In this paper, new symmetry reductions and similarity solutions for Burgers equation with moving boundary are obtained by means of Lie’s method of infinitesimal transformation groups, for a linearly moving boundary as well as a parabolically moving boundary. By using discrete symmetries, new analytical solutions for the problem under consideration are presented, for two cases of the moving boundary: one moving with constant velocity and another one rapidly oscillating.

1.
P. L.
Sachdev
,
Nonlinear Diffusive Waves
(
Cambridge University Press
,
1987
).
2.
G. B.
Whitham
,
Linear and Nonlinear Waves
,
Pure and Applied Mathematics
(
John Wiley & Sons, Inc
,
1974
).
3.
F.
Calogero
and
S. d.
Lillo
, “
The Burgers equation on the semi-infinite and finite intervals
,”
Nonlinearity
2
,
37
43
(
1989
).
4.
F.
Calogero
and
S. D.
Lillo
, “
Burgers equation on the semiline
,”
Inverse Probl.
5
,
L37
L40
(
1989
).
5.
K. T.
Joseph
, “
Burgers’ equation in the quarter plane, a formula for the weak limit
,”
Commun. Pure Appl. Math
41
,
133
149
(
1988
).
6.
M. J.
Ablowitz
and
S.
de Lillo
, “
Forced and semiline solutions of the Burgers equation
,”
Phys. Lett. A
156
,
483
487
(
1991
).
7.
M. J.
Ablowitz
and
S.
de Lillo
, “
Forced initial boundary value problems for Burgers equation
,” in
Solitons and Chaos
,
Research Reports in Physics
, edited by
I.
Antoniou
and
F. J.
Lambert
(
Springer-Verlag
,
1991
), pp.
292
297
.
8.
F.
Calogero
and
S.
de Lillo
, “
The Burgers equation on the semiline with general boundary conditions at the origin
,”
J. Math. Phys.
32
,
99
105
(
1991
).
9.
K. T.
Joseph
and
P. L.
Sachdev
, “
Exact analysis of Burgers equation on semiline with flux condition at the origin
,”
Int. J. Non-Linear Mech.
28
,
627
639
(
1993
).
10.
G.
Biondini
and
S.
de Lillo
, “
Semiline solutions of the Burgers equation with time dependent flux at the origin
,”
Phys. Lett. A
220
,
201
204
(
1996
).
11.
D. O.
Besong
, “
A new transformation of Burger’s equation for an exact solution in a bounded region necessary for certain boundary conditions
,”
Appl. Math. Comput.
215
,
3455
3460
(
2010
).
12.
S. I.
Zaki
, “
Solitary waves of the Korteweg–de Vries–Burgers’ equation
,”
Comput. Phys. Commun.
126
,
207
218
(
2000
).
13.
I.
Dağ
,
B.
Saka
, and
A.
Boz
, “
B-spline Galerkin methods for numerical solutions of the Burgers’ equation
,”
Appl. Math. Comput.
166
,
506
522
(
2005
).
14.
S.-S.
Xie
,
S.
Heo
,
S.
Kim
,
G.
Woo
, and
S.
Yi
, “
Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function
,”
J. Comput. Appl. Math.
214
,
417
434
(
2008
).
15.
X. H.
Zhang
,
J.
Ouyang
, and
L.
Zhang
, “
Element-free characteristic Galerkin method for Burgers’ equation
,”
Eng. Anal. Boundary Elem.
33
,
356
362
(
2009
).
16.
J.
Crank
,
Free and Moving Boundary Problems
(
Oxford Science Publications, The Clarendon Press, Oxford University Press
,
1984
).
17.
A.
Friedman
,
Partial Differential Equations of Parabolic Type
(
Prentice-Hall, Inc
,
1964
).
18.
A.
Friedman
,
Variational Principles and Free-Boundary Problems
,
Pure and Applied Mathematics
(
John Wiley & Sons, Inc
,
1982
).
19.
C. M.
Elliott
and
J. R.
Ockendon
, “
Weak and variational methods for moving boundary problems
,” in
Research Notes in Mathematics
(
Pitman
,
1982
), Vol.
59
.
20.
M. E.
Hubbard
,
M. J.
Baines
, and
P. K.
Jimack
, “
Consistent dirichlet boundary conditions for numerical solution of moving boundary problems
,”
Appl. Numer. Math.
59
,
1337
1353
(
2009
).
21.
L.
Vrankar
,
E. J.
Kansa
,
L.
Ling
,
F.
Runovc
, and
G.
Turk
, “
Moving-boundary problems solved by adaptive radial basis functions
,”
Comput. Fluids
39
,
1480
1490
(
2010
).
22.
S. D.
Lillo
, “
Moving boundary problems for the Burgers equation
,”
Inverse Probl.
14
,
L1
(
1998
).
23.
G.
Biondini
and
S.
de Lillo
, “
On the Burgers equation with moving boundary
,”
Phys. Lett. A
279
,
194
206
(
2001
).
24.
L. I.
Rubenšteĭn
,
The Stefan Problem
,
Translations of Mathematical Monographs
(
American Mathematical Society
,
1971
), Vol.
27
, translated from the Russian by A. D. Solomon.
25.
M. J.
Ablowitz
and
S.
de Lillo
, “
Solutions of a Burgers-Stefan problem
,”
Phys. Lett. A
271
,
273
276
(
2000
).
26.
M. J.
Ablowitz
and
S. D.
Lillo
, “
On a Burgers-Stefan problem
,”
Nonlinearity
13
,
471
478
(
2000
).
27.
L. V.
Ovsiannikov
, in
Group Analysis of Differential Equations
, edited by
W. F.
Ames
(
Academic Press, Inc
,
1982
), translated from the Russian by Y. Chapovsky. Translation.
28.
P. J.
Olver
,
Applications of Lie Groups to Differential Equations
,
Graduate Texts in Mathematics
(
Springer-Verlag
,
1986
), Vol.
107
.
29.
G. W.
Bluman
and
S.
Kumei
,
Symmetries and Differential Equations
,
Applied Mathematical Sciences
(
Springer-Verlag
,
1989
), Vol.
81
.
30.
C.
Rogers
and
W. F.
Ames
, “
Nonlinear boundary value problems in science and engineering
,” in
Mathematics in Science and Engineering
(
Academic Press, Inc
,
1989
), Vol.
183
.
31.
F.
Oliveri
, “
Lie symmetries of differential equations: Classical results and recent contributions
,”
Symmetry
2
,
658
706
(
2010
).
32.
F. S.
Ibrahim
,
M. A.
Mansour
, and
M. A. A.
Hamad
, “
Lie-group analysis of radiative and magnetic field effects on free convection and mass transfer flow past a semi-infinite vertical flat plate
,”
Electron. J. Differential Equations
2005
,
39
.
33.
H.
Azad
and
M. T.
Mustafa
, “
Symmetry analysis of wave equation on sphere
,”
J. Math. Anal. Appl.
333
,
1180
1188
(
2007
).
34.
M.
Nadjafikhah
and
R.
Bakhshandeh-Chamazkoti
, “
Symmetry group classification for general Burgers’ equation
,”
Commun. Nonlinear Sci. Numer. Simul.
15
,
2303
2310
(
2010
).
35.
T.
Ozis
and
I.
Aslan
, “
Similarity solutions to Burgers’ equation in terms of special functions of mathematical physics
,”
Acta Phys. Pol., B
48
,
1349
1369
(
2017
).
36.
P. E.
Hydon
,
Symmetry Methods for Differential Equations. A Beginner’s Guide
,
Cambridge Texts in Applied Mathematics
(
Cambridge University Press
,
2000
).
37.
J. D.
Crawford
,
M.
Golubitsky
,
M. G. M.
Gomes
,
E.
Knobloch
, and
I. N.
Stewart
, “
Boundary conditions as symmetry constraints
,” in
Singularity Theory and its Applications, Part II
,
Lecture Notes in Mathematics
(
Springer
,
1991
), Vol.
1463
, pp.
63
79
.
38.
G.
Gaeta
and
M. A.
Rodriguez
, “
Determining discrete symmetries of differential equations
,”
Il Nuovo Cimento B
111
,
879
891
(
1996
).
39.
G.
Gaeta
and
M. A.
Rodriguez
, “
Discrete symmetries of differential equations
,”
J. Phys. A: Math. Gen.
29
,
859
880
(
1996
).
40.
G. J.
Reid
,
D. T.
Weih
, and
A. D.
Wittkopf
, “
A point symmetry group of a differential equation which cannot be found using infinitesimal methods
,” in
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics
(
Kluwer Academic Publishers
,
1993
), pp.
311
316
.
41.
P. E.
Hydon
, “
How to construct the discrete symmetries of partial differential equations
,”
Eur. J. Appl. Math.
11
,
515
527
(
2000
).
42.
H.
Yang
,
Y.
Shi
,
B.
Yin
, and
H.
Dong
, “
Discrete symmetries analysis and exact solutions of the inviscid Burgers equation
,”
Discrete Dyn. Nat. Soc.
2012
,
908975
.
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