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Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates
Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
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Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates M. Rosa, M.S. Bruzón, M.L. Gandarias ⇑ Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11500 Puerto Real, Cádiz, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 August 2014 Received in revised form 4 January 2015 Accepted 19 January 2015 Available online 29 January 2015 Keywords: Partial differential equations Lie symmetries Optimal systems Exact solutions
In this paper, a generalized Fisher equation is studied from the point of view of the theory of symmetry reductions in partial differential equations. A group classification is obtained. All the reductions are derived from the optimal system of subalgebras. Some of the reduced equations admit Lie symmetries which yield to further reductions. By applying direct methods such as the simplest equation method we derive some exact wave solutions. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The Fisher equation
ut ¼ uxx þ uð1 uÞ;
0 6 uðx; tÞ 6 1;
ð1Þ
which shows the spread of an advantageous gene into a population, was proposed for population dynamics in 1930. The analysis and study of the Fisher equation is used to model heat and reaction–diffusion problems applied to mathematical biology, physics, astrophysics, chemistry, genetics, bacterial growth problems as well as to the development and growth of solid tumours [15]. For some special wave speeds the equation is shown to be of Painlevé type and the general solution for these wave speeds were found in ref [1]. Generalizations of (1) are needed to more accurately model complex diffusion and reactions effects found in many biological systems [5]. There are many models that use nonlinear dispersal to describe the tendency for diffusion to increase due to overcrowding [15]. The Fisher equation with the diffusive term generalized to yield a nonlinear diffusion equation with a reaction term
ut ¼ ðDðuÞux Þx þ uð1 uÞ;
ð2Þ
has been considered by Hayes, Murray and Witelski [15]. The Kolmogorov–Petrovskii–Piskunov equation [19] given by
ut ¼ uxx þ f ðuÞ;
ð3Þ
provides a different generalization to the Fisher equation. This equation is reduced to the well known Huxley equation for f ðuÞ ¼ u2 ð1 uÞ and has been studied by Hodgkin, Huxley and Kolmogorov. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Rosa), [email protected] (M.S. Bruzón), [email protected] (M.L. Gandarias). http://dx.doi.org/10.1016/j.cnsns.2015.01.010 1007-5704/Ó 2015 Elsevier B.V. All rights reserved.
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
75
Another important equation of this class is, for f ðuÞ ¼ uð1 u2 Þ, the Fitzhugh-Nagumo equation which arises in the study of nerve cells [10,18]. Over the last two decades a lot of attention has been paid on using Lie point symmetry methods to exploit the invariance of the generalized equation
ut ¼ ðAðuÞux Þx þ BðuÞux þ CðuÞ:
ð4Þ
A complete Lie symmetry classification for the non-linear heat Eq. (4) with B ¼ C ¼ 0 was described by Ovsiannikov in [17]. Several papers [6,7,9], were devoted to extension of Ovsiannikov’s result to Eq. (4) with B ¼ 0 and B – 0. In [8] the form-preserving transformations were applied to derive the complete Lie symmetry classification of (4). Reaction–diffusion equations arise from modeling densities of particles such as substances and organisms which disperse through space as a result of the irregular movement of every particle. Eq. (4) with BðuÞ ¼ 0, becomes the so called density dependent equation
ut ¼ f ðuÞ þ ðgðuÞux Þx ;
ð5Þ
which has been considered by J.D. Murray in [15]. Due to the great interest in getting conservation laws and conserved quantities in a recent paper [11] applying the concept of nonlinear self-adjointness and a theorem related to conservation laws due to Ibragimov [12] as well as the multiplier method due to Anco and Bluman [2] we have analyzed the functions for which Eq. (5) becomes nonlinear self-adjoint and we have derived nontrivial conservation laws for Eq. (5). There is no existing general theory for solving nonlinear partial differential equations (PDEs) and the machinery of the Lie group theory provides the systematic method to search for the special group-invariant solutions. For PDEs with two independent variables, a single group reduction transforms the PDE into ordinary differential equations (ODEs), which are generally easier to solve than the original PDE. The knowledge of the optimal system of subalgebras gives the possibility to construct the optimal system of solutions and permits the generation of new solutions starting from invariant or non-invariant solutions. The list of the disjoint classes is called the optimal system of solutions. The equation analyzed in this paper is a generalized Fisher equation in cylindrical coordinates using x for radial variable with assumption of radial symmetry, gðuÞ is the diffusion coefficient depending on the variable uand f ðuÞ an arbitrary function
1 ut ¼ f ðuÞ þ ½xgðuÞux x : x
ð6Þ
Eq. (6) has attracted a lot of attention from researchers and been analyzed in different and particular cases by other authors that we will detail below. The transient heat conduction equation with a heat source term following a power law in a rectangular, cylindrical or spherical coordinate system has been considered by Moitsheki in [14] by using Lie classical symmetries. In [3] Bohkari et al. have considered the following particular case of (6) where gðuÞ ¼ u and f ðuÞ ¼ uð1 uÞ
1 ut ¼ uð1 uÞ þ ½xuux x ; x
ð7Þ
by using Lie classical reductions the authors derived an exact solution in term of the Bessel functions. In [4] Bohkari et al. have considered Eq. (6) but only when gðuÞ is a linear function gðuÞ ¼ a1 u þ a2 . They state that a classification of (6) can only be achieved when g is linear in u. When f and g follow a power law they give an stationary solution. The aim of this paper is to study the generalized Fisher equation in cylindrical coordinates using x for radial variable (6) from the point of view of the theory of symmetry reductions in partial differential equations. The paper is organized as follows: In Section 2 we give the Lie symmetries of Eqs. (6) by means of the classical Lie direct approach. In Section 3 we report the reductions obtained from the optimal system of subalgebras. In Section 4 we analyze some exact solutions of Eq. (6). 2. Lie symmetries In this section we perform Lie symmetry analysis for Eq. (6). Let us consider a one-parameter Lie group of infinitesimal transformations in ðx; t; uÞ given by
x ¼ x þ enðx; t; uÞ þ Oðe2 Þ; t ¼ t þ esðx; t; uÞ þ Oðe2 Þ;
ð8Þ
u ¼ u þ e/ðx; t; uÞ þ Oðe2 Þ; where e is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of equation (4). This yields to the overdetermined, linear system of eleven equations for the infinitesimals nðx; t; uÞ, sðx; t; uÞ and /ðx; t; uÞ. The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form
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v¼n
@ @ @ þs þ/ : @x @t @u
ð9Þ
Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition
Un
@u @u þs / ¼ 0: @x @t
ð10Þ
From the determining system, we get that n ¼ nðx; tÞ, equations:
s ¼ sðtÞ; / ¼ /ðx; t; uÞ where n; s; /; g, and f must satisfy the following
f st x g /x x x þ f /u x þ /t x f u / x þ g /x ¼ 0; 2 g x nx þ g st x þ g u / x ¼ 0; 2 g u x2 nx g u st x2 g /u u x2 g u /u x2 g u u / x2 ¼ 0; g x2 nx x g x nx þ x2 nt g n þ 2 g u /x x2 þ 2 g /u x x2 þ g st x þ g u / x ¼ 0:
ð11Þ
From the second equation we get that as g u – 0
/¼
ð st 2 n x Þ g gu
and the third equation becomes
ðst 2 nx Þ
g gu u u ðg u Þ2
2 g ðg u u Þ2 ðg u Þ3
g þ uu gu
! ¼ 0:
In order to find a complete classification we observe that this condition will be satisfied only in the following two cases either
st 2 nx ¼ 0 or
g k1 u k2 ¼ 0: gu We can distinguish: Case 1. If n ¼ s2t x þ d with d ¼ dðtÞ. By substituting into the remaining Eqs. (6), we get d ¼ 0 and
n ¼ 0;
s ¼ k3 ; / ¼ 0:
For f ðuÞ and gðuÞ arbitrary, the only symmetry that is admitted by (6) is
v1 ¼
@ : @t
Case 2. If g is a solution of n
gðuÞ ¼ kðu þ k2 Þ ;
g gu
k1 u k2 ¼ 0. The solutions without loosing generality are: cu
gðuÞ ¼ de :
The functional forms of f and g which yield extra symmetries are given in Table 1. 3. Optimal systems and reductions In order to construct the one-dimensional optimal system, following Olver in [16], we construct the commutator tables and the adjoint tables, appearing in the appendix, which show the separate adjoint actions of each element in v i ; i ¼ 1 . . . n, as Table 1 Functions and generators. i
fi
gi
vk
1 2
um
un un
3
c2 unþ1 c1nu
un
4 5 6
c1nu c1 u
un
c2 enu cn1
u1 nu de
7
c1 n
de
8
c2 enu
de
v 1 ; v 2 ¼ ðn m þ 1Þx@ x þ 2ð1 mÞt@ t þ 2u@ u v 1 ; v 3 ¼ nt@ t u@ u v 1 ; v 4 ¼ nec t @ t c1 ec t u@ u v1 ; v2 ; v4 v 1 ; v 2 ; v 4 ; v 5 ¼ ðx log ðxÞ xÞ@ x 2 u logðxÞ@ u v 1 ; v 6 ¼ nec t @ t c1 ec t @ u v 1 ; v 6 ; v 7 ¼ nx@ x þ 2@ u v 1 ; v 8 ¼ nt@ x @ u
unþ1 nþ1
nu nu
1
1
1
1
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
it acts on all other elements. This construction is done easily by summing the Lie series. The corresponding generators of the optimal system of subalgebras are given in Table 2. Here i indicates the corresponding functions f i ; g i . In Table 3 and Table 4 respectively, similarity solutions and reductions of the Eq. (6) to ODEs are obtained using the generators of the optimal system. Here i indicates the corresponding functions f i ; g i ; j indicates the corresponding similarity reductions and k the generators of the corresponding optimal systems. For f ðuÞ and gðuÞ arbitrary functions the similarity variable and similarity solution are
z ¼ x;
u ¼ hðzÞ:
Substituting into (6) we obtain the reduced ODE
g hx þ g hx x þ f ¼ 0; x with f ¼ f ðhÞ and g ¼ gðhÞ arbitrary functions. 4. Some further reductions and exact solutions In this section we give the exact solutions for some of the reduced equations, by some method of integration or reducing the order again using Lie classical method. 4.1. Mass preserving solutions. Source and Sink solutions ODE1 Among our reduced equations we have one equation related to this type of solution. For the reduced ODE1 with nmþ1
z ¼ t2 ðm1Þ x;
u¼
hðzÞ 1
t m1
:
In fact if we choose the parameter n ¼ m 3 the solution has the form
z¼
x 1
tm1
;
u¼
hðzÞ 1
t m1
: Table 2 Optimal systems. i 1 2 3 4 5 6 7 8
v1 ; v2 v1 ; v3 v1 ; v4 v 1 þ av 2 ; av 2 þ bv 4 ; v 2 ; v 4 v 1 þ v 2 ; av 2 þ bv 4 ; v 1 þ av 5 ; v 4 þ bv 5 ; v 5 v1 ; v6 v 1 þ av 7 ; av 6 þ bv 7 ; v 6 ; v 7 v1 ; v8
Table 3 Similarity solutions. i
j
wk
1
1
v2
zj t
2
2
3
3
4
4
4 4
5 6
5
7
5
8
v3 v4 v2 v 1 þ av 2 v 2 þ bv 4 v5 v 1 þ av 5
5
9
v 4 þ bv 5
e
v6 v 1 þ av 7 av 6 þ bv 7 v8
x
6
10
7
11
7
12
8
13
uj
nmþ1 2 ðm1Þ
x
hðzÞ 1
t m1
x
hðzÞt n
x
hðzÞ e
1 c1 t n
t
2 n
ea n t x ec1 t
e t
b c1
x
hðzÞ e
c1 t
2be c
1 n
c1 t n
h x2 ðlog x1Þ2
log ðlog x1Þa t a c1 t
c1
log ðlog x1Þ b
ea n t x b ec1 t c1
e x
hðzÞx hðzÞe2 a t
hðzÞ ðx log xxÞ2 2 e
b ec1 t b z c t c1 2 b ec 1 þc1 1
hðzÞ e hðzÞ c1n t
x hðzÞ þ 2 log n c
x
t
hðzÞ 2 bce1 n1 c1n t t hðzÞ log n
t
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Table 4 Reduced ODEs. i
j
ODEj
1
1
Þz hzz ðhzhÞ hz 2ðnþm1 hn ðm1Þ
2
2
3
3
4
4
4h
4
5
a hz n z n h
4
6
hz z n bh
5 5
7 8
hz c 1 h 2 ¼ 0 2 2 3 2 h hzz ðhz Þ þ a2 h a h hz þ a2 c1 h þ 2 a2 h ¼ 0
5
9
h hzz e2 b z ðhz Þ e2 b z þ b h hz e2 b z þ e2 b h hz 2 e2 b h ¼ 0
6
10
7
11
d ðhz Þ n z þ d hzz z þ c2 z þ d hz ¼ 0 2 eh n d ðhz Þ n2 þ d hzz n z2 þ 5 d hz n z þ 4 d c1 ¼ 0 2 eh n d ðhz Þ n2 þ d hzz n z þ d hz n 2 b z ¼ 0 2 eh n d ðhz Þ n2 þ ðd hzz þ c2 Þ n z þ d hz n þ z ¼ 0
2
12
8
13
1 x u¼ 1 h 1 tm1 tm1
þ
h0 z
þ
h nþ1
þ
2
ðh0 Þ h nþ1
n
ðhz Þ n h
þ
1n
h
n
n
1n
h m1 h
mn
hz z ¼ 0
þ hz z ¼ 0
00
þ h þ c2 h ¼ 0
ðnþ1Þ n2
þ c1nh þ hz ¼ 0 1n
þ hzz h
2
þ hzz þ ðhzhÞ
n
ð2 a nþc1 Þ n
2
þ ðhzhÞ
n
þ hz z ¼ 0
1n
2 bh n þ hz z ¼ 0
2
7
Consequently the quantity
hz z
2
2
2
3
3
2
R1 1
uðx; tÞdx ¼
R1 1
hðzÞdz does not depend on t and the total mass is conserved for the solutions
For m > 1 it is clear that uðx; tÞ ! dðxÞ as t ! 0 and the similarity solution is a source solution. ODE5 An analogue situation is found in reduction ODE5 for the particular case n ¼ 2
uðx; tÞ ¼ e2at hðxe2at Þ the total mass is conserved for these solutions. Moreover we have that
If If
a < 0 uðx; tÞ ! dðxÞ as t ! 1 a > 0 uðx; tÞ ! dðxÞ as t ! þ1
so, the similarity solution is in the first case a source solution while in the second case is a sink solution. ODE3
0 2 0 h n h 00 þ þ h þ c2 h ¼ 0: z h
ð12Þ
Eq. (12) admits the symmetry generator
w¼
@ : @h
ð13Þ
Taking into account the invariants of its first prolongation and using the new variables 0
f ¼ z;
w¼
h ; h
ð14Þ
we obtain that Eq. (12) can be reduced to the first order Riccati equation
w þ wz þ n w2 þ w2 þ c2 ¼ 0: z
ð15Þ
In order to solve Eq. (15) we substitute
w¼
y0 ðn þ 1Þ y
to get
yz þ yz z þ c2 ðn þ 1Þ y ¼ 0; z which is a Bessel equation whose general solution is
y ¼ k1 J 0 ðkzÞ þ k2 Y 0 ðkzÞ:
ð16Þ
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
79
Consequently 1=nþ1
h ¼ ðk1 J 0 ðkzÞ þ k2 Y 0 ðkzÞÞ
;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ c2 ðn þ 1Þ. Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by c1 t
u ¼ e n ðk1 J 0 ðkxÞ þ k2 Y 0 ðkxÞÞ
1=nþ1
:
The asymptotic behavior of the solution can be understood from the well-known asymptotic behavior of the modified Bessel functions which is
ex I0 ðxÞ pffiffiffiffiffiffiffiffiffi 2px
ex K 0 ðxÞ ¼ qffiffiffi 2 px
for large x. This solution for the particular case n ¼ 1 has been derived in [3]. ODE4 The reduced equation
4h
nþ1
ðn þ 1Þ c1 h þ þ hz ¼ 0 n2 n
ð17Þ
can be integrated by quadrature, the solution is:
hðzÞ ¼
1 4 þ 4 n þ ec1 z k1 nc1 n ; nc1
ð18Þ
where k1 2 R. Using the substitutions given in Table 3 we obtain an exact solution of (6) in cylindrical coordinates, given by
uðx; tÞ ¼ x2=n
1 4 þ 4 n þ ec1 t k1 nc1 n ; nc1
ð19Þ
where k1 2 R. ODE5 2
hz ðhz Þ n þ þ hz z ¼ 0 h z
ð20Þ
For the reduced Eq. (20), we realize that the ODE admits the group corresponding to the generator
w¼z
@ @ þh : @z @h
ð21Þ
Taking into account the invariants of its first prolongation and using the new variables
wðfÞ ¼ hz ;
f¼
h ; z
ð22Þ
we obtain that Eq. (20) can be reduced to the first order ODE
ðw fÞwf þ w þ
nw2 ¼ 0; f
ð23Þ
whose implicit solution is:
fþ
1 wðt Þ k1 ðwðfÞÞnþ1 ¼ 0; n
where k1 2 R. Setting h ¼ y1=ðnþ1Þ into (20), we get
yz þ yz z ¼ 0; z whose solution is
y ¼ k1 log z þ k2 ; ð
1
Þ
h ¼ ðk1 log z þ k2 Þ nþ1 : From Table 3 we get
ð24Þ
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83 c1 t
ð
1
Þ
u ¼ e n ðk1 log x þ k2 Þ nþ1 : ODE8 4.2. Simplest equation method The reduced equation ODE7
2 2 3 2 h hzz ðhz Þ þ a2 h a h hz þ a2 c1 h þ 2 a2 h ¼ 0
ð25Þ
is an ODE that admits the symmetry generator
w¼
@ ; @h
ð26Þ
which does not yield any symmetry reduction. However Eq. (25) is an autonomous equation and we can apply the modified method of the simplest equation which has been developed by Kudryashov [13]. The Kudryashov version of the method contains a procedure analogous to the first step of the test for the Painleve property. In the modified method of the simplest equation [11,12], this procedure is substituted by the concept for the balance equation. Frequently this method is applied for travelling wave solutions. We apply the modified Kudryashov method [20] to Eq. (25), taking the Riccati equation as the simplest equation, we suppose that the solutions can be expressed by a polynomial in y in the form
h¼
n X
ai ðyÞi ;
ð27Þ
i¼0
where y ¼ yðzÞ satisfies the Riccati equation
yz k2 y2 k1 y k0 ¼ 0;
ð28Þ
ai ; i ¼ 0; . . . ; n; ki i ¼ 1; . . . ; 2 are constants to be determined later. Considering the homogeneous balance between highest derivative term and the highest power law term, we get n ¼ 1 we can write (27) as
h ¼ a0 þ a1 y;
a1 – 0:
ð29Þ
By substituting (29) with
yz ¼ k2 y2 þ k1 y þ k0 ; yzz ¼ 2k2 y þ k1 ; into (25), we get setting equal zero the coefficients of the different powers of y the following conditions:
a21 k2 k2 þ a2 a1 ¼ 0; 2
2 a0 k2 þ a1 k1 k2 þ 2 a2 a0 a1 k2 a a1 k2 þ a2 a21 k1 þ a2 a21 c1 ¼ 0; 3 a0 k1 k2 þ a2 a20 k2 a a0 k2 þ 2 a2 a0 a1 k1 a a1 k1 þa2 a21 k0 þ 3 a2 a0 a1 c1 þ 2 a2 a1 ¼ 0; 2 a0 k0 k2 þ
2 a0 k1
ð30Þ
a1 k0 k1
þa2 a20 k1 a a0 k1 þ 2 a2 a0 a1 k0 a a1 k0 þ 3 a2 a20 c1 þ 4 a2 a0 ¼ 0; 2
a0 a1 k0 k1 a21 k0 þ a2 a20 a1 k0 a a0 a1 k0 þ a2 a30 c1 þ 2 a2 a20 ¼ 0: By solving this system we get
a ¼ c1 ; k2 ¼ c21 a1 ; k0 ¼ a0 c1 ðaa01 c1 þ2Þ ; k1 ¼ 2 a0 c21 2 c1 : The Ricatti Eq. (28) becomes
a0 c1 ða0 c1 þ 2Þ yz þ a1 c21 y2 2 a0 c21 2 c y þ ; a1 whose solution is
y¼
ða0 c1 þ 2Þ e2 c1 z2 k c1 a0 a1 c1 e2 c1 z2 k c1 a1
ð31Þ
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
and
h¼
2 ; e2 c1 zþ2 k c1 c1
where k is the integrating constant. Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by
u¼
2 2
ðx log x xÞ ðe2 c1 k2 ðlog ðlog x1Þþc1 tÞ c1 Þ
:
For c1 < 0 and x > 0 it is a bounded solution. ODE10 The reduced equation 2
d ðhz Þ n z þ d hzz z þ c2 z þ d hz ¼ 0
ð32Þ
is an ODE that admits the symmetry generator
w¼
@ ; @h
ð33Þ 0
which does not yield any reduction however in Eq. (32) the terms in h are missing allowing the substitution h ¼ w it becomes the Riccati equation:
c w wz þ nw2 þ þ ¼ 0: d z The solution to Eq. (32) can be given in term of Bessel functions as:
hðzÞ ¼ being p ¼
2 2 2 ln d nðJ0 ðpÞk1 k2 Y0 ðpÞÞ c1 2 z ðJ1 ðpÞY0 ðpÞ Y1 ðpÞJ0 ðpÞÞ 2n
qffiffiffiffiffiffi
c2 n ; k1 ; d
;
ð34Þ
k2 2 R.
Using the corresponding substitution given in Table 3 we obtain that an exact solution of (6) is
u ¼ hðxÞ
c1 t n
where hðxÞ given by (34). ODE11 The reduced equation
eh n
2 d ðhz Þ n2 þ d hzz n z2 þ 5 d hz n z þ 4 d c1 ¼ 0;
ð35Þ
can be integrated by quadrature, the solution is:
hðzÞ ¼ ln
2 c1 z þ 4 k1 d n 4 k2 d n ln ðzÞ 1 n ; 2 4dz
ð36Þ
where k1 ; k2 2 R. Using the corresponding substitution given in Table 3 we obtain that an exact solution of (6) is
u ¼ hðzÞ þ
2 log x ; n
where z ¼ ea n t x and hðzÞ given by (36). ODE13
eh n
2 d ðhz Þ n2 þ ðd hzz þ c2 Þ n z þ d hz n þ z ¼ 0:
ð37Þ
The solution to Eq. (37) can be given in term of Bessel functions as
hðzÞ ¼ being H ¼
ln ðHÞ1 ; n
ð38Þ p
p
J0 ðpÞzY1 ðpÞþJ0 ðpÞk1 d n z zJ1 ðpÞY0 ðpÞk2 d n z Y0 ðpÞ ; zc2 nðJ1 ðpÞY0 ðpÞþY1 ðpÞJ0 ðpÞÞ
p¼
qffiffiffiffiffiffi c2 n d
and k1 ; k2 2 R.
Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by
uðx; tÞ ¼ hðxÞ
logt n
where hðxÞ is given by (38).
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
5. Conclusions In this paper we have classified the Lie symmetries of the generalized Fisher Eq. (6) in cylindrical coordinates with radial symmetry. We have studied the functions f ðuÞ and gðuÞ for which the principal Lie algebra is extended. In general, the groups that leave (6) invariant depend on several parameters, to each one-parameter subgroup there will correspond a family of group-invariant solutions. We wanted to minimize the search for group-invariant solutions in order to find non-equivalent branches of solutions, which leads to the concept of optimal systems of group-invariant solutions, from which, every other solution can be derived. To obtain the one-dimensional optimal systems of solutions, we have searched for the one-dimensional optimal systems of subalgebras. We then constructed all the invariant solutions with respect to the one-dimensional optimal system of subalgebras, as well as all the ODEs to which (6) are reduced. Finally, some of these ODEs are invariant under a symmetry group and we have reduced the order of these equations. We also find some exact solutions by using a nonlinear simple Riccati equation as well as some solutions in term of the well known Bessel equations. Some of the solutions we have found conserve the mass through time and exhibit instantaneous sources and sinks.
Acknowledgments The authors acknowledge the support of DGICYT project MTM2009–11875, FEDER and Junta de Andalucía FQM-201 group.
Appendix A See Tables A.1–A.6. Table A.1 Commutator table for Case 4. ½v i ; v j
v1
v2
v4
v1 v2 v4
0 0 c1 v 4
0 0 0
c1 v 4 0 0
Table A.2 Adjoint table for Case 4. Ad
v1 v2 v4
v1 v1 v1 v 1 þ e c1 v 4
v2 v2 v2 v2
v4 ec1 e v 4
v4 v4
Table A.3 Commutator table for Case 5. ½v i ; v j
v1
v2
v4
v5
v1 v2 v4 v5
0 0 c1 v 4 0
0 0 0 v 2
c1 v 4 0 0 0
0
v2 0 0
Table A.4 Adjoint table for Case 5. Ad
v1 v2 v4 v5
v1 v1 v1 v 1 þ e c1 v 4 v1
v2 v2 v2 v2 ee v 2
v4 ec1 e v 4
v4 v4 v4
v5 v5 v5 e v2 v5 v5
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
83
Table A.5 Commutator table for Case 7. ½v i ; v j
v1
v6
v7
v1 v6 v7
0 c1 v 6 0
c1 v 6 0 0
0 0 0
Table A.6 Adjoint table for Case 7. Ad
v1 v6 v7
v1 v1 v 1 þ e c1 v 6 v1
v6 ec1 e v 6
v6 v6
v7 v7 v7 v7
References [1] Ablowitz MJ, Zeppetella A. Explicit solutions of Fisher’s equation for a special wave speed. Bull Math Biol 1979;41:835–40. [2] Anco SC, Bluman G. Direct construction method for conservation laws for partial differential equations Part II: general treatment Euro. J Appl Math 2002;41:567–85. [3] Bokhari AH, Mustaf MT, Zaman FD. An exact solution of a quasilinear Fisher equation in cylindrical coordinates. Nonlinear Anal 2008;69:4803–5. [4] Bokhari AH, Al-Rubaee RA, Zaman FD. On a generalized Fisher equation. Commun Nonlinear Sci Numer Simulat 2011;16:2689. [5] Britton NF. Aggregation and the competitive exclusion principle. J Theor Biol 1989;136:57–66. [6] Cherniha R, Serov M. Symmetries, Anstzae and exact solutions of nonlinear second-order evolution equations with convection term. Eur J Appl Math 1998;9:527–42. [7] Cherniha R, Serov M. Symmetries, Anstzae and exact solutions of nonlinear second-order evolution equations with convection term II. Eur J Appl Math 2006;17:597–605. [8] Cherniha R, Serov M, Rassokha I. Lie symmetries and form-preserving transformations of reaction–diffusion–convection equations. J Math Anal Appl 2008;342:136–1379. [9] Dorodnitsyn VA. On invariant solutions of nonlinear heat conduction with a source. USSR Comput Math Phys 1982;22:115–22. [10] Fitzhugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1961;1:445–66. [11] Gandarias ML, Bruzón MS, Rosa M. Nonlinear self-adjointness and conservation laws for a generalized Fisher equation. Commun Nonlinear Sci Numer Simul, 1007-5704 2013;18(7):1600–6. [12] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28. [13] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations Chaos. Solitons Fractals 2005;24:1217–31. [14] Moitsheki RJ, Makinde OD. Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage. Appl Math Comput 2010;216:251–60. [15] Murray JD. Mathematical biology. New York, Berlin, Heidelberg: Springer-Verlag; 2002. [16] Olver PJ. Applications of Lie groups to differential equations. Berlin: Springer; 1986. [17] Ovsiannikov LV. Group relations of the equation of nonlinear heat conductivity. Dokl Akad Nauk SSSR 1959;125:492–5. [18] Nagumo JS, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc IRE 1962;50:2061–71. [19] Kolmogorov A, Petrovskii I, Piscounov N. Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem. In: Pelce P, editor. Dynamics of curved fronts. Boston Academic Press; 1988. [20] Vitanov NK. Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Commun Nonlinear Sci Numer Simul 2011;16:1176–85.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates M. Rosa, M.S. Bruzón, M.L. Gandarias ⇑ Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11500 Puerto Real, Cádiz, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 August 2014 Received in revised form 4 January 2015 Accepted 19 January 2015 Available online 29 January 2015 Keywords: Partial differential equations Lie symmetries Optimal systems Exact solutions
In this paper, a generalized Fisher equation is studied from the point of view of the theory of symmetry reductions in partial differential equations. A group classification is obtained. All the reductions are derived from the optimal system of subalgebras. Some of the reduced equations admit Lie symmetries which yield to further reductions. By applying direct methods such as the simplest equation method we derive some exact wave solutions. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The Fisher equation
ut ¼ uxx þ uð1 uÞ;
0 6 uðx; tÞ 6 1;
ð1Þ
which shows the spread of an advantageous gene into a population, was proposed for population dynamics in 1930. The analysis and study of the Fisher equation is used to model heat and reaction–diffusion problems applied to mathematical biology, physics, astrophysics, chemistry, genetics, bacterial growth problems as well as to the development and growth of solid tumours [15]. For some special wave speeds the equation is shown to be of Painlevé type and the general solution for these wave speeds were found in ref [1]. Generalizations of (1) are needed to more accurately model complex diffusion and reactions effects found in many biological systems [5]. There are many models that use nonlinear dispersal to describe the tendency for diffusion to increase due to overcrowding [15]. The Fisher equation with the diffusive term generalized to yield a nonlinear diffusion equation with a reaction term
ut ¼ ðDðuÞux Þx þ uð1 uÞ;
ð2Þ
has been considered by Hayes, Murray and Witelski [15]. The Kolmogorov–Petrovskii–Piskunov equation [19] given by
ut ¼ uxx þ f ðuÞ;
ð3Þ
provides a different generalization to the Fisher equation. This equation is reduced to the well known Huxley equation for f ðuÞ ¼ u2 ð1 uÞ and has been studied by Hodgkin, Huxley and Kolmogorov. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Rosa), [email protected] (M.S. Bruzón), [email protected] (M.L. Gandarias). http://dx.doi.org/10.1016/j.cnsns.2015.01.010 1007-5704/Ó 2015 Elsevier B.V. All rights reserved.
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
75
Another important equation of this class is, for f ðuÞ ¼ uð1 u2 Þ, the Fitzhugh-Nagumo equation which arises in the study of nerve cells [10,18]. Over the last two decades a lot of attention has been paid on using Lie point symmetry methods to exploit the invariance of the generalized equation
ut ¼ ðAðuÞux Þx þ BðuÞux þ CðuÞ:
ð4Þ
A complete Lie symmetry classification for the non-linear heat Eq. (4) with B ¼ C ¼ 0 was described by Ovsiannikov in [17]. Several papers [6,7,9], were devoted to extension of Ovsiannikov’s result to Eq. (4) with B ¼ 0 and B – 0. In [8] the form-preserving transformations were applied to derive the complete Lie symmetry classification of (4). Reaction–diffusion equations arise from modeling densities of particles such as substances and organisms which disperse through space as a result of the irregular movement of every particle. Eq. (4) with BðuÞ ¼ 0, becomes the so called density dependent equation
ut ¼ f ðuÞ þ ðgðuÞux Þx ;
ð5Þ
which has been considered by J.D. Murray in [15]. Due to the great interest in getting conservation laws and conserved quantities in a recent paper [11] applying the concept of nonlinear self-adjointness and a theorem related to conservation laws due to Ibragimov [12] as well as the multiplier method due to Anco and Bluman [2] we have analyzed the functions for which Eq. (5) becomes nonlinear self-adjoint and we have derived nontrivial conservation laws for Eq. (5). There is no existing general theory for solving nonlinear partial differential equations (PDEs) and the machinery of the Lie group theory provides the systematic method to search for the special group-invariant solutions. For PDEs with two independent variables, a single group reduction transforms the PDE into ordinary differential equations (ODEs), which are generally easier to solve than the original PDE. The knowledge of the optimal system of subalgebras gives the possibility to construct the optimal system of solutions and permits the generation of new solutions starting from invariant or non-invariant solutions. The list of the disjoint classes is called the optimal system of solutions. The equation analyzed in this paper is a generalized Fisher equation in cylindrical coordinates using x for radial variable with assumption of radial symmetry, gðuÞ is the diffusion coefficient depending on the variable uand f ðuÞ an arbitrary function
1 ut ¼ f ðuÞ þ ½xgðuÞux x : x
ð6Þ
Eq. (6) has attracted a lot of attention from researchers and been analyzed in different and particular cases by other authors that we will detail below. The transient heat conduction equation with a heat source term following a power law in a rectangular, cylindrical or spherical coordinate system has been considered by Moitsheki in [14] by using Lie classical symmetries. In [3] Bohkari et al. have considered the following particular case of (6) where gðuÞ ¼ u and f ðuÞ ¼ uð1 uÞ
1 ut ¼ uð1 uÞ þ ½xuux x ; x
ð7Þ
by using Lie classical reductions the authors derived an exact solution in term of the Bessel functions. In [4] Bohkari et al. have considered Eq. (6) but only when gðuÞ is a linear function gðuÞ ¼ a1 u þ a2 . They state that a classification of (6) can only be achieved when g is linear in u. When f and g follow a power law they give an stationary solution. The aim of this paper is to study the generalized Fisher equation in cylindrical coordinates using x for radial variable (6) from the point of view of the theory of symmetry reductions in partial differential equations. The paper is organized as follows: In Section 2 we give the Lie symmetries of Eqs. (6) by means of the classical Lie direct approach. In Section 3 we report the reductions obtained from the optimal system of subalgebras. In Section 4 we analyze some exact solutions of Eq. (6). 2. Lie symmetries In this section we perform Lie symmetry analysis for Eq. (6). Let us consider a one-parameter Lie group of infinitesimal transformations in ðx; t; uÞ given by
x ¼ x þ enðx; t; uÞ þ Oðe2 Þ; t ¼ t þ esðx; t; uÞ þ Oðe2 Þ;
ð8Þ
u ¼ u þ e/ðx; t; uÞ þ Oðe2 Þ; where e is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of equation (4). This yields to the overdetermined, linear system of eleven equations for the infinitesimals nðx; t; uÞ, sðx; t; uÞ and /ðx; t; uÞ. The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form
76
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
v¼n
@ @ @ þs þ/ : @x @t @u
ð9Þ
Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition
Un
@u @u þs / ¼ 0: @x @t
ð10Þ
From the determining system, we get that n ¼ nðx; tÞ, equations:
s ¼ sðtÞ; / ¼ /ðx; t; uÞ where n; s; /; g, and f must satisfy the following
f st x g /x x x þ f /u x þ /t x f u / x þ g /x ¼ 0; 2 g x nx þ g st x þ g u / x ¼ 0; 2 g u x2 nx g u st x2 g /u u x2 g u /u x2 g u u / x2 ¼ 0; g x2 nx x g x nx þ x2 nt g n þ 2 g u /x x2 þ 2 g /u x x2 þ g st x þ g u / x ¼ 0:
ð11Þ
From the second equation we get that as g u – 0
/¼
ð st 2 n x Þ g gu
and the third equation becomes
ðst 2 nx Þ
g gu u u ðg u Þ2
2 g ðg u u Þ2 ðg u Þ3
g þ uu gu
! ¼ 0:
In order to find a complete classification we observe that this condition will be satisfied only in the following two cases either
st 2 nx ¼ 0 or
g k1 u k2 ¼ 0: gu We can distinguish: Case 1. If n ¼ s2t x þ d with d ¼ dðtÞ. By substituting into the remaining Eqs. (6), we get d ¼ 0 and
n ¼ 0;
s ¼ k3 ; / ¼ 0:
For f ðuÞ and gðuÞ arbitrary, the only symmetry that is admitted by (6) is
v1 ¼
@ : @t
Case 2. If g is a solution of n
gðuÞ ¼ kðu þ k2 Þ ;
g gu
k1 u k2 ¼ 0. The solutions without loosing generality are: cu
gðuÞ ¼ de :
The functional forms of f and g which yield extra symmetries are given in Table 1. 3. Optimal systems and reductions In order to construct the one-dimensional optimal system, following Olver in [16], we construct the commutator tables and the adjoint tables, appearing in the appendix, which show the separate adjoint actions of each element in v i ; i ¼ 1 . . . n, as Table 1 Functions and generators. i
fi
gi
vk
1 2
um
un un
3
c2 unþ1 c1nu
un
4 5 6
c1nu c1 u
un
c2 enu cn1
u1 nu de
7
c1 n
de
8
c2 enu
de
v 1 ; v 2 ¼ ðn m þ 1Þx@ x þ 2ð1 mÞt@ t þ 2u@ u v 1 ; v 3 ¼ nt@ t u@ u v 1 ; v 4 ¼ nec t @ t c1 ec t u@ u v1 ; v2 ; v4 v 1 ; v 2 ; v 4 ; v 5 ¼ ðx log ðxÞ xÞ@ x 2 u logðxÞ@ u v 1 ; v 6 ¼ nec t @ t c1 ec t @ u v 1 ; v 6 ; v 7 ¼ nx@ x þ 2@ u v 1 ; v 8 ¼ nt@ x @ u
unþ1 nþ1
nu nu
1
1
1
1
77
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
it acts on all other elements. This construction is done easily by summing the Lie series. The corresponding generators of the optimal system of subalgebras are given in Table 2. Here i indicates the corresponding functions f i ; g i . In Table 3 and Table 4 respectively, similarity solutions and reductions of the Eq. (6) to ODEs are obtained using the generators of the optimal system. Here i indicates the corresponding functions f i ; g i ; j indicates the corresponding similarity reductions and k the generators of the corresponding optimal systems. For f ðuÞ and gðuÞ arbitrary functions the similarity variable and similarity solution are
z ¼ x;
u ¼ hðzÞ:
Substituting into (6) we obtain the reduced ODE
g hx þ g hx x þ f ¼ 0; x with f ¼ f ðhÞ and g ¼ gðhÞ arbitrary functions. 4. Some further reductions and exact solutions In this section we give the exact solutions for some of the reduced equations, by some method of integration or reducing the order again using Lie classical method. 4.1. Mass preserving solutions. Source and Sink solutions ODE1 Among our reduced equations we have one equation related to this type of solution. For the reduced ODE1 with nmþ1
z ¼ t2 ðm1Þ x;
u¼
hðzÞ 1
t m1
:
In fact if we choose the parameter n ¼ m 3 the solution has the form
z¼
x 1
tm1
;
u¼
hðzÞ 1
t m1
: Table 2 Optimal systems. i 1 2 3 4 5 6 7 8
v1 ; v2 v1 ; v3 v1 ; v4 v 1 þ av 2 ; av 2 þ bv 4 ; v 2 ; v 4 v 1 þ v 2 ; av 2 þ bv 4 ; v 1 þ av 5 ; v 4 þ bv 5 ; v 5 v1 ; v6 v 1 þ av 7 ; av 6 þ bv 7 ; v 6 ; v 7 v1 ; v8
Table 3 Similarity solutions. i
j
wk
1
1
v2
zj t
2
2
3
3
4
4
4 4
5 6
5
7
5
8
v3 v4 v2 v 1 þ av 2 v 2 þ bv 4 v5 v 1 þ av 5
5
9
v 4 þ bv 5
e
v6 v 1 þ av 7 av 6 þ bv 7 v8
x
6
10
7
11
7
12
8
13
uj
nmþ1 2 ðm1Þ
x
hðzÞ 1
t m1
x
hðzÞt n
x
hðzÞ e
1 c1 t n
t
2 n
ea n t x ec1 t
e t
b c1
x
hðzÞ e
c1 t
2be c
1 n
c1 t n
h x2 ðlog x1Þ2
log ðlog x1Þa t a c1 t
c1
log ðlog x1Þ b
ea n t x b ec1 t c1
e x
hðzÞx hðzÞe2 a t
hðzÞ ðx log xxÞ2 2 e
b ec1 t b z c t c1 2 b ec 1 þc1 1
hðzÞ e hðzÞ c1n t
x hðzÞ þ 2 log n c
x
t
hðzÞ 2 bce1 n1 c1n t t hðzÞ log n
t
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
Table 4 Reduced ODEs. i
j
ODEj
1
1
Þz hzz ðhzhÞ hz 2ðnþm1 hn ðm1Þ
2
2
3
3
4
4
4h
4
5
a hz n z n h
4
6
hz z n bh
5 5
7 8
hz c 1 h 2 ¼ 0 2 2 3 2 h hzz ðhz Þ þ a2 h a h hz þ a2 c1 h þ 2 a2 h ¼ 0
5
9
h hzz e2 b z ðhz Þ e2 b z þ b h hz e2 b z þ e2 b h hz 2 e2 b h ¼ 0
6
10
7
11
d ðhz Þ n z þ d hzz z þ c2 z þ d hz ¼ 0 2 eh n d ðhz Þ n2 þ d hzz n z2 þ 5 d hz n z þ 4 d c1 ¼ 0 2 eh n d ðhz Þ n2 þ d hzz n z þ d hz n 2 b z ¼ 0 2 eh n d ðhz Þ n2 þ ðd hzz þ c2 Þ n z þ d hz n þ z ¼ 0
2
12
8
13
1 x u¼ 1 h 1 tm1 tm1
þ
h0 z
þ
h nþ1
þ
2
ðh0 Þ h nþ1
n
ðhz Þ n h
þ
1n
h
n
n
1n
h m1 h
mn
hz z ¼ 0
þ hz z ¼ 0
00
þ h þ c2 h ¼ 0
ðnþ1Þ n2
þ c1nh þ hz ¼ 0 1n
þ hzz h
2
þ hzz þ ðhzhÞ
n
ð2 a nþc1 Þ n
2
þ ðhzhÞ
n
þ hz z ¼ 0
1n
2 bh n þ hz z ¼ 0
2
7
Consequently the quantity
hz z
2
2
2
3
3
2
R1 1
uðx; tÞdx ¼
R1 1
hðzÞdz does not depend on t and the total mass is conserved for the solutions
For m > 1 it is clear that uðx; tÞ ! dðxÞ as t ! 0 and the similarity solution is a source solution. ODE5 An analogue situation is found in reduction ODE5 for the particular case n ¼ 2
uðx; tÞ ¼ e2at hðxe2at Þ the total mass is conserved for these solutions. Moreover we have that
If If
a < 0 uðx; tÞ ! dðxÞ as t ! 1 a > 0 uðx; tÞ ! dðxÞ as t ! þ1
so, the similarity solution is in the first case a source solution while in the second case is a sink solution. ODE3
0 2 0 h n h 00 þ þ h þ c2 h ¼ 0: z h
ð12Þ
Eq. (12) admits the symmetry generator
w¼
@ : @h
ð13Þ
Taking into account the invariants of its first prolongation and using the new variables 0
f ¼ z;
w¼
h ; h
ð14Þ
we obtain that Eq. (12) can be reduced to the first order Riccati equation
w þ wz þ n w2 þ w2 þ c2 ¼ 0: z
ð15Þ
In order to solve Eq. (15) we substitute
w¼
y0 ðn þ 1Þ y
to get
yz þ yz z þ c2 ðn þ 1Þ y ¼ 0; z which is a Bessel equation whose general solution is
y ¼ k1 J 0 ðkzÞ þ k2 Y 0 ðkzÞ:
ð16Þ
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
79
Consequently 1=nþ1
h ¼ ðk1 J 0 ðkzÞ þ k2 Y 0 ðkzÞÞ
;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ c2 ðn þ 1Þ. Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by c1 t
u ¼ e n ðk1 J 0 ðkxÞ þ k2 Y 0 ðkxÞÞ
1=nþ1
:
The asymptotic behavior of the solution can be understood from the well-known asymptotic behavior of the modified Bessel functions which is
ex I0 ðxÞ pffiffiffiffiffiffiffiffiffi 2px
ex K 0 ðxÞ ¼ qffiffiffi 2 px
for large x. This solution for the particular case n ¼ 1 has been derived in [3]. ODE4 The reduced equation
4h
nþ1
ðn þ 1Þ c1 h þ þ hz ¼ 0 n2 n
ð17Þ
can be integrated by quadrature, the solution is:
hðzÞ ¼
1 4 þ 4 n þ ec1 z k1 nc1 n ; nc1
ð18Þ
where k1 2 R. Using the substitutions given in Table 3 we obtain an exact solution of (6) in cylindrical coordinates, given by
uðx; tÞ ¼ x2=n
1 4 þ 4 n þ ec1 t k1 nc1 n ; nc1
ð19Þ
where k1 2 R. ODE5 2
hz ðhz Þ n þ þ hz z ¼ 0 h z
ð20Þ
For the reduced Eq. (20), we realize that the ODE admits the group corresponding to the generator
w¼z
@ @ þh : @z @h
ð21Þ
Taking into account the invariants of its first prolongation and using the new variables
wðfÞ ¼ hz ;
f¼
h ; z
ð22Þ
we obtain that Eq. (20) can be reduced to the first order ODE
ðw fÞwf þ w þ
nw2 ¼ 0; f
ð23Þ
whose implicit solution is:
fþ
1 wðt Þ k1 ðwðfÞÞnþ1 ¼ 0; n
where k1 2 R. Setting h ¼ y1=ðnþ1Þ into (20), we get
yz þ yz z ¼ 0; z whose solution is
y ¼ k1 log z þ k2 ; ð
1
Þ
h ¼ ðk1 log z þ k2 Þ nþ1 : From Table 3 we get
ð24Þ
80
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83 c1 t
ð
1
Þ
u ¼ e n ðk1 log x þ k2 Þ nþ1 : ODE8 4.2. Simplest equation method The reduced equation ODE7
2 2 3 2 h hzz ðhz Þ þ a2 h a h hz þ a2 c1 h þ 2 a2 h ¼ 0
ð25Þ
is an ODE that admits the symmetry generator
w¼
@ ; @h
ð26Þ
which does not yield any symmetry reduction. However Eq. (25) is an autonomous equation and we can apply the modified method of the simplest equation which has been developed by Kudryashov [13]. The Kudryashov version of the method contains a procedure analogous to the first step of the test for the Painleve property. In the modified method of the simplest equation [11,12], this procedure is substituted by the concept for the balance equation. Frequently this method is applied for travelling wave solutions. We apply the modified Kudryashov method [20] to Eq. (25), taking the Riccati equation as the simplest equation, we suppose that the solutions can be expressed by a polynomial in y in the form
h¼
n X
ai ðyÞi ;
ð27Þ
i¼0
where y ¼ yðzÞ satisfies the Riccati equation
yz k2 y2 k1 y k0 ¼ 0;
ð28Þ
ai ; i ¼ 0; . . . ; n; ki i ¼ 1; . . . ; 2 are constants to be determined later. Considering the homogeneous balance between highest derivative term and the highest power law term, we get n ¼ 1 we can write (27) as
h ¼ a0 þ a1 y;
a1 – 0:
ð29Þ
By substituting (29) with
yz ¼ k2 y2 þ k1 y þ k0 ; yzz ¼ 2k2 y þ k1 ; into (25), we get setting equal zero the coefficients of the different powers of y the following conditions:
a21 k2 k2 þ a2 a1 ¼ 0; 2
2 a0 k2 þ a1 k1 k2 þ 2 a2 a0 a1 k2 a a1 k2 þ a2 a21 k1 þ a2 a21 c1 ¼ 0; 3 a0 k1 k2 þ a2 a20 k2 a a0 k2 þ 2 a2 a0 a1 k1 a a1 k1 þa2 a21 k0 þ 3 a2 a0 a1 c1 þ 2 a2 a1 ¼ 0; 2 a0 k0 k2 þ
2 a0 k1
ð30Þ
a1 k0 k1
þa2 a20 k1 a a0 k1 þ 2 a2 a0 a1 k0 a a1 k0 þ 3 a2 a20 c1 þ 4 a2 a0 ¼ 0; 2
a0 a1 k0 k1 a21 k0 þ a2 a20 a1 k0 a a0 a1 k0 þ a2 a30 c1 þ 2 a2 a20 ¼ 0: By solving this system we get
a ¼ c1 ; k2 ¼ c21 a1 ; k0 ¼ a0 c1 ðaa01 c1 þ2Þ ; k1 ¼ 2 a0 c21 2 c1 : The Ricatti Eq. (28) becomes
a0 c1 ða0 c1 þ 2Þ yz þ a1 c21 y2 2 a0 c21 2 c y þ ; a1 whose solution is
y¼
ða0 c1 þ 2Þ e2 c1 z2 k c1 a0 a1 c1 e2 c1 z2 k c1 a1
ð31Þ
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M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
and
h¼
2 ; e2 c1 zþ2 k c1 c1
where k is the integrating constant. Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by
u¼
2 2
ðx log x xÞ ðe2 c1 k2 ðlog ðlog x1Þþc1 tÞ c1 Þ
:
For c1 < 0 and x > 0 it is a bounded solution. ODE10 The reduced equation 2
d ðhz Þ n z þ d hzz z þ c2 z þ d hz ¼ 0
ð32Þ
is an ODE that admits the symmetry generator
w¼
@ ; @h
ð33Þ 0
which does not yield any reduction however in Eq. (32) the terms in h are missing allowing the substitution h ¼ w it becomes the Riccati equation:
c w wz þ nw2 þ þ ¼ 0: d z The solution to Eq. (32) can be given in term of Bessel functions as:
hðzÞ ¼ being p ¼
2 2 2 ln d nðJ0 ðpÞk1 k2 Y0 ðpÞÞ c1 2 z ðJ1 ðpÞY0 ðpÞ Y1 ðpÞJ0 ðpÞÞ 2n
qffiffiffiffiffiffi
c2 n ; k1 ; d
;
ð34Þ
k2 2 R.
Using the corresponding substitution given in Table 3 we obtain that an exact solution of (6) is
u ¼ hðxÞ
c1 t n
where hðxÞ given by (34). ODE11 The reduced equation
eh n
2 d ðhz Þ n2 þ d hzz n z2 þ 5 d hz n z þ 4 d c1 ¼ 0;
ð35Þ
can be integrated by quadrature, the solution is:
hðzÞ ¼ ln
2 c1 z þ 4 k1 d n 4 k2 d n ln ðzÞ 1 n ; 2 4dz
ð36Þ
where k1 ; k2 2 R. Using the corresponding substitution given in Table 3 we obtain that an exact solution of (6) is
u ¼ hðzÞ þ
2 log x ; n
where z ¼ ea n t x and hðzÞ given by (36). ODE13
eh n
2 d ðhz Þ n2 þ ðd hzz þ c2 Þ n z þ d hz n þ z ¼ 0:
ð37Þ
The solution to Eq. (37) can be given in term of Bessel functions as
hðzÞ ¼ being H ¼
ln ðHÞ1 ; n
ð38Þ p
p
J0 ðpÞzY1 ðpÞþJ0 ðpÞk1 d n z zJ1 ðpÞY0 ðpÞk2 d n z Y0 ðpÞ ; zc2 nðJ1 ðpÞY0 ðpÞþY1 ðpÞJ0 ðpÞÞ
p¼
qffiffiffiffiffiffi c2 n d
and k1 ; k2 2 R.
Finally using the substitutions given in Table 3 we obtain an exact solution of the generalized Fisher equation in cylindrical coordinates, given by
uðx; tÞ ¼ hðxÞ
logt n
where hðxÞ is given by (38).
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5. Conclusions In this paper we have classified the Lie symmetries of the generalized Fisher Eq. (6) in cylindrical coordinates with radial symmetry. We have studied the functions f ðuÞ and gðuÞ for which the principal Lie algebra is extended. In general, the groups that leave (6) invariant depend on several parameters, to each one-parameter subgroup there will correspond a family of group-invariant solutions. We wanted to minimize the search for group-invariant solutions in order to find non-equivalent branches of solutions, which leads to the concept of optimal systems of group-invariant solutions, from which, every other solution can be derived. To obtain the one-dimensional optimal systems of solutions, we have searched for the one-dimensional optimal systems of subalgebras. We then constructed all the invariant solutions with respect to the one-dimensional optimal system of subalgebras, as well as all the ODEs to which (6) are reduced. Finally, some of these ODEs are invariant under a symmetry group and we have reduced the order of these equations. We also find some exact solutions by using a nonlinear simple Riccati equation as well as some solutions in term of the well known Bessel equations. Some of the solutions we have found conserve the mass through time and exhibit instantaneous sources and sinks.
Acknowledgments The authors acknowledge the support of DGICYT project MTM2009–11875, FEDER and Junta de Andalucía FQM-201 group.
Appendix A See Tables A.1–A.6. Table A.1 Commutator table for Case 4. ½v i ; v j
v1
v2
v4
v1 v2 v4
0 0 c1 v 4
0 0 0
c1 v 4 0 0
Table A.2 Adjoint table for Case 4. Ad
v1 v2 v4
v1 v1 v1 v 1 þ e c1 v 4
v2 v2 v2 v2
v4 ec1 e v 4
v4 v4
Table A.3 Commutator table for Case 5. ½v i ; v j
v1
v2
v4
v5
v1 v2 v4 v5
0 0 c1 v 4 0
0 0 0 v 2
c1 v 4 0 0 0
0
v2 0 0
Table A.4 Adjoint table for Case 5. Ad
v1 v2 v4 v5
v1 v1 v1 v 1 þ e c1 v 4 v1
v2 v2 v2 v2 ee v 2
v4 ec1 e v 4
v4 v4 v4
v5 v5 v5 e v2 v5 v5
M. Rosa et al. / Commun Nonlinear Sci Numer Simulat 25 (2015) 74–83
83
Table A.5 Commutator table for Case 7. ½v i ; v j
v1
v6
v7
v1 v6 v7
0 c1 v 6 0
c1 v 6 0 0
0 0 0
Table A.6 Adjoint table for Case 7. Ad
v1 v6 v7
v1 v1 v 1 þ e c1 v 6 v1
v6 ec1 e v 6
v6 v6
v7 v7 v7 v7
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