characteristic movable singularity manifold defined by u0 = 0
and is a positive integer.
Through the leading order analysis, we found that = 1 and
u0 = 2g(t)
f(t)x. (2.5)
The recursion relations are found to be
uj2,t + (j 2)uj1t + fj
m=0
ujm [um1,x + (m 1)xum] + g(t) [uj2,xx
+2(j 2)uj1,xx + (j 2)xxuj1 + (j 1)(j 2)uj2
x
+ l(t)uj2 = 0(2.6)
Collecting terms involving uj, it is found that
2x(j 2)(j + 1)uj = F(uj1,...,u0, t, x, xx,...)forj = 0, 1, 2,... (2.7)
We note that the recursion relations (2.7) are not defined when j = 1, 2. These
values of j are called the resonances of the recursion relation and correspond to
points where arbitrary functions of (x, t) are introduced into the expansion. j = 1
corresponds to the arbitrary singularity manifold ( = 0).On the other hand the
resonance at j=2 introduces an arbitrary function u2 and a compatability condition
on the functions (, u0, u1) that requires the right hand side of (2.7) vanish identically.
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For Burgers equation,we find from (2.6)
j = 0 u0 = 2g(t)
f(t)
x (2.8)
j = 1 u1 = 1f x
(t + gxx) (2.9)
j = 2 u0,t + f(t)(uu0)x + g(t)u0,xx + l(t)u0 = 0 (2.10)
Taking account of (2.8), (2.9) and (2.10)(which represents a compatibility condition
since it involves already determined quantities) reduces to
gft + l
g
f = 0 (2.11)
from which it follows that equation (2.1) possesses the Painleve property if and only
if
g(t) = c1f(t)e
l(t)dt. (2.12)
2.4 Conclusions
The variable-coefficient nonlinear evolution equations, although their coefficient func-
tions often make the studies hard, are of current interests since they are able to
describe the real situations in many fields of physical and engineering sciences. In
this chapter we have considered a generalized Burgers equation containing arbitrary
functions f(t), g(t) and l(t). The Painleve analysis leads to the explicit constraint
on the variable coefficient on the variable coefficients for such a equation to pass the
Painleve test.
We have applied directly the PDE Painleve test to equation (2.1) and provided
constrain (2.12) which is the necessary and sufficient condition for equation (2.1)
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to have the Painleve property. Thus, equation (2.1) is predicted to be completely
integrable if and only if the variable coefficients of equation (2.1) satisfy constraint
(2.12).
References
1. J. Weiss, M. Tabor, G. Carnevale, The Painleve property for partial differential
equations J. Math. Phys. 24 (1983) 522-526.
2. J. Weiss, The Painleve property for partial differential equations, J. Math. Phys.
24 (1983) 1405-1413.
3. P. A. Clarkson, Painleve analysis and the complete integrability of a generalized
variable-coefficient Kadomtsev-Petviashvili equation, IMA J. Appl. Math. 44
(1990) 27-53
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Chapter III
Auto-Backlund transformation andExact Solutions of generalized Burg-ers equation
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3.1 Introduction
It has been shown that for the generalized Burgers equation of the form
ut + uux + g(t)uxx. (3.1)
which is a well established nonlinear equation for the study of acoustic and shock
waves. Backlund transformations exist only if g(t) is constant when (3.1) reduces to
the ordinary Burgers equation. The equation has not been solved for non-constant
a(t). However, Doyle and Englefield(1990) have found that similarity solutions of
(3.1) for specific g(t), (i.e). et or e1/t for which is infinitesimal invariant transfoma-
tions exist. Later, Kington and Sophocleous(1991) have shown that in addition to the
finite point transformation, a reciprocal point transformation as well as transforma-
tions relating equations with different functions g(t) exist. In this work, we further
generalize the equation (3.1)is of the form
ut + f(t)uux + g(t)uxx + l(t)u = 0. (3.2)
The generalized Burgers equation with the nonlinear f(t), damping term l(t) and
dispersion term a(t) can model propagation of a long shock-wave in a two layer
shallow liquid. Malomed and Shira(1991) have qualitatively demonstrated that for
the special case of the GBE with g(t) = -1 a shock wave solution reverses its velocity
and disintegrate after the passage of the critical point where f(t) changes its sign.
However, any analytic solitary wave solutions for the GBE have not been found. In
the following, we make use of both the truncated Painleve expansion and symbolic
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computation method to obtain an auto-Backlund transformation and certain soliton-
typed explicit solutions with some constraints between f(t) and g(t).
A non-linear pde is said to possess the Painleve property when the solutions of
the NPDE are single valued about the movable singularity manifold which is non-
characteristic. To be more precise, if the singularity manifold is determined by
(z1, z2,...,zn) = 0. (3.3)
and u = u(z1, z2,...,zn) is a solution of the NPD, then it is required that
u = j=0
ujj (3.4)
where u0 = 0, = (z1, z2,...,zn), uj = uj(z1, z2,...,zn) are analytic function of (zj)
in a neighbourhood of the manifold and is a negative rational number. Substitution
of equation(3.4) into the NPDE determines the allowed values of , and defines the
recursion relation for uj, j=0,1,2,...When the equation (3.4) is correct, the NPDE is
said to possess the Painleve property and is conjectured to be integrable.
3.2 Auto-Backlund Transformation
In order to find the solitonic solutions for equation (3.2), we truncate the Painleve
expansion (3.4) at the constant-level term in the senses of Tian and Gao(1995) and
Hong and Jung(1999)
u(x, t) = j(x, t)jl=0
ut(x, t)t(x, t). (3.5)
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On balancing the highest-order contributions from the linear term (uxx) with the
highest-order contributions from the nonlinear terms (uux), we get J = 1, so that
u(x, t) =u0(x, t) + u1(x, t)(x, t)
(x, t)(3.6)
We will stay with the general assumption then x = 0 but will not initially impose
any constraint on the model parameter g(t) and f(t). When substituting the above
expression into (3.2), we let the coefficients of like powers of to vanish so as to get
the set of Painleve- Backlund equations,
3 : f(t)u20x + 2g(t)u02x = 0 (3.7)
2 : f(t)u0u0,x f(t)u1u0x 2g(t)xu0, x g(t)u0xx u0t = 0. (3.8)
1 : u0,t + b(t)u0u1,x + f(t)u1u0,x + g(t)u0,xx + lu0 = 0. (3.9)
0 : u1,t + f(t)u1u1,x + g(t)u1,xx + iu1 = 0 (3.10)
The set of equations (3.6)-(3.10)contitutes an auto-backlund transformation, if
the set is solvable with respect to (x, t), u0(x, t) and u1(x, t). Equation (3.7) brings
out two solutions:
u0(x, t) =
2g(t)(x)
f(t) or u0(x, t) = 0 (3.11)
After substituting the non trivial solution into (3.8), we obtain
u1(x, t) = g(t)xx + txf(t)
. (3.12)
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Subsequently, we find the constraint equation for variable coefficients g(t) and f(t)
from equation (3.9)
g(t) = c1f(t)e
l(t)dt (3.13)
Where c1 is an arbitrary constant.Thus, we are able to find a family of exact analytic
solutions to Eq.(3.2) as follows.
u(x, t) = g(t)xx + txf(t)
+
2g(t)x
f(t)
(x, t)1 (3.14)
with the constraint equation for (x, t) and g(t)
22xg(t)txx + 2xg(t)tt 4txxg(t)2xx 2txxg(t)t
ddt
g(t)2xt 4g(t)3xxxxxx + 3g(t)33xx
+4g(t)22xxt 2txg(t)2xxx
+g(t)xx2
t + g(t)3
2
xxxx + g(t)l2
xt = 0 (3.15)
we note that once a Backlund transformation discovered, and a set of seed solutions
is given, one will be able to find an infinite number of solutions by the repeated
applications of the transformation, (i.e.) to generate a heirarchy of solutions with
increasing complexity. In the rest, we will find a family of exact analytic solutions to
(3.2).
Sample solution:
(x, t) = 1 + e[A(t)x+B(t)] (3.16)
Is subtituted into the constraint Eq.(3.15). Equating to zero the coefficients of
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like powers of x yield
x1 :ddt
A(t) ddt
a(t)
a(t) d2
dt2A(t) + 2
( ddt
A(t))2
A(t) lA(t) = 0 (3.17)
x0 :ddt
B(t) ddt
a(t)
a(t) d
2
dt2B(t) + 2
ddt
B(t) ddt
A(t)
A(t) lB (t) = 0 (3.18)
By setting A(t) = e1t a a trial function to Eq.(3.17),we find an ordinary differ-
ential equation for g(t) as
d
dta(t)1 + g(t)
21 = 0
g(t) = c2e
1tel(t)dt (3.19)Finally, B(t) is obtained from Eq.(3.18) as
B(t) = c4 +
c5e1te
l(t)dtdt (3.20)
where ci are all non-zero arbitrary constants. Combining all terms, we find a family
of the analytical solutions of Eq.(3.2) as
u(x, t) = e1t
c2c1
c2e
l(t)dt + 1xe
2l(t)dt + c5e
l(t)tl(t)dt
+2
c1e
l(t)dt e[1t+e
1tx+c5e1te
l(t)dt
dt+c4] 1(x, t) (3.21)
References
1. J. Doyle, M.J. Englefield, IMAJ Appl. Math. 44 (1990). 145.
2. J.G. Kingston, C. Sophocleous, Phys. Lett. A 155 (1991). 15.
3. N. Joshi, Phys. Lett. A 125 (1987). 456.
4. W.P. Hong, Y.D. Jung, Phys. Lett. A 257 (1999). 149.
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5. B.A. Malomed, V.I. Shrira, Physica D 53 (1991). 1.
6. B. Tian, Y.T. Gao, Phys. Lett. A 209 (1995). 297.
7. W.P. Hong, Nuovo Cimeto B 114 (1999). 845.
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Chapter IV
Exact Linearization of GeneralizedBurgers Equation via the General-ized Cole-Hopf Transformations
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4.1 Introduction
The search of exact solutions of nonlinear partial differential equations is of great
importance, because these equations appear in complex physics phenomena, mechan-
ics, chemistry, biology and engineering.
In this chapter we obtain new solutions for the generalized Burgers equation
ut + f(t)uux + g(t)uxx + l(t)u = 0, (3.22)
Equation (3.22) is a generalization of the well-known Burgers equation(Burgers (1974))
ut + uux = uxx. (3.23)
Owing to the assumptions of the constant coefficients and unforced turbulence, the
physical situations in which the classical Burgers equation arises tend to be highly
idealized. In practice, the generalized Burgers equation (3.22) may provide us with
more realistic models in many different physical contextslike the long-wave propaga-
tion in an inhomogeneous two-layer shallow liquid, directed polymers in a random
medium, pinning of vortex lines in superconductors etc.
Harry Bateman(1915) considered a nonlinear equation whose steady solutions
were thought to describe certain viscous flows. This equation, modeling a diffusive
nonlinear wave, is now widely known as the Burgers equation, and is given by
ut + uux = uxx. (3.24)
where is a constant measuring the viscosity of the fluid. It is nonlinear parabolic
equation, simply describing a temporal evolution where nonlinear convection and
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linear diffusions are combined, and it can be derived as a weakly nonlinear approxi-
mation to the equations of gas dynamics. Although nonlinear, Eq.(3) is very simple,
and interest in it was revived in the 1940s, when Dutch physicist Jan Burgers pro-
posed it to describe a mathematical model of turbulence in gas(Burgers(1940)). As
a model for gas dynamics, it was then studied extensively by (Burgers(1948)), Eber-
hard Hopf(1950), Julian Cole(1951), and others, in particular; after the discovery of a
coodinate transformation that maps it to the heat equation. While as a model for gas
turbulence the equation was soon rivaled by more complicated models, the linearizing
transformation just mentioned added importance to the equation as a mathematical
model, which has since been extensively studied. The limit 0 is an hyperbolic
equation, called the inviscid Burgers equation.
ut + uux = 0. (3.25)
This limiting equation is important because it provides a simple example of a con-
servation law, capturing the crucial phenomenon of shock formation. Indeed, it was
oriinally introduced as a model to describe the formation of shock waves in gas dy-
namics. A first-order partial differential equation for u(x, t) is called a conservation
law if it can be written in the form ut+(f(u))x = 0. For equation (3.24), f(u) = u2/2.
Such conservation laws may exhibit the formation of shocks, which are discontinuities
appearing in the solution after a finite time and then propagating in a regular manner.
When this phenomenon arises, an initially smooth wave becomes steeper and steeper
as time progresses, until it forms a jumb discontinuity-the shock.
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Nowadays, the Burgers equation is used as a simplified model of a kind of hydro-
dynamic turbulence(Case & Chiu(1969)), called Burgers turbulence. Burers himself
wrote a treatise on the equation now known by his name (Burgers (1974)), where
several variants are proposed to describe this particular kind of turbulence. Equa-
tion (3.23) was originally derived to describe the propagation of nonlinear waves in
dissipative media, where (> 0) is the kinematic viscosity, and u(x, t) represents the
fluid velocity field. It plays an active role in explaining two fundamental effects char-
acteristic of any turbulence: the nonlinear redistribution of energy over the spectrum
and the action of viscosity in small scales. Over the decades, the Burgers equation
has been widely used to model a large calss of physical systems in which the non-
linearity is fairly weak(quadratic) and the dispersion is negligible compared to the
linear damping. Hopf (1950) and Cole(1951) independently discovered a transforma-
tion that reduces the Burgers equation (3.23) to a linear diffusion equation. First, we
write (3.23) in a form similar to a conservation law
ut +
x
1
2u2 ux
= 0. (3.26)
This may be regarded as the compatibility condition for a function ti exist, such
that
u = x, (3.27)
ux 12
u2 = t. (3.28)
We substitue the value of u from (3.27) in (3.28) to obtain
xx 12
x2 = t. (3.29)
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Next, we introduce = 2log so that
u = x = 2x
. (3.30)
This is called the Cole-Hopf transformation which, by differentiating, gives
xx = 2
x
2 2
xxandt = 2t
. (3.31)
Consequently, (3.29) reduces to the linear heat equation
t = xx. (3.32)
The relation between the Burgers and the heat equation was already mentioned in
an earlier book(Forsyth(1906), but the former had not been recognized as physically
releant; hence, the importance of this connection was seemingly not noticed at the
time. Using the transformaion of Equation (3.23), known as the Cole-Hopf trans-
formation, it is easy to solve the initial value problem for this equation. Recently,
a generalization of the Cole-Hopf transformation has been successfully used to lin-
earize the boundary value problem for the Burgers equation posed on the semiline
x > 0Calogero(1989).
Many solutions of equation (3.32) are well-known in the literature. For a more
complete exposition, we exhibit some of them in next sections.
To solve equation (3.23), we simply substitute the given solution for in (3.30).
Motivated by this idea, we intend to extend the Cole-Hopf transformaion to lin-
earize equation (3.22). This is our aim in the Second section.
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This chapter is organized as follows: In section 4.2 we successfully apply the
generalized Cole-Hopf transformation to generalized Burgers equation and we reduce
the problem of solving this equation to solve the linear heat equation. Section 4.3
is dedicated to show some exact solutions to linear heat equation by transformation
groups. At the end, we give some conclusions.
4.2 Exact Linearization of Generalized Burgers Equation
In this section we construct the generalized Cole-Hopf transformation from equa-
tion (3.23) to the standard heat equation. In order to look for soltions to generalized
Burgers equation (3.22), we apply the generalized Cole-Hopf transformation given by
u(x, t) = A(x, t)G(z, ) + B(x, t), z = (t)x + (t), = (t), (3.33)
where A(x, t) = 0, B(x, t), (t), (t) and (t) are some functions to be determinedlater. We use the above ansatz into (1) to obtain
G[AT] + GGz[f A
2] + Gzz [gA2] + G[At + f BAx + f ABx + lA]
+Gz[A(x + ) + fBA + 2gAx] + [Bt + f BBx + gAxx + gBxx + lB] = 0, (3.34)
Now we assume the following conditions
f A2
A= 1 (3.35)
gA2
A= 1 (3.36)
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At + f BAx + f ABx + lA = 0 (3.37)
A(x + ) + fBA + 2gAx = 0, (3.38)
Bt + f BBx + gAxx + gBxx + lB = 0, (3.39)
f AAx = 0 (3.40)
In view of (14)-(18), equation (13) reduces to the Burgers equation
G + GGz + Gzz = 0 (3.41)
Application of x to (17)gives
Bx =f
(3.42)
Equations (18) and (19) lead to
(
f) + f(
f)2 l
f = 0 (3.43)
(
f) + f(
(f)2) l
f = 0 (3.44)
from which we find that
(t) =c1
c0 +
f eldtdt
(3.45)
(t) =
c2f 2e
ldtdt (3.46)
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Inserting (3.46) in equations (3.33) and (3.38) we have
z = (c1
c0 +
f eldtdt
)x + c2c2
1 f e
ldtdt
[c0 +
f eldtdt]2
(3.47)
B =e
ldt[x c2c1]
c0 +
f eldtdt
(3.48)
Using (3.42) in (3.37) and solving for A(t), we get
A(t) = c3e
l(t)dt (3.49)
Now (3.35) gives = A, and its solution is (after inserting for A, )
= c3c21
e
ldtdt
[c0
f eldtdt]2
+ c4 (3.50)
Substituting all the known terms into (3.33) we find that
u = c3c1
c0 + f eldtdte
l(t)dtG(, z) +
eldt[x c2c1]
c0 + f eldtdt (3.51)
If we replace G in (3.51) by
G =z(, z)
(, z)(3.52)
We find that (3.41) and (3.51) become
= zz (3.53)
which is the linear heat equation and
u =e
ldt
[c0 +
f eldtdt]
[c1c3z(, z)
(, z)+ (x c2c1)] (3.54)
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4.3 Exact Solutions to linear heat equation
Suppose that = (x, t) is a solution to equation
t = xx (3.55)
Then = (x,t) is a solution to heat equation
t = xx (3.56)
Conversely, if = (x, t) is a solution to (3.56), then = (x, t/) is a solution to
(3.55).Thus, equation (3.55) and (3.56) are the same. We shall call Equation (3.55)
the normalized heat equation. We already obtained some solutions to equation (3.56)
in a form of travelling wave.In thi section we give solutions to this equation by uing
Lie group theory[12].
Definition : Let be a system of differential equations. A symmetry group of
the system is a local group of transformations G acting on an open subset M of
the space of independent and dependent variables for the system with the property
that whenever u = f(x) is a solution of, and whenever g.f is defined for g G,
then u = g.f(x) is also a solution of the system. (By Solution we mean any smooth
solution u = f(x) defined on any subdomain X).
Following are Symmetry groups of the heat equation (1):
G1 (x + ,t,u),
G2 (x, t + , u),
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G3 (x,t, exp(e)u),
G4 (exp()x, exp(2)t, u),
G5 (x + 2t), t , u exp(x 2t),
G6
x
1 4t ,t
1 4t , u
1 4t exp x2
1 4t
,
G (x,t,u + (x, t))
This means that if = (x, t)is a solution of the heat equation (1), so are the functions
(1) = (x , t),
(2) = (x, t ),
(3) = exp()(x, t),
(4) = (exp()x, exp(2)t) ,
(5) = exp(x + 2t)(x 2t,t),
(6) =1
1 + 4texp[
x21 + 4t
](x
1 + 4t,
t
1 + 4t),
(7) = (x, t) + x(x, t)
Where is any real number and (x, t)is any other solution to the heat equation. The
symmetry groups G3 and G thus reflect the linearity of the heat equation;we can add
solutions and multiply them by constants. The groups G1
and G2
demonstrate the
time- and space-in variance of the equation, reflecting the fact that the heat equation
has constant coefficients.The well-known scaling symmetry turns up in G4, while G5
represents a kind of Galilean boost to a moving coordinate frame. The last group G6
is a genuinely local group of transformations. Its appearance is far from obvious from
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basic physical principles, but it has the following nice consequence If we let = c be
just a constant solution, then we immediately conclude that the function
=c
1 + 4texp(
x21 + 4t
), (3.57)
is also a solution. In particular, if we set c =
/ we obtain the fundamental solution
to the heat equation at the point (x0, y0) = (0, (1/(4)))To obtain the fundamental
solution
=14t
exp(x2
4t), (3.58)
We need to translate this solution in t using the group G2 (with replaced by (1/4))
It can be shown that the most general solution obtainable from a given solution
= (x, t) by group transformations is of the form
(x, t) =1
1 + 46texp[3 5x + 6x
2 5t1 + 46t
]
(exp(4)(x 25t)
1 + 46 1, exp(24)t
1 + 46t 2) + (x, t) (3.59)
Where 1,...,6are real constants and = (x, t) an arbitrary olution to the heat
equation.
Previous considerations allow to obtain particular solutions to the heat equations
t = xx.Some of them are
w(t) = k = constant, (3.60)
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w(x) = A(x) + B, (3.61)
w(x, t) = A(x2 + 2t) + B, (3.62)
w(x, t) = A(x3 + 6tx) + B, (3.63)
w(x, t) = A(x4 + 12tx2 + 122t2) + B, (3.64)
w(x, t) = x2n +n
k=1
(2n)(2n 1)...(2n 2k + 1)(t)kx2n2kk!
(3.65)
w(x, t) = x2n+1 +n
k=1
(2n + 1)(2n)...(2n 2k + 2)(t)kx2n2k+1k!
(3.66)
w(x, t) = Aexp(2
t x) + B, (3.67)w(x, t) = A
1t
exp( x2
4t) + B, (3.68)
w(x, t) = A exp(2tcos(x + B), (3.69)
w(x, t) = A exp(2tcos(x + B) + C, (3.70)
w(X, t) = A exp(x)cos(x 22t + B) + K, (3.71)
w(x, t) = A erf(x
2t ) + B, (3.72)
Where A,B,C and are arbitrary constants, n is a positive integer,
erf(z) 2
z0
exp(2)d, (3.73)
is the error function ( probability integral).
These solutions are useful in solving generalized Burgers equation (3.22).
4.4 Conclusions
We successfully applied the generalized Cole-Hopf transformation to generalized Burg-
ers equation. As a particular case, we obtained some solutions to linear heat equation.
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We think that the application of the generalized Cole-Hopf transformation is a useful
tool in the search of solutions to other generalized burgers equation.
References
1. M. Scott, Encyclopedia of Nonlinear Science, Taylor and Francis, 2005.
2. H. Bateman, Some recent research on the motion of fluids, Monthly Weather
Review 43 (1915) 163-170.
3. J. Burgers, Application of a model system to illustrate some points of the sta-
tistical theory of free turbulence, Proceedings of the Nederlandse Akademie van
Wetenschappen 43 (1940) 2-12.
4. J. Burgers, A mathematical model illustrating the theory of turbulence, Ad-
vances in Applied Mechanics 1 (1948) 171-199.
5. E. Hopf, The partial differential equation ut uux luxx, Communications in
Pure and Applied Mathematics 3 (1950) 201-230.
6. J. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quar-
terly Journal of Applied Mathematics 9 (1951) 225-236.
7. K.M. Case, S.C. Chiu, Burgers turbulence models, Physics of Fluids 12 (1969)
1799-1808.
8. J. Burgers, The Nonlinear Diffusion Equation: Asymptotic Solutions and Sta-
tistical Problems, Reidel, Dordrecht and Boston, 1974.
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9. A.R. Forsyth, Theory of Differential Equations, Cambridge University Press,
Cambridge, 1906.
10. F. Calogero, S. De Lillo, The Burgers equation on the semiline, Inverse Problems
5 (1989) L37.
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Chapter V
SIMILARITY SOLUTIONS OF THE GENERALIZED
BURGERS EQUATION ut + unux +
+ j
2t
u +
+
x
un+1 =
(t)2 uxx
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5.1 Introduction
The generalized Burgers equation
ut + unux +
+
j
2t
u +
+
x
un+1 =
(t)
2uxx, (5.1)
where , , are non negative constants, n is a positive integer, j = 0, 1, 2 and
(t) is the variable viscosity, has been studied recently by several authors. When
n = 1, = = = j = 0 and (t) = , a constant, (5.1) becomes the well-known
Burgers equation
ut + uux =
2uxx. (5.2)
The Hopf-Cole transformation (Hopf (1950), Cole (1951)) changes (5.2) to the linear
heat equation. When n = 1, = = = 0 and (t) = , a constant, (5.1) becomes
the nonplanar Burgers equation
ut + uux +j
2tu =
2uxx. (5.3)
When n = 2, = = = j = 0 and (t) = , a constant, (5.1) becomes the
modified Burgers equation
ut + u2ux =
2uxx. (5.4)
When n = 1, = = j = 0 and (t) = , a constant, (5.1) becomes the Burgers
equation with linear damping term:
ut + uux + u =
2uxx. (5.5)
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In fact equation (5.1), with (t) = , a constant, includes as special cases the equa-
tions
Ur + 12a20
U U +jU
2r=
2a30
U, (5.6)
vr + 12c20
vv +v
r=
b
2c200v, (5.7)
ut + uux +ju
2t=
2uxx and ut + uux + u =
2uxx, (5.8)
ut + uux + u =
2uxx, (5.9)
vx +
jv
2x vv = v, (5.10)ut + uux + H(x,t,u,ux) = 0, (5.11)
ut + uux +
j
2tu =
2uxx, (5.12)
studied respectively by Scott (1981), Enflo (1985), Sachdev, Enflo, Srinivasa Rao,
Mayil Vaganan and Poonam Goyal (2003), Lardner and Arya (1980), Crighton and
Scott [4], Nimmo and Crighton (1979), and Sachdev and Nair (1987). The Burgers
equation with linear damping (5.9) is recently investigated by Mayil Vaganan and
Senthil Kumaran (2004) for its symmetries by the Lies classical method (Lie (1967),
Bluman and Kumei (1989)).
When (t) = , a constant, (5.1) becomes
ut + unux + ( +
j
2t)u + (+
x)un+1 =
2uxx. (5.13)
Sachdev, Joseph and Mayil Vaganan (1996) studied (5.13) for exact N-wave solutions.
Mayil Vaganan and Asokan (2003) obtained similarity solutions of (5.13) by the direct
method of Clarkson and Kruskal (1989).
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But the viscosity is actually a function of t (Lighthill (1956)). In this case the
Burgers equation takes the form
ut + uux =(t)
2uxx. (5.14)
Doyle and Englefield (1990) determined similarity reductions of the generalized Burg-
ers equation (GBE) (5.14) using the method for defining an optimal system of group-
invariant solutions (Olver (1995)). Mayil Vaganan and Senthil Kumaran (2003) ob-
tained invariant solutions of (5.14) by the direct method. Scott (1981) obtained the
long time asymptotics of solutions of (5.14) when (t) = t, > 0. Parker (1980)
also derived large-time asymptotics of the generalized Burgers equation
V
t+ (t)V (t)V V
x= (t)
2V
x2, (5.15)
subject to the initial condition
V(x, 0) = r(x) = A1 sin(x/l), 0 < x < l. (5.16)
Later Sachdev, Nair and Tikekar (1988) introduced the Euler-Painleve transcendents
in the form
HH + aH2
+ A(z)HH + B(z)H2 + bH + c = 0, (5.17)
as the self-similar reductions of the GBE
ut + uux =
2g(t)uxx, (5.18)
when g(t) is given by either (1 + t)n or emt.
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Mayil Vaganan and Senthil Kumaran (2004, 2005, 2005) in a series of papers
obtained similarity solutions in terms of exponential, error and Kummer functions of
the GBEs
ut + unux =
2uxx, (5.19)
ut + unux + u =
2uxx, (5.20)
ut + uux + u =
2uxx. (5.21)
This chapter gives detailed account of group-invariant solutions of the generalized
Burgers equation (5.1), recover previously obtained solutions and determine new so-
lutions and this is carried out in section 2. The results and discussions of the present
chapter are contained in section 3.
2. Similarity solutions of (5.1)
If (5.1) is invariant under a one parameter group of infinitesimal transformations
x = x + X(x,t,u) + (2),
t = t + T(x,t,u) + (2),
u = u + U(x,t,u) +
(2), (5.22)
then
u
jT
2t2 T
j
2t+
(Uu 2Xx)
j
2t+
+ut
T
Tt +
2Txx + 2Xx
+ unux
T
+ Xx
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un+1T
+
x
(Uu 2Xx)
+
x
X
x2
+un1ux [nU] + U j2t
+ + Ut
2Uxx
+un
U(n + 1)
+
x
+ Ux
+ ux
Xt
2(2Uxu Xxx)
+uxut
2Txu + 2Xu
+ unut[Tx] + unu2x[2Xu] + u2x
2(Uuu 2Xxu)
+u3x
2Xuu
+ u2xut
2Tuuuux
3Xu
j
2t+
+ un+1ux
3Xu
+
x
+uxt[Tx] + uut
Tu
j
2t+
+ un+1ut
Tu
+
x
+ uxuxt[Tu] = 0.(5.23)
We consider two cases: 1) The case Xu = Tu = Uu = 0 and 2) The general case.
1: The case Xu = Tu = Uu = 0.
Here (5.23) simplifies to
u T j
2t2
T
j
2t
+ (Uu
2Xx)
j
2t
+ +ut
T
Tt +
2Txx + 2Xx
+ unux
T
+ Xx
+un+1T
+
x
(Uu 2Xx)
+
x
X
x2
+nun1uxU +
U
j
2t+
+ Ut
2Uxx
+un
U(n + 1)
+
x
+ Ux
+ ux
Xt
2(2Uxu Xxx)
unutTx + uxtTx = 0. (5.24)
Equating the coefficients of uxt, un1ux to zero in (5.24), we get Tx = U = 0. Setting
the coefficients of u, ut, unux, u
n+1, ux in (5.24) equal to zero, we have
jT2t2
+
j
2t+
2Xx T
= 0, (5.25)
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T
Tt + 2Xx = 0, (5.26)
T
+ Xx = 0, (5.27)2Xx T
+
x
X
x2= 0, (5.28)
Xt + 2
Xxx = 0. (5.29)
We obtain from (5.26) and (5.27) that
Tt = Xx. (5.30)
Using (5.27) and (5.30) in equation (5.25) we get
jTt jt
T = 0. (5.31)
On inserting (5.27), equation (5.28) becomes
+
x
Xx
x2X = 0. (5.32)
Now we consider three cases: In the foregoing analysis c0, r1, x0, t0, f0, c1, r2, c2, A0, B0, r3, r0,
C0, D0, r4, r5, r6, r7, r8 are arbitrary constants and the relation between f and H is
f = H1/n.
Case 1.1: = = 0
Equation (5.1) with = = 0 is
ut + unux +
j
2tu +
xun+1 =
(t)
2uxx. (5.33)
In this case the solutions of (5.30)- (5.32) are
T = c0t, X = c0x. (5.34)
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The characteristic equations are
dx
c0x
=dt
c0t
=du
0
. (5.35)
Solving (5.35) we obtain
z =x
tand u = f(z). (5.36)
Substituting (5.34) in (5.27) and solving for , we get
(t) = r1t. (5.37)
Using (5.36) and (5.37) in (5.33), we thus arrive at
f +2
r1zf 2
r1fnf j
r1f 2
r1zfn+1 = 0. (5.38)
Through the transformation f = H1/n equation (5.38) is changed to
HH
n + 1
n
H
2+
2
r1zH H 2
r1H +
jn
r1H2 +
2n
r1zH = 0. (5.39)
We call (5.39) the generalized Euler-Painleve transcendent because it contains a term
2nr1z
H which is not present in the general form of the Euler-Painleve transcendent
(5.17).
Case 1.2 = = j = 0
For this case equation (5.1) is
ut + unux + u
n+1 =(t)
2uxx. (5.40)
The infinitesimals are
X = x0, T = t0 and U = 0. (5.41)
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Substituting (5.41) in (5.27) and solving for , we get
= , a constant. (5.42)
Thus the similarity form of equation (5.40) is
u = f(z), z = x x0t0
t. (5.43)
Inserting (5.43) in (5.40) and using (5.42) we find that f(z) satisfies the equation
f 2x0t0
f 2
fnf 2
fn+1 = 0. (5.44)
A solution of equation (5.44) is
f = f0e2x0z/t0, (5.45)
provided = 2x0t0
.
Under the transform f = H1/n equation (5.44) changes again to a generalized
Euler-Painleve transcendent(GEPT)
HH
n + 1
n
H
2 2x0t0
HH 2
H +2n
H = 0. (5.46)
It follows from f = H1/n and the solution (5.45) of (5.44) that the corresponding
solution of the GEPT (5.46) is
H =1
fn0e2nx0z/t0. (5.47)
The reduction of (5.33) with (t) given by (5.37) to the GEPT (5.46) facilitates the
determination of an exact closed form solution in terms of an exponential function.
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The extra term 2n
H in (5.46) (recall the EPT (5.17)) together with 2
H helps
to find an exact solution. As for (5.44) the terms f,2x0t0
f and 2
fnf,2
fn+1
separately lead to this exponential solution.
Case 1.3 = = = 0
In this case equation (5.1) takes the form
ut + unux +j
2tu =(t)
2 uxx. (5.48)
The infinitesimals are
T = c0t, X = c0x + c1, U = 0. (5.49)
Inserting (5.49) in (5.27) and solving for , we get
(t) = r2t. (5.50)
The similarity transformation of (5.48) is
u = f(z), z =c0x + c1
t. (5.51)
Putting (5.50) and (5.51) in (5.48), the latter reduces to
f +2
r2c20zf 2
r2c0fnf j
r2c20f = 0. (5.52)
Using the transformation f = H1/n, equation (5.52) becomes an EPT
HH
n + 1
n
H
2+
2
r2c20zH H 2
r2c0H +
jn
r2c20H2 = 0. (5.53)
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2. The general case
Now setting the coefficients of uxuxt, uxt in (5.23) equal to zero, we have
Tu = Tx = 0. (5.54)
Equating the coefficients of u2x, uxut in (5.23) to zero and using (5.54), we obtain
Xu = Uuu = 0. (5.55)
Equation (5.55) lead to
U = A(x, t)u + B(x, t). (5.56)
Again equating the coefficients ofux, ut and the remaining terms in (5.23) to zero, we
have
unT
+ Xx
+ nun1UXt
2(2Uxu Xxx) = 0,(5.57)
T
Tt + 2Xx = 0,(5.58)u
jT
2t2 T
j
2t+
(Uu 2Xx)
j
2t+
+ U
j
2t+
+Ut 2
Uxx + un+1
T
+
x
(Uu 2Xx)
+
x
X
x2
+un
U(n + 1)
+
x
+ Ux
= 0.(5.59)
Using (5.56) and (5.58) in (5.57) and setting the coefficients of un, un1 and u0 to
zero, we get
Tt Xx + nA = 0, (5.60)
B = 0, (5.61)
Xt Ax + 2
Xxx = 0. (5.62)
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Substituting (5.56) and (5.58) in (5.59) and equating the coefficient of un+1 and u to
zero, we have
T
+
x
(A 2Xx)
+
x
X
x2+ A(n + 1)
+
x
+ Ax = 0,(5.63)
jT2t2
T
j
2t+
+ 2Xx
j
2t+
+ At
2Axx = 0.(5.64)
Inserting (5.58) in (5.63) and (5.64), we get
Xxx + n
+
x
Xx n
x2x = 0, (5.65)
j2t +
T j2t2 T + At Axx = 0. (5.66)
Integration of equation (5.60) with respect to x gives
X = (T + nA) x + C1(t), (5.67)
where C1(t) is the function of integration. Substituting (5.67) in (5.62) and equating
the coefficients of x and x0 to zero, we get
T + nAt = 0, (5.68)
C1 + Ax = 0. (5.69)
Solving (5.68) for A, we get
A = T
n+ c2. (5.70)
Using (5.70) in (5.69), we find that
C1(t) = c1. (5.71)
In view of (5.70) and (5.71), (5.67) becomes
X = nc2x + c1. (5.72)
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On inserting (5.70), equation (5.66) becomes
T
nj
2t +
T
+
jn
2t2 T = 0. (5.73)
Equations (5.56) and (5.61) lead to
U = Au. (5.74)
We reduce (5.1) to ODEs under the cases (1) j = = 0, (2) = = = 0, (3)
= = 0, (4) j = = 0, (5) = = j = 0 and (6) = j = 0. For brevity we
suppress the details of the determination of X , T , U .
Case 2.1 j = = 0
For this case equation (5.1) is
ut + unux + u + u
n+1 = (t)2
uxx. (5.75)
Substituting j = 0 in (5.73) and solving for T, we get
T = A0ent + B0. (5.76)
On using (5.76), equation (5.70) gives
A = A0ent + c2. (5.77)
Putting = 0 in (5.65), we get
Xxx + nXx = 0. (5.78)
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Inserting (5.72) in (5.78), we get c2 = 0. Therefore (5.77) and (5.72) can be written
as
A = A0ent, X = c1. (5.79)
Using (5.79) in (5.76), we get
U = A0entu. (5.80)
Substituting (5.76) and (5.79) in (5.58), we have
= A0nent
A0ent + B0. (5.81)
The exact solution of equation (5.81) is
(t) = r3
A0ent + B0
1
(5.82)
The characteristic differential equation dxX
= dtT
= duU
takes the form
dx
c1=
dt
A0ent + B0=
du
A0entu, A0 = 0. (5.83)
The case A0 = 0 is studied in case 2.1(b).
The similarity form of (5.75) with (t) given by (5.82) namely
ut
+ unux
+ u + un+1 =r3
2 (A0ent + B0)uxx
, (5.84)
is
u =
A0ent + B0
1/n
f(z), (5.85)
z = x c1nB0
log
ent
A0ent + B0
, B0 = 0. (5.86)
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The case B0 = 0 is subsequently considered in Case 2.1(a).
Substituting (5.86) in (5.84), we get the ordinary differential equation
f +2c1r3
f 2r3
fnf 2r3
fn+1 2B0r3
f = 0. (5.87)
A solution of equation (5.87) is
f(z) = expc1
r3 1
r3
c21 + 2B0r3
z
. (5.88)
Through f = H1/n equation (5.87) is transformed again to an GEPT
HH
n + 1
n
H
2+
2c1r3
HH 2r3
H +2n
r3H+
2nB0r3
H2 = 0. (5.89)
The solution of (5.89) corresponding to (5.88) is
H(z) = exp
nc1r3 n
r3
c21 + 2B0r3
z
. (5.90)
Case 2.1(a) B0 = 0
In this case the infinitesimals are
T = A0ent, X = c1, U = A0entu. (5.91)
The viscosity is
(t) = r0ent, (5.92)
The similarity form of equation (5.75) when (t) is given by (5.92) namely
ut + unux + u + u
n+1 =r02
entuxx, (5.93)
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is
u = etf(z), z(x, t) = x +c1
A0nent. (5.94)
Here f is governed by
f 2c1r0A0
f +2
r0fnf 2
r0fn+1 = 0, (5.95)
whose solution is
f(z) = e2c1r0A0
z, (5.96)
provided that = 2c1r0A0 .
Through the transform f = H1/n equation (5.95) becomes a GEPT
HH
n + 1
n
H
2 2c1A0r0
HH +2
r0H +
2n
r0H = 0, (5.97)
with the solution
H(z) = e2nc1r0A0
z. (5.98)
Case 2.1(b) A0 = 0
The infinitesimals and (t) are
T = B0, X = c1, U = 0, (t) = r1. (5.99)
The similarity transformation is
u = f(z), z = x c1B0
t, (5.100)
where f satisfies
f 2r1
fnf +2c1
B0r1f 2
r1f
r1fn+1 = 0. (5.101)
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A solution of equation (5.101) is
f(z) = exp c1
B0r1 c21
B0r12+
2
r1 z , (5.102)
provided that =
c1
B0r1
c21B0r12
+ 2r1
.
Through the transform f = H1/n equation (5.101) becomes a GEPT
HH
n + 1
n
H
2+
2c1B0r1
H +2
r1H2 +
2
r1H = 0. (5.103)
Therefore the solution of (5.103) is
H(z) = exp
n
c1
B0r1
c21B0r12
+2
r1
z
. (5.104)
Case 2.2 = = = 0
Equation (5.1) reduces to
ut + unux +
j
2tu =
(t)
2uxx. (5.105)
The infinitesimals are
T = C0 + D0tjn2 , X = nc2x + c1, U =
c2 c0
n
D0j
2tjn21
u. (5.106)
The viscosity equation is
=
2nc2
C0 +jnD02
tjn21
C0t + D0tjn/2. (5.107)
The general solution of equation (5.107) is
(t) = r4t2nc2C0
1C0 + D0t
jn2 1
2nc2C0[ jn2 1]
1
. (5.108)
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The similarity form of the equation (5.105) with (t) is given by (5.108), viz.,
ut + unux +
j
2tu =
r4
2t2nc2C0
1 C0 + D0t jn2 1
2nc2
C0[ jn2 1]1
uxx, (5.109)
is
u = tc2C0
1n
C0 + D0t
jn21 c2
C0( jn2 1)+ 1n
f(z), (5.110)
z = (c2x + c1)C0 + D0t
jn21 nc2C0[ jn2 1] . (5.111)
For this case the reduced ODE is
f 2C0r4(n + 1)
jn
2 1
zf +
2C0r4(n + 1)
jn
2 1
fnf 2C0
r4(n + 1)
jn
2 1
f = 0.
(5.112)
The first integral of equation (5.112) with c2 = C0n(n+1)jn2 1
is
f 2C0r4(n + 1)
jn
2 1
zf +
2C0r4(n + 1)2
jn
2 1
fn+1 = 0. (5.113)
The transformation f = H1/n linearizes (5.113) to
H 2nC0r4(n + 1)
1 jn
2
zH +
2nC0r4(n + 1)2
1 jn
2
= 0. (5.114)
The general solution of equation (5.114) is
H(z) =
H(0)
nC0(2jn)
2r4(n + 1)2erf
nC0(2jn)2r4(n + 1)
z
exp
nC0(2jn)
2r4(n + 1)z2
.
(5.115)
The corresponding solution of (5.112) with c2 = C0n(n+1)jn2 1
is
f(z) =
H(0)
nC0(2jn)
2r4(n + 1)2erf
nC0(2jn)
2r4(n + 1)z
1/n
exp
C0(2jn)2r4(n + 1)
z2
.
(5.116)
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Thus an exact closed form solution of (5.109) is given by using (5.111); (5.116) in
(5.110):
u(x, t) = tc2C0
1n
C0 + D0t
jn21 c2C0( jn2 1)
+ 1n
H(0)
nC0(2jn)
2r4(n + 1)2
erf
nC0(2jn)
2r4(n + 1)(c2x + c1)
C0 + D0t
jn21 nc2C0[ jn2 1]
1/n
exp
C0(2jn)
2r4(n + 1)(c2x + c1)
2C0 + D0t
jn2 1
2nc2C0[ jn2 1]
. (5.117)
Case 2.2(a) c2 = 0
When c2 = 0, (5.106), (5.108) reduce to
T = C0t+D0tjn2 , X = c1 U =
C0n
+D0j
2tjn21
u, (t) = r41
C0t + D0tjn2
1.
(5.118)
The corresponding similarity transformation is
u = t1/nC0 + D0t
jn211/n
f(z), (5.119)
z = x c1C0
jn2 1
log tjn21
C0 + D0tjn21
, (5.120)
where f(z) is any solution of
f +2
r4fnf +
2c1
r4f
2C0
r4j
2 1
n f = 0. (5.121)
The equation (5.121) changes to an EPT
HH
n + 1
n
H
2+
2c1r4
HH 2r4
H +2C0r4
H2 = 0, (5.122)
through f = H1/n. The case n = 2, j = 1 is investigated in case 2.2(d).
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Case 2.2(b) c2 = C0 = 0
Setting c2 = C0 = 0 in (5.107) and solving we obtain
(t) = r4tjn2 . (5.123)
The infinitesimals are
T = D0tjn2 , X = c1, U = D0j
2tjn2 1u, (5.124)
give rise to the similarity transformation
u = tj/2f(z), z = x +c1
D0jn2 1
t1 jn2 . (5.125)
The reduced ODE for this case is
f 2r4
fnf +2C1
D0r4
f = 0, (5.126)
whose first integral is the Bernoullis equation
f 2r4(n + 1)
fn+1 +2C1
D0r4f = 0, (5.127)
where we have taken the constant of integration equal to zero. The transformation
f = H1/n linearizes (5.127) to
H 2nc1D0r4
H+2n
r4(n + 1)= 0. (5.128)
The general solution of (5.128) is
H(z) = H(0)e2nc1D0r4
z D0c1(n + 1)
. (5.129)
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The corresponding solution of (5.126) is
f(z) =
H(0)e
2nc1D0r4
z
D0
c1(n + 1)1/n
. (5.130)
In fact (5.129) is a solution of
HH
n + 1
n
H
2+
2c1D0r4
HH 2r4
H = 0, (5.131)
which is derived from (5.126) through f = H1/n.
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Case 2.2(c) D0 = 0
The infinitesimals and viscosity are
T = C0t, X = nc2x + c1, U =
c2 C0n
u, (t) = r4t
2nc2C0
1
. (5.132)
The similarity transformation is
u = tc2C0
1n f(z), z = (nc2x + c1) t
nc2/C0. (5.133)
The similarity function f is governed by
f 2nc2r4
fnf 2nc2C0r4
zf
j
n2c22r4+
2(nc2 C0)n3C0c22r4
f = 0. (5.134)
A first integral of (5.134) with c2 =C0(2jn)2n(n+1)
is the Bernoullis equation
f 4C0r4(2jn) f
n+1 +4(n + 1)
C20r4(2jn)zf = 0, n = 2, j = 1. (5.135)
Equation (5.135) is lenearized via f = H1/n to
H 4(n + 1)nC20r4(2jn)
zH +4n
C0r4(2jn) = 0, (5.136)
whose general solution is
H(z) =
H(0)
2n
r4(n + 1)(2jn)erf
2n(n + 1)
C20r4(2jn)z
exp
2n(n + 1)
C20r4(2jn)z2
.
(5.137)
Therefore the corresponding solution of (5.134) is
f(z) =
H(0)
2n
r4(n + 1)(2jn)erf
2n(n + 1)
C20r4(2jn)z
1/n
exp
2(n + 1)
C20r4(2jn)z2
. (5.138)
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Case 2.2(d) c2 = D0 = 0
The variable viscosity and the infinitesimals are
(t) =r4t
, T = C0t, X = c1, U = C0n
u. (5.139)
The similarity form is
u = t1/nf(z), z = x c1C0
log t, (5.140)
where f is satisfied by
f 2r4
fnf +2c1
C0r4f +
1
r4
2jn
2
f = 0. (5.141)
When n = 2, j = 1 (5.141) is integrated to (after setting the constant of integration
equal to zero)
f 2r4(n + 1) fn+1 + 2c1C0r4
f = 0, (5.142)
a Bernoulli equation and therefore linearizes through f = H1/n to
H 2nc1C0r4
H+2n
r4(n + 1)= 0. (5.143)
The general solution of (5.143) is
H(z) =
H(0)e2nc1C0r4
z + C0c1(n + 1)
, (5.144)
and the corresponding solution of equation (5.141) with n = 2, j = 1 is
f(z) =
H(0)e
2nc1C0r4
z+
C0c1(n + 1)
1/n
. (5.145)
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When n = 2, j = 1, (5.105) becomes the modified cylindrical Burgers equation with
variable viscosity
ut + u2ux + 1
2tu = r
4
2tuxx. (5.146)
Its closed form solution is written by using (5.140), (5.145):
u(x, t) = t1/n
H(0)tc1/C0e(4c1/C0r4)x +C03c1
1/2. (5.147)
Case 2.2(e) C0 = 0
The infinitesimals are
T = D0tjn2 , X = nc2x + c1, U =
c2 D0j
2tjn21
. (5.148)
The viscosity is
(t) = r4tjn2 exp 4nc2
D0(2jn)t1
jn2 , n = 2, j = 1. (5.149)
The similarity form is
u = exp
2c2
D0(2jn)t1 jn2
t
j2 f(z), (5.150)
z = (nc2x + c1)exp
2nc2D0(2jn) t
1 jn2
. (5.151)
The reduced ODE for this case is
f 2nc2r4
fnf +2
nc2D0r4zf 2
n2c2D0r4f = 0. (5.152)
Through f = H1/n equation (5.152) changes to
HH
n + 1
n
H
2+
2
nc2D0r4zH H 2
nc2r4H +
2
n2c2r4D0H2 = 0. (5.153)
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We note that for (5.105) we have found exact closed form solutions in terms of an
exponential and error functions.
Case 2.3 = = 0
In this case equation (5.1) becomes
ut + unux +
j
2tu + un+1 =
(t)
2uxx. (5.154)
The infinitesimals are
T = C0t + D0tjn2 , X = c1, U =
C0
n D0j
2tjn21
u. (5.155)
The function (t) is governed by
= C0 +
jnD02
tjn2 1
C0t + D0tjn2
. (5.156)
The general solution of equation (5.156) is
(t) =r5
C0t + D0tjn2
. (5.157)
We remark that a special case of (5.154) with = 0, namely (5.105) also admits (t)
in the form same as (5.157) (see equation (5.118) in Case 2.2(a)).
The similarity form of solutions of (5.154) is
u = t1/n
C0 + D0tjn211/n
f(z), (5.158)
z = x c1(jn2 1)C0
logtjn2 1
C0 + D0tjn2 1
, (5.159)
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where f(z) satisfies
f
2
r5 f
n
f
2c1
r5 f
+
2
r5 f
n+1
2C0
r5j
2 1
n
f = 0. (5.160)
A solution of (5.160) is
f(z) = exp
c1
r5c21
r25+
2C0r5
j
2 1
n
z , (5.161)
provided that = c1r5
c21r25
+ 2C0r5
j2 1
n
.
Through f = H1/n equation (5.160) is transformed to a GEPT
HH
n + 1
n
H
2 2c1r5
HH 2r5
H +2n
r5H+
2nC0r5
j
2 1
n
H2 = 0. (5.162)
Therefore the solution of (5.162) is
H(z) = exp
n
c1r5
c21r25
+2C0r5
j
2 1
n
z
. (5.163)
A special but physically interesting situation arises if we take j = n = 2. Under this
assumption (5.154) become the modified spherical Burgers equation with nonlinear
damping term and variable viscosity:
ut + u2ux +
1
tu + u3 =
r5t2
uxx. (5.164)
Case 2.3(a) C0 = 0
In this case the infinitesimals and (t) are
T = D0tjn2 , X = c1, U = D0j
2tjn21u, (t) = r5t
jn2 . (5.165)
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The similarity transformations are
u = t
j/2
f(z), z = x +
c1
D0jn2 1
t
1 jn2
. (5.166)
The f- equation is found to be
f 2r5
fnf +2c1
D0r5f 2
r5fn+1 = 0, (5.167)
for which we obtain the solution
f(z) = e2c1
D0r51 z, (5.168)
under the condition = 2c1D0r5
. We assume that c1/D0 < 0 to ensure that > 0.
Equation (5.167) is transformed via f = H1/n to GEPT
HH
n + 1
n
H
2+
2c1D0r5
HH 2r5
H +2n
r5H = 0, (5.169)
with the solution
H(z) = e2c1nD0r51
z. (5.170)
Case 2.3(b) D0 = 0
In this case the infinitesimal transformations are
T = C0t, X = c1, U = C0n
u. (5.171)
Inserting D0 = 0 in (5.156) and solving, we obtain
(t) =r5t
. (5.172)
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The similarity transformations are
u = t1/nf(z), z = x
c1
C0log t. (5.173)
Equations satisfied by f and H are
f 2r5
fnf +2c1
C0r5f +
1
r5
2jn
n
f 2
r5fn+1 = 0.(5.174)
HH
n + 1
n
H
2+
2c1C0r5
HH 2r5
H
2jnr5
H2 +
2n
r5H = 0. (5.175)
It is easily verify that
f(z) = exp
c1
C0r5 c21
C20r25
+jn 2
nr5
z
. (5.176)
is a solution of (5.174) provided = c1C0r5
c21C20r
25
+ jn2nr5
and therefore the solution
of (5.175) is
H(z) = exp
n
c1C0r5
c21C20r
25
+jn 2
nr5
z
. (5.177)
All the 3 reductions of (5.154) given in Case 2.3, 2.3(a), 2.3(b) are GEPTs for which
exact closed form solutions in terms of exponential function is found.
Case 2.4 = = 0
Putting = = 0, (5.1) takes the form
ut + unux +
j
2tu +
xun+1 =
(t)
2uxx. (5.178)
The infinitesimals are
T = C0t + D0tjn2 , X = nc2x, U =
c2 C0
n
D0j
2tjn21
u. (5.179)
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The viscosity equation is
=
2nc2 C0 + D0jn2
tjn21
C0t + D0t jn2 . (5.180)
Solving equation (5.180), we get
(t) = r6t2nc2C0
1
C0 + D0tjn21 2nc2
C0(jn2 1)
+1
. (5.181)
The similarity transformation is
u = t c2C0 1n
C0 + D0tjn21 c2
C0( jn2 1)+1/n
f(z), (5.182)
z = xtnc2C0
C0 + D0t
jn21 nc2C0(
jn2
1) . (5.183)
The f and H equations are
f 2r6
fnf +2nc2r6C0
zf 2r6z
fn+1 2r6
c2 C0
n+
jC02
f = 0,(5.184)
HH
n + 1n H2 +2nc2r6C0 zH H
2
r6 H
+2n
r6zH+2n
r6
c2 C0n +
jC02
H
2
= 0.(5.185)
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Case 2.4(a) C0 = 0
The infinitesimal transformations are
T = D0tjn2 , X = nc2x, U =
c2 D0j
2tjn2 1
u. (5.186)
The general solution of (5.180) with C0 = 0 is
(t) = r6exp
4nc2
D0(2jn)t1 jn2
t
jn2 . (5.187)
The form of the similarity solution of (5.178) is
u = exp
2c2
D0(2jn)t1 jn2
t
jn2 f(z),
z = xexp
nc2
D0(2jn)t1
jn2
, (5.188)
where f(z) satisfies
f 2r6
fnf +2nc2D0r6
zf 2r6z
fn+1 2c2D0r6
f = 0. (5.189)
The H equation is
HH
n + 1
n
H
2 2nc2r6D0
zH H 2r6
H 2nr6z
H+2nc2D0r6
H2 = 0. (5.190)
Case 2.5 = = j = 0
Equation (5.1) with = = j = 0 is
ut + unux +
xun+1 =
(t)
2uxx. (5.191)
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The function (t) is any solution of
=
2c2
s1
s1t + s2 . (5.192)
Solving (5.192), we get
(t) = r7 (s1t + s2)2c2s1
s1 . (5.193)
The infinitesimals are
T = s1t + s2, X = c2x, U = c2 s1
n u. (5.194)
The similarity transformations are
u = (s1t + s2)(c2s1)/(ns1) f(z), z = x (s1t + s2)
c2/s1 , (5.195)
where f(z) is satisfied by
f
2
r7 fn
f
+
2c2r7 zf
2
r7zfn+1
2
r7c2 s1s1 f = 0. (5.196)
The H equation is a GEPT :
HH
n + 1
n
H
2+
2c2r7
zH H 2r7
H +2n
r7zH+
2n
r7
c2 s1
s1
H2 = 0. (5.197)
Case 2.5(a) s1 = 0
In this case the infinitesimals, (t) and similarity transformation are
T = s2, X = c2x, U =c2n
u, (t) = r7e2c2s2
t, u = e
c2ns2
tf(z), z = xe
c2s2t.
(5.198)
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The f and H equations are
f
2
r7 f
n
f
+
2c2
s2r7 zf
2
r7zf
n+1
2c2
ns2r7 f = 0,(5.199)
HH
n + 1
n
H
2+
2c2s2r7
zH H 2r7
H +2
r7zH+
2n
r7
c2 s1
s1
H2 = 0.(5.200)
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Case 2.6 = j = 0
Equation (5.1) when = j = 0 is of the form
ut + unux + u +
xun+1 =
(t)
2uxx. (5.201)
The infinitesimals are
T = A0ent + B0, X = nc2x, U =
A0ent + c2
u. (5.202)
The viscosity equation is
=
2nc2 A0entA0ent + B0
, (5.203)
which admits the following general solution
(t) = r8e2nc2B0
tA0e
nt + B0
2c2B0
+1
. (5.204)
The similarity transformation is
u = ec2B0
tA0e
nt + B0
c2
nB0+1/n
, z = xe
nc2B0
tA0e
nt + B0 c2B0 , (5.205)
where f and H = fn satisfy
f 2r8
fnf +2nc2
r8zf 2
r8zfn+1 2
r8(c2 + B0) f = 0.(5.206)
HH
n + 1
n
H
2+
2nc2r8
zH H 2r8
H +2n
r8zH+
2n
r8(c2 + B0) H
2 = 0.(5.207)
Case 2.6(a) B0 = 0
The infinitesimals are
T = A0ent, X = nc2x, U =
A0ent + c2
u. (5.208)
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The function (t) and the similarity transformation are
(t) = r8exp
2nc2
A0 e
nte
nt
, (5.209)
u = etexp c2
nA0ent
f(z), z = x exp
c2
A0ent
. (5.210)
Here f and H are governed by
f 2r8
fnf +2nc2A0r8
zf 2r8z
fn+1 2c2r8A0
f = 0, (5.211)
HH
n + 1
nH
2+
2nc2
A0r8
zH H
2
r8
H +2
r8z
H+2c2
r8A0
H2 = 0. (5.212)
3. Result and Discussions
We applied the Lies classical method to equation (5.1) and considered two case
Tu = Xu = Uu = 0 and the general case separately. We list below the conditions
imposed on the parameters appearing in the GBE
ut + unux + ( +
j
2t)u + (+
x)un+1 =
(t)
2uxx, (5.213)
and all possible (t)s obtained for each such condition:
1. = = j = 0
1 = .
2. j = = 0
2 = r3
A0ent + B0
1
, 3 = r0ent, 4 = r1.
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3. = = = 0
5
= r4t2nc2C0
1 C0
+ D0tjn21
2nc2
C0[ jn2 1]1
, 6
= r4C
0t + D
0tjn2 1 ,
7= r
4t
jn2 ,
8 = r4t2nc2C0
1, 9 =
r4t
, 10 = tjn2 exp
4nc2
D0(2jn)t1 jn2
, 11 = r2t.
4. = = 0
12 = r5
C0t + D0tjn2
1, 13 = r5t
jn2 , 14 =
r5t
.
5. = = 0
15 = r6t2nc2C0
1
C0 + D0tjn2 1
2nc2C0(
jn2 1)
1, 16 = r6t
jn2 exp
4nc2
D0(2jn)t1 jn2
,
18 = r1t.
6. = = j = 0
19 = r7 (s1t + s2)2c2s1
s1 , 20 = r7e2c2s2
t.
7. = j = 0
21 = r8e2nc2B0
tA0e
nt + B0
2c2B0
+1
, 23 = r8exp
2nc2
A0ent
et.
An important contribution of the present work is the generalization of the Euler-
Painleve transcendent (5.17) to a generalized EPT in the form
HH + aH2
+ A(z)HH + B(z)H2 + bH + C(z)H+ c = 0, (5.214)
(see for instance (5.39), (5.46),(5.89) (5.97)). Indeed the similarity reductions of
ut + unux +
j
2tu +
xun+1 =
r1t
2uxx,
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ut + unux +
j
2tu +
xun+1 =
r62
t2nc2C0
1
C0 + D0tjn21 2nc2
C0(jn2 1)
+1
uxx,
ut + un
ux +j
2tu +
xun+1
=r62 exp
4nc2D0(2jn) t
1 jn
2
tjn
2 uxx,
ut + unux +
j
2tu +
xun+1 =
r62
t2nc2C0
1uxx,
ut + unux + u
n+1 =
2uxx,
ut + unux + u + u
n+1 =r32
A0e
nt + B01
uxx,
ut + unux + u + u
n+1 =r02
uxx,
ut + unux + u + un+1 = r02
entuxx,
ut + unux +
j
2tu + un+1 =
r52
tjn/2uxx,
ut + unux +
j
2tu + un+1 =
r52t
uxx,
ut + unux +
xun+1 =
r72
(s1 + s2)2c2s11
uxx,
ut + unux +
xun+1 =
r72
e2c2s2
tuxx,
ut + unux + u +
x
un+1 = r8
2e2nc2B0
tA0e
nt + B0
2c2B0
+1
uxx,
ut + unux + u +
xun+1 =
r82
exp2nc2
A0ent
etuxx,
are transformed via f = H1/n to (5.214). Equation (5.214) is more general than
(5.17) in the sense that the former contains an extra term C(z)H. And when
A(z), B(z), C(z) are constants then an exact solutions of (5.214) in terms of an ex-
ponential function can be determined. It is also evident from the general form of the
f-equation viz.,
f + p(z)f + qf + rfnf + s(z)fn+1 = 0. (5.215)
The condition A,B,C are constants is equivalent to p(z), s(z) are constants. And in
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this case by splitting (5.215) into two equations f+p(z)f+qf = 0, rfnf+sfn+1 = 0
one can easily see that f-equation admits an exponential solution. We also obtain
solutions of f-equations and hence those of H-equations in terms of error functions
by linearizing the f equation when p(z) is linear in z and s(z) is a constant in (5.215).
Here we first obtain an intermediate integral of the f-equation in terms of Bernoullis
equation and is easily linearized. The linear first order equation is shown to have
exact solutions in terms of an error function.(see (5.116), (5.136)).
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